Flecnodal and LIE-curves of ruled surfaces

Dissertation

Zur Erlangung des akademischen Grades

Doktor rerum naturalium (Dr. rer. nat.)

vorgelegt

der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden

von

Ashraf Khattab geboren am 12. März 1971 in Alexandria

Gutachter: Prof. Dr. G. Weiss, TU Dresden Prof. Dr. G. Stamou, Aristoteles Universität Thessaloniki Prof. Dr. W. Hanna, Ain Shams Universität Kairo

Eingereicht am: 13.07.2005

Tag der Disputation: 25.11.2005 Acknowledgements

I wish to express my deep sense of gratitude to my supervisor Prof. Dr. G. Weiss for suggesting this problem to me and for his invaluable guidance and help during this research. I am very grateful for his strong interest and steady encouragement. I benefit a lot not only from his intuition and readiness for discussing problems, but also his way of approaching problems in a structured way had a great influence on me. I would like also to thank him as he is the one who led me to the study of line geometry, which gave me an access into a large store of unresolved problems.

I am also grateful to Prof. Dr. W. Hanna, who made me from the beginning interested in geometry as a branch of mathematics and who gave me the opportunity to come to Germany to study under the supervision of Prof. Dr. Weiss in TU Dresden, where he himself before 40 years had made his Ph.D. study.

My thanks also are due to some people, without whose help the research would probably not have been finished and published: Prof. Dr. W. Degen, who helped me to find a missed part of the unpublished work of Gerrit Bol [8], Frau Ing. Helga Mettke for the marvellous finishing and organization of the thesis, and Michaela Bercikova for her kind help in the translation from Czech language.

I also take this opportunity to express my gratitude to DAAD for awarding me a scholarship that enabled me to study in Germany at the institute of Geometry, TU Dresden.

I would like to use this occasion to congratulate all the members of the institute of Geometry, TU Dresden for the family and friendly atmosphere.

I wish of course to thank my parents for having raised me in such a stable and loving environment, which has enabled me to come so far.

Last but not least, I want to thank my friends, who helped me maintain my sanity during my stay in Germany and during my Ph.D. study, as I felt myself drawing in this over ambitious thesis. Preface

Let us first consider a space curve c and the complex K(3) of all straight lines through c. Then we may select a (not necessarily unique) ruled surface Σ⊂K(3) by demanding that c ⊂Σ has certain projective or affine or Euclidean geometric properties on Σ. It even might happen that the demanded conditions can only be fulfilled, if c is somehow special.

To give an example, we consider Σ⊂K(3) such that c is the (Euclidean) striction curve on Σ. It is well known that to any strip through c there still exists a one parameter set of ruled surfaces Σ such that c has this property. So the solution set {Σ} is too big and thus not interesting. But if we add the assumption that c should be an asymptotic curve, we will receive the one-parameter PIRONDINI set of surfaces Σ to the osculating strip of c only.

To restrict the set of solution surfaces Σ, we also might start with two curves c1, c2 and (2) (2) the congruence K of lines hitting both curves ci and ask for surfaces Σ⊂K such that ci have certain properties on Σ. Demanding for example that Σ consists of common main normals of ci will lead to the well known BERTRAND curves problem and thus showing an example, where the demanded assumptions to Σ (and ci ) can not be fulfilled by a pair of arbitrary curves ci in general.

As many of the classical geometric properties of a curve c lead to a quadratic functional along the generators of Σ, it is some how natural to start with a pair of curves c1 and c2 and construct Σ under certain assumptions on properties of with ci . Let ci depend on parameters

uIii∈⊂\, then Σ is described by a correspondence (ut12(), u()t),tJ∈ ⊂ \, between the point sets of ci . A certain demanded property of Σ and ci ⊂Σ should allow to develop the functions uti (), at least in the vicinity of an arbitrarily chosen parameter value ui= ui,0 , as well as it should deliver TAYLOR-representations of ci , too. The topic we want to deal with belongs therefore to local differential geometry and in most cases the canonical representation of ci by curvature functions (with respect to the considered euclidean, affine or projective) geometry) will be the proper tool for such a treatment.

In the following, projective differential geometric properties of ci ⊂Σ will be the main topic of consideration. So, this dissertation deals with local projective differential geometry of ruled surfaces of a real projective 3-space. The single chapters of this doctoral thesis collect and present the different treatments of ruled surfaces throughout history, as they were found in literature. In more detail the content is as follows:

Chapter 1 is an introductory chapter with an explanation of the problem and giving basic definitions and presenting the aim of the dissertation.

Chapter 2 provides the method of WILCZYNSKI and of his followers, who gave some examples for the effectiveness of that calculus. WILCZYNSKI’s general scheme in studying a given geometric surface is to set up a system of one or more linear homogenous differential equations such that the fundamental set of solutions of the system determine the surface uniquely up to projective transformations. The independent and the dependent variables, which appear in the differential equations and also in the parametric equations of the surface, can be subjected to certain transformations which leave the surface invariant and do not Preface iii change the form of the system of differential equations, but such transformations will in general change the coefficients of the differential equations. A function of the transformed coefficients and their derivatives and of the new dependent variables and their derivatives, which is equivalent to the same function of the original coefficients and variables, is called a covariant. A covariant which does not contain the dependent variables or their derivatives is called an invariant. WILCZYNSKI used a complete and independent system of invariants and covariants as the basis for each of his geometric studies. His general method of calculating the invariants and covariants is by means of the LIE-theory of continuos groups. Since the method is purely analytical, there is no certainty that the invariants and covariants will readily show their geometric significance.

Chapter 3 presents the method of CECH and FUBINI for treating ruled surfaces from a projective geometric point of view. FUBINI’s method is based on studying projective differential geometric properties of surfaces by means of differential forms (see Appendix A). However, to obtain results with a geometrical interpretation by this method, it is necessary to pass from the differential forms to a system of differential equations of the type used by WILCZYNSKI. Thus, in CECH’s theory of ruled surfaces, the coefficients of the system of differential equations of WILCZYNSKI are decomposed into terms, from which the invariants of a ruled surface Σ become obvious.

Chapter 4 treats the flecnode- and LIE-curves of ruled surfaces in the affine space. Using the affine geometric natural equations of a space curve c in the affine space, O. MAYER studied ruled surfaces Σ which have a given curve c as an asymptotic line, and he deduced the equations of its flecnodes. MAYER used the deduced formulae to study some special cases like the case of c being a TZITZEICA’s curve, the case when Σ has distinct planar flecnodes, and the case of two flecnodes lying in parallel planes. MAYER defines also a ‘harmonic transformation’ and applies it on an R-family of curves on Σ . Such an R-family (or cross ratio set) is a set of curves that intersect the generators of Σ in projectively coupled rows of points. Particularly, he called the R-family which contains the flecnode-, the LIE-curves and the so called involute curves the principal family of Σ.

Chapter 5 deals with BOL’s treatment of ruled surfaces. G. BOL developed the projective differential geometry of ruled surfaces using a new point of view, which depends on HESSE’s correspondence principle. Using this principle BOL gave a geometric projective classification of ruled surfaces depending on its characteristic curve, which is an image of the flecnode curve under a so called hyperbolic mapping. BOL deduced the differential equations of a ruled surface firstly with respect to an arbitrary pair of curves q1, q2 , which is normalized in such a way that ()qq12,,qq1′′,2= const.≠ 0, and later on q1, q2 are specialised to the pair of flecnode curves. Applying what he called *-transformation, a process of differentiating and norming projective coordinates at the same time, BOL deduced a system of projective invariants, by which he explained some geometric concepts related to ruled surfaces like cross ratio sets of curves, the flecnode strip, the principal system of curves, the fundamental mapping and the sequel of flecnode tangents’ surfaces.

Chapter 6 gathers J. KLAPKA’s [32] geometric constructions of the flecnode points and LIE- th points on the generators of a ruled surface Σ of 4 degree. For a surface ΣIII of type III (according to STURM’s classification), which lies in the congruence of chords of a twisted Preface iv cubic, the construction of the flecnode points is based on the following theorem: “On an arbitrary generator p of a ruled surface ΣIII , which lies in the congruence of chords of a twisted cubic, there exists a well defined involution of second degree which contains the pair of flecnode points F1, F2 ”. Through p pass two bi-tangent planes of ΣIII , who intersect ΣIII in two conic components, which of intersect p in a pair of points Q1, T2 and Q2 , T1 each. The self-corresponding points of the involution determined by the pairs T1, Q1 and T2 , Q2 are the

LIE-points L1, L2 of p. The previous theorem is valid also for the construction of the th flecnodes on 4 degree surfaces of STURM’s types IX and IV, which are dual to each other. A modified construction is given for the flecnodes of a gererator of a surface of STURM’s type V . The flecnodes on a surface of type VI can be constructed as the intersection of this surface with another surface of 4th degree with rectilinear directrices. The surfaces of type XI th are dual to those of type VI. The other types of 4 degree surfaces, according to STURM’s classification, belong to linear congruences, therefore their flecnode curves coincide with the directrices of these congruences.

Chapter 7 shows a treatment of flecnode and LIE-curves of ruled surfaces in the KLEIN- model of line space. Using KLEIN’s correspondence principle, J. KANITANI [28], [29] studied the projective differential geometry of ruled surfaces in the projective space of three 3 2 dimensions Π as curves on KLEIN’s hyperquadric M4 in the projective space of five dimensions Π5. In the differential relations deduced by KANITANI appear, as coefficients, five fundamental quantities, which are not invariant by themselves. But applying a parameter transformation and a special norming of the projective coordinates one can deduce a system of invariants as functions of those fundamental quantities. These invariants are interpreted as anharmonic ratios, in which mainly the following surface elements appear: The intersection points of the osculating quadric along a generator p with the ‘neighbouring’ generator that is infinitely near to p, (thus these points are the flecnodes of p ), the flecnode tangents of Σ , the flecnode curves’ tangents and the flecnode strips’ limit lines. If the 5 fundamental quantities are given as functions of a known parameter u, then one can determine, up to a projective transformation, a ruled surface, to which the given fundamental quantities do belong. We will give a summary of KANITANI’s most important results as far as they are concerned with the flecnode curves on a ruled surface. Finally, we are going to give a new treatment, which allows to explicitly calculate LIE- and flecnode curves of an arbitrarily parameterised ruled surface Σ based on preceding work and lectures of G. WEISS [72], [73], what could not be done before. Closely related is F. ANZBÖCK’s treatment, which uses a natural moving frame of Σ and BOL’s concepts of projective arc lenght and curvatures. Finally we also list some still open problems concerning the topics of the thesis.

Appendix A contains a summary of FUBINI’s method as applied to surfaces in ordinary space (cf. LANE [40]).

A comprehensive bibliography∗) containing references to related and previous work on the treated subject concludes the thesis.

∗) The numbers in square brackets after author names refer to the bibliography at the end of the thesis. Table of Contents

Acknowledgements ...... i Preface ...... ii

1 Introduction ...... 1 1.1 The aim of the dissertation ...... 1 1.2 Flecnodal points (flecnodes) ...... 2 1.3 LIE-points ...... 2 1.4 Involute points ( harmonic points) ...... 3 1.5 Ruled surfaces belonging to a linear line congruence ...... 3 1.6 Ruled surfaces belonging to a special linear complex ...... 4 1.7 BERTRAND curves ...... 4

2 The Method of WILCZYNSKI ...... 5 2.1 The complete system of invariants and covariants ...... 5 2.2 The ruled surface referred to its flecnode curves ...... 8 2.3 The ruled surface referred to its LIE-curves ...... 10 2.4 The osculating quadric – The flecnode congruence ...... 11 2.5 The derivative cubic ...... 14 2.6 The differential equations of curves situated on a ruled surface ...... 16 2.7 CARPENTER’s surfaces with planar flecnode curves ...... 27 2.8 Ruled surfaces whose LIE-curves have planar branches ...... 31 2.9 Ruled surfaces with coincident plane branches of the flecnode curve ...... 32 2.10 Ruled surfaces with rectilinear branches of the flecnode curve ...... 33 2.11 The primary and secondary flecnode cubics and LIE-cubics ...... 36 2.12 The flecnode-LIE tetrahedron ...... 38 2.13 The complex-permute quadric – The flecnode-reciprocal quadric ...... 42 2.14 Ruled surfaces whose flecnode curves belong to linear complexes ...... 45 2.15 The flecnode sequel ...... 47 2.16 The metric properties of the flecnode curves ...... 48

3 The Method of CECH ...... 51 3.1 The fundamental equations ...... 51 3.2 The projective linear element of a ruled surface ...... 53 3.3 The pangeodesic ...... 57 3.4 The intrinsic fundamental forms ...... 58 3.4.1 The quadratic intrinsic form f ()t ...... 58 3.4.2 The bilinear intrinsic form f (,t τ ) ...... 60 3.5 The flecnode congruence – The normalization of coordinates ...... 63 3.6 The fundamental system of invariants ...... 64 3.6.1 Ruled surfaces with distinct flecnode curves ...... 64 3.6.2 Ruled surfaces with coincident flecnode curves ...... 67 3.7 The osculating linear complex ...... 71 3.8 Further studies on a hyperbolic ruled surface with non rectilinear directrices ..... 72 3.8.1 The first case: ε =1 ...... 73 3.8.2 The second case: ε =−1, h > 0 ...... 75

Table of Contents vi

3.9 The flecnode surfaces ...... 77 3.10 WILCZYNSKI’s principal surface of the flecnode congruence ...... 81

3.11 Some remarks on the ruled surface ΣF ...... 83 3.12 The conjugate and the bi-conjugate pair of curves on a ruled surface ...... 89

4 Flecnode curves and LIE-curves in affine space ...... 93 4.1 The fundamental equations – The flecnode curves ...... 93 4.2 TZITZEICA’s curves ...... 96 4.3 Ruled surfaces with distinct planar flecnode branches ...... 98 4.4 A ruled surface with one rectilinear flecnode branch ...... 99 4.5 A ruled surface whose flecnodes are situated in two parallel planes ...... 101 4.6 The harmonic transformation ...... 103 4.7 The R-family of curves on a ruled surface ...... 104 4.8 The principal family of curves on a ruled surface ...... 107 4.9 The G-surfaces ...... 108 4.10 The Centro-affine group ...... 110 4.11 Space curves of constant affine curvature ...... 113

5 The Method of BOL ...... 117 5.1 The fundamental equations ...... 117 5.2 The osculating quadric – the flecnode congruence ...... 120 5.3 The ruled surfaces in line coordinates – The osculating congruence ...... 123 5.4 A projective classification of ruled surfaces – The characteristic curve ...... 126 5.5 The osculating linear complex ...... 127 5.6 The Plane hyperbolic geometry ...... 130 5.7 DV-set of curves on a ruled surface ...... 135 5.8 A curve-pair on a ruled surface ...... 140 5.9 Ruled surfaces through surface strips ...... 143 5.10 Ruled surfaces with two distinct flecnode curves ...... 151 5.11 Ruled surfaces with coincident flecnode curves ...... 157 5.12 The fundamental mapping – The circular ruled surfaces ...... 158 5.13 The flecnode sequel of surfaces ...... 162

6 The flecnodal and LIE-curves on a ruled surface of fourth degree ...... 163 6.1 Ruled surfaces of fourth degree and STURM-type III ...... 163 6.2 An application on a general normal surface of a quadric along a plane section .... 169 6.3 STURM-type IX and IV ...... 171 6.4 STURM-type V ...... 171 6.4.1 An application on the arched surface of an oblique passage ...... 173 6.5 STURM-type VI and XI ...... 179

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space ...... 182 7.1 Explicit method of determining flecnode and LIE curves ...... 182 7.1.1 The KLEIN-model of line space and the differential geometry of curves ...... 182 7.1.2 The flecnodes and the LIE-curves of a ruled surface ...... 185 7.2 The method of ANZBÖCK ...... 196

Table of Contents vii

7.3 The method of KANITANI ...... 200 7.3.1 KANITANI’s fundamental quantities ...... 200

7.3.2 The KANITANI-invariants of ΣI ...... 204 7.3.3 The flecnode transformation ...... 205 7.3.4 Applications of KANITANI’s calculus of ruled surfaces ...... 206 7.4 Final remarks ...... 208

Appendix A ...... 209 References ...... 212 Index ...... 216

1 Introduction

Two space curves c1 and c2 can be related to each other point for a point in different ways.

Each such assignment determines a correspondence or a relation, by which a space curve c1 is transformed into the other one c2. Depending on the kind of this assignment, there exist between the data of the two curves more or less simple relations, whose analytic derivation prepares from the given conditions no difficulties. If the correspondence is too particularly selected, then it can occur that there are either no curves, which suffice it, or only a special class of space curves, for which then the rule can be regarded as a defining characteristic. If one requires for example a space curve c2 , which has common tangents with another given space curve c1, then c1 and c2 must coincide together: the selected relation is not suitable to relate two different curves one to the other. On the other hand, the demand that two curves are to have common principal normal, can be fulfilled only be a special class of curves and therefore can be used for the definition of these BERTRAND’s curves. If one selects less special relations, then one has the advantage to connect curves of unknown characteristics with those which admit more exactly known ones, and from this connection the possibility to get a helping tool for research.

1.1 The aim of the dissertation

A general skew ruled surface Σ in a three-dimensional projective space Π3 possesses a pair of flecnodal curves and a pair of LIE-curves, which are invariantly connected with Σ with respect to projective transformations in Π3. So the topic belongs to the projective differential geometry of ruled surfaces in Π3. The original aim of this dissertation was to investigate, whether two arbitrarily given curves may serve as a pair of flecnodal curves or LIE-curves of a ruled surface or not. If not, what are the conditions, which the curves and their mutual position have to fulfil, such that they become flecnodal curves or LIE curves on a connecting ruled surface?

Throughout the research it turned out that two curves cannot be chosen freely such that there is a correspondence between them delivering a connecting ruled surface with the given curves as flecnodal curves or LIE-curves. When collecting references it turned out that this result is not at all new, but hidden in partly unpublished articles.

These facts might justify the change of the main goal of this dissertation: I want to present the different approaches to the described problem throughout history, add explicit examples, and connect the problem with another group of analogous classical problems: the BERTRAND curves pairs.

Throughout history ruled surfaces and flecnodal and LIE-curves are treated in three slightly different ways: by the method of E. J. WILCZYNSKI, the method of E. CECH and finally by the projective geometric calculus of G. BOL. These three persons stand centrally for three academic schools in treating ruled surfaces independently (!) in a projective resp. an affine 3-space.

Introduction 2

1.2 Flecnodal points (flecnodes)

It seems that SALMON [56] was the first to notice and publish the fact that a tangent may meet a surface in four consecutive points. This appeared in 1849 in the Cambridge and Dublin Mathematical Journal. In 1852, and in the same journal, CAYLEY [17] published an article “On the Singularities of Surfaces“, and denoted a point, at which a four-point tangent can be drawn, as a flecnode. It’s called so, because at such a point the tangent plane of the surface intersects the surface in a curve having at the point an inflexion and a node. Since four skew lines in projective space have in general two straight line intersectors (as solutions of a quadratic equation), it follows that there are in general two flecnode points on each generator of a ruled surface Σ in ordinary space. These two flecnode points may, of course, be real and distinct, real and coincident, or imaginary. The four-points-tangents shall be called the flecnode tangents; the locus of the flecnodes on Σ is it’s flecnode curve1 and the locus of the flecnode tangents, a ruled surface of two sheets, is the flecnode surface of Σ. Note that, in general, the flecnode tangent is not tangent to the flecnode curve, never, in fact, unless the latter degenerates into a straight line.

1.3 Lie-points

Five consecutive generators of a ruled surface define, in general, a linear complex. A regular generator gt(0 ) of Σ and its derivatives ggg ,, ,g thus span, in general, the (hyperosculating) accompanying linear complex K of Σ . If K is not special2, what means that K defines a regular null-polarity ν (g), then the restriction of ν (t0 ) to gt(0 ) is a projective correlation of the points of gt(0 ) to the planes through gt(0 ).  Similarly, ν |(gt0 ) defines also a linear complex K and its null-polarity, restricted to gt(0 ), delivers another projective correspondence, the so-called CHASLES-correlation, mapping the points of gt(0 ) to the tangent planes of Σ along gt(0 ). The product of ν |(gt0 ) and the

CHASLES-correlation turns out to be an involutoric projectivity of the point set of gt(0 ), which is called the KLEIN-involution γ along gt(0 ), see [36]. If γ is hyperbolic, there are two real and distinct fixed points Lt10(),L2(t0) of γ , which are named the LIE-points of gt(0 ). Actually, in English literature and according to E.J. WILCZYNSKI [78] these points are called the “complex points” of gt(0 ), and their locus on Σ is “the pair of complex curves” of the 3 ruled surface. As these concepts allow misinterpretations , we use the concept LIE-points and LIE-curves instead.

Note that the pair of flecnodal points of gt(0 ) suits into the KLEIN involution γ of gt(0 ) and therefore is harmonic to the LIE-points Lt10(), L2(t0) of gt(0 ). It seems that A. VOSS [70] was the first to notice that the LIE-curves and the flecnode curves of a rules surface intersect each generator in harmonic pairs of points.

1 In view of the fact that the flecnode curve on a ruled surface ordinarily crosses each generator twice and consequently appears in the neighbourhood of a generator like two curves, we may speak of the flecnode curves on a ruled surface when we are considering only the neighborhood of a generator. 2 A special linear complex K consist of all lines meeting a fixed line, the axis of K. If K is not special, then K consists of the fixed lines of a null-polarity ν and ν and K define each other mutually.

3 For example a complex curve usually is understood as a real curve whose tangents belong to a linear complex K. Introduction 3

Already in 1872 S. LIE [43] stated that if (and only if) the accompanying linear complex is fixed, i.e. K()tt= K(0 ) ∀∈tI⊂\ , then the LIE-curves are asymptotic lines of Σ, which means that the osculating planes of these curves are tangent planes of Σ and their tangents are osculating tangents of Σ. Such asymptotic LIE-curves are indeed “complex curves” in the sense of footnote 3.

1.4 Involute points (harmonic points)

There exists, in the complex extension of a real line, always a pair of points harmonically conjugate to two arbitrarily given pairs of points, namely the pair of fixed points of the involutoric projectivity ι defined by the two given pairs. This fixed pair is real if the given (real) pairs do not separate each other, otherwise it is imaginary, cf. H. BRAUNER [9].

On a generator g of Σ the flecnodes and the LIE-points are two pairs, which, in general, define such an involutoric projectivity ι, which is invarianty connected with Σ up to collineations. As real flecnodes and real LIE-points are harmonic pairs thus separating each other, ι is a hyperbolic involution only if one of the two pairs, flecnodes or LIE-points, are imaginary. WILCZYNSKI [78] calls the fixed points of ι the “involute points” of g and their locus on Σ the “involute curves” of Σ. In the following we restrict our considerations to the flecnodal curves and the LIE curves of Σ, and if needed use the complex extension of the real 3-space Π3.

1.5 Ruled surfaces belonging to a linear line congruence

A ruled surface Σ is said to be a “congruence surface”, if all its generators belong to a linear line congruence C. The focal lines of C are met by all generators of C and they act as the flecnodal curves of Σ, as they are flecnodal tangents of Σ in each of their points. We distinguish three cases of C : elliptic congruences with two conjugate high imaginary focal lines, parabolic congruences with a coinciding pair of focal lines and a CHASLES-correlation along this one and only focal line, and finally hyperbolic congruences with two real and skew focal lines. So the problem of finding a surface Σ to a given pair of real, coinciding or imaginary straight flecnodal lines is trivial and not determined by the focal lines alone.

The generators of a congruence C belong to a one-parameter set of linear complexes K, they are the common lines of a pencil of linear complexes. Thus a congruence surface Σ can be interpreted as a complex surface, too. The LIE-curves of Σ with respect to a regular complex K of the mentioned pencil of complexes is a pair of asymptotic curves of Σ . Due to H. NEUDORFER [51] one can construct the asymptotic curves of a congruence surface without integration with ruler and compass alone! He found that, on a generator g, besides the flecnodes, the contour point U with respect to a central projection with centre Z and the intersection point V of g with the null-plane of Z with respect to the null-polarity ν to a th chosen K form a second pair (U, V) of KLEIN’s involution along g. At the end of the 19 and at the beginning of the 20th century direct integral free “geometric” solutions of differential equations constituted an important topic of research. NEUDORFER’s idea is an example of that type of research. Introduction 4

The problem of finding a congruence surface Σ to a given LIE-curve l1 can be solved as follows: Necessarily l1 must be complex curve of a linear complex K, and the flecnodal lines must be a pair of (real or imaginary) reciprocal polar lines (in case of a hyperbolic resp. elliptic congruence) or a null-line (in case of a parabolic congruence) of K. These conditions are already sufficient, and they define Σ uniquely.

1.6 Ruled surfaces belonging to a special linear complex

In the following we deal with surfaces Σ having one straight directrix line a, but do not belong to a linear congruence of lines. Such a surface Σ is a “complex surface”, whereby the Γ 25 linear complex Ks spanned by Σ is special. (KLEIN’s image curve cΣ := Σ⊂M4 ⊂Π spans 5 2 a hyperplane in Π , which is tangential to the PLÜCKER-KLEIN quadric M4 ).

A ruled surface Σ of a special linear complex with complex axis a has this axis as (one) flecnodal curve and as coinciding LIE-curves. By arbitrarily giving a second flecnodal curve

f2 , there exist always a solution surface Σ (cf. Theorem 2.9.1 chapter 2, Theorem 3.6.2.1 chapter 3 and ANZBÖCK [1], p. 111).

1.7 BERTRAND curves

A pair of BERTRAND curves is defined by having common main normals, c.f. e.g. K.STRUBECKER [62]. BERTRAND curves are classical differential geometric objects of the euclidean 3-space. A curve c is of BERTRAND type, iff there holds a linear equation between its (euclidean) curvature κ(s) and its torsion τ (s), s being the arc length parameter of c.

We are interested in the ruled surface Σ formed by the common main normals gs( ) of the BERTRAND pair cc1, 2. The curves c1, c2 are asymptotic curves on Σ, as their osculating planes are tangent to Σ. In addition, as their tangents are asymptotic tangents of Σ being normal to gs( ), BERTRAND curves constitute the locus of vanishing mean curvature on Σ. In case Σ is a helicoidal conoid (right helicoid), this property is trivial.

The latter very special ruled surface Σ gives rise to connecting the BERTRAND problem to the LIE-curve problem: Σ is a hyperbolic congruence surface, such that the asymptotic curves are LIE-curves with respect to any regular complex K containing the hyperbolic congruence. The LIE-curves are helices and thus complex curves. The LIE-curves of Σ intersect the generators orthogonally, as their tangents are asymptotic tangents and Σ is a minimal surface with over all vanishing mean curvature. So the LIE-curves turn out to be BERTRAND curves, too. This in mind we may already state as a conjecture, that will be described and proved in the following chapters in details:

Conjecture:

Two arbitrarily given curves, in general, cannot serve as a pair of flecnodal curves or LIE- curves of a ruled surface. They have to fulfil certain conditions, which involve (projective differential geometric) curvature and torsion of the given curves. 2 The Method of WILCZYNSKI

As a basis for the theory of ruled surfaces, WILCZYNSKI [78] has used a system of ordinary linear homogenous differential equations of the form yp′′ ++y′ pz′ +qy+qz=0 11 12 11 12 (A) zp′′ ++21 y′ p22 z′ +q21 y+q22 z=0 with coefficients pik , qik functions of an independent variable x, to describe a ruled surface in affine space. Where (yzi, i) for (i =1, 2, 3, 4 ) be any four systems of solutions of (A), and dy d2 y y′ = , y′′ = , … etc. dx dx2 A transformation of the form η =+αyzβ, ζ =+γyzδ, ξ = f (x) (2.1) where α, βγ, , δ, f are arbitrary functions of x, converts it into another system of the same form. We can build a complete system of invariants and covariants to describe a ruled surface depending on the rule saying that: Any projective property of a curve will be expressed by an invariant equation or a system of such equations, and any invariant equation or system of equations will be the adequate analytical expression of some projective property of the curve.

2.1 The complete system of invariants and covariants

Firstly, to build the complete system of invariants and covariants which describes the projective properties of a ruled surface, we are going to use the following rules and properties:

– There are no rational covariants of odd degree for a binary system of linear homogenous differential equations.

– If Cλ,ω is an integral, rational, irreducible covariant of degree λ and of weight ω, it is transformed by the transformation (2.1) in accordance with the equation

λ ()αδ − βγ 2 Γ=λ,ωCλ,ω (2.2) ()ξ′ ω where its degree λ is necessarily even.

– For invariants λ = 0. From two invariants, an absolute invariant can always be formed. Similarly, from three covariants, an absolute covariant may be obtained.

– An absolute rational invariant is the quotient of two relative integral invariants of the same weight.

– Let θν be an integral rational invariant of weight ν. Then the considered transformation

(2.1) will convert it into θν , where 1 θν = θν . (2.3) ()ξ′ ν 2 The Method of WILCZYNSKI 6

– From two invariants of weights µ and λ, we can always form a new invariant of weight λ ++µ 1, viz.:

θλ++µµ1 =−µθ θλ′λθλθµ′, (2.4)

which is the Jacobian of θµ and θλ .

– We may form from every invariant of weight m, its quadriderative∗), of weight (2m + 2), where

21 22 θmm.1 =−2(mmθθ′′ m2+1)(θm′ )+2 mIθm (2.5)

is the required quadriderative of θm. Now let us put

2 u11 =−24p1′1 qpp11 +11 +12 p21, up=−24′ q+p(p+p), 12 12 12 12 11 22 (2.6) up21 =−242′1 q21 +p21(p11 +p22 ), 2 u22 =−24p22′ qpp22 +22 +12 p21, and

v11 =+2,up11′ 12u21 −p21u12 v =+2(up′ −p)u−p(u−u), 12 12 11 22 12 12 11 22 (2.7) v 21 =−2(up21′ 11 −p22 )u21 +p21(u11 −u22 ),

v 22 =−2,up22′ 12u21 +p21u12 and

wp11 =+2,vv11′ 12 21 −p21v12

wp12 =+2,vv12′ ()11 −p22 12 −p12 (v11 −v22 ) (2.8) wp21 =−2,vv21′ ()11 −p22 21 +p21 (v11 −v22 )

wp22 =−2,vv22′ 12 21 +p21v12 with pik , qik from (A). Then the expressions I =+uu, J=uu−uu 11 22 11 22 12 21 (2.9) KL=−vv11 22 v12v21, =w11w22 −w12w21 turn out to be semi-invariants. Where vv11 + 22 = 2,I′ ww11 +=22 24(vv11′′+22 ) =I′′. Using the previous rules, we can deduce the following expressions for the invariants, viz.:

2 22 θ4 =−I 4J, θ4.1 =−89θθ4′′ 4 (θ4′ )+8Iθ4

uu11 − 22 u12 u21

θ9 =∆, where ∆= vv11 − 22 v12 v21 (2.10)

ww11 − 22 w12 w21

221 θθ10 =−4 ()KI′′+4 (θ4 ).

∗) see FORSYTH, “Invariants, covariants and quotient-derivatives associated with linear differential equations”, London Phil. Trans. Vol. 179, 1888 2 The Method of WILCZYNSKI 7

To express the covariants in a simple manner, we can write

ρ =+2,yp′ 11 y+p12 z σ =+2,zp′ 21 y+p22 z (2.11) whence we find the following covariants:

22 Cz1 =−ρ yσ , Cu21=−2zu21y+(u11−u22) yz

1 24()uu11 −+22 ρσu12 +(vv11 −22 ) y+v12 zy C = 2 (2.12) 3 1 42uu21ρσ−−()11 u22 +vv21 y−2 (11 −v22 )zz

22 Cu5=⎡⎤()11−u22vv12−−(112v2)u12z+⎡(u11−u22)v21−−(v112v2)u21⎤y ⎣⎦⎣⎦ +−2()uu12vv21 21 12 yz

22 22 Cz7 =−⎣⎦⎡⎤vv12 21 y+()v11 −v22 yzθ4 −⎣⎡u12 z−u21 y+(u11 −u22 ) yz⎦⎤θ4′ where the indices indicate the weight of the invariant or covariant considered.

Now, if ( yzk, k) for k = 1, 2, 3, 4 are four pairs of functions satisfying (A), and for which the determinant

yz11yz11′ ′ yzyz′ ′ D = 22 22 (2.13) yz33yz3′ 3′

yz44yz4′ 4′ does not vanish, the most general solutions of (A) are given by

4 4 y= ∑ckky, z= ∑ckkz, (2.14) k=1 k=1 with four arbitrary constants cc14,,… .

Let y1,,… y4 be interpreted as the homogenous coordinates of a point Py ; z1,,… z4 as the coordinates of another point Pz . As x changes, these points describe two curves cy and cz which are the integral curves of the system of form (A). Let g yz be the line joining Py to Pz .

The points of the integral curves cy and cz , moreover, are put into a definite correspondence with one another, those being corresponding points which belong to the same value of x. It is necessary and sufficient that the tangents of the two curves, constructed at corresponding points, shall not intersect, i.e. the integrating ruled surface of a system (A) is never a developable. The developable surfaces are excluded by the condition D ≠ 0. The part which the transformation (2.1) plays in this theory in now quite obvious. The points

Py and Pz are changed by it into two other points Pη and Pζ of g yz , i.e. the curves cy and cz on Σ are converted into any two others cη and cζ , while the parameter x is, at the same time, transformed in an arbitrary manner. As the invariants of a linear homogenous differential equation characterize the projective properties of its integral curve, we can investigate the geometrical relations related to a ruled surface through studying the invariants and covariants of the system of ordinary linear 2 The Method of WILCZYNSKI 8 homogenous differential equations describing this surface, and finally we can characterize the projective invariant curves of a ruled surface by very simple conditions. Of course we are not going to explain all the results of WILCZYNSKI’s work about projective differential geometry of ruled surfaces, but we are going to concentrate on the results, especially those, which related to flecnode curves and LIE-curves.

First of all we will state the following theorem, which is according to WILCZYNSKI, the fundamental theorem of the theory of ruled surfaces:

Theorem 2.1.1: If θ4 ,θ4.1 ,θ9 and θ10 are given as arbitrary functions of x, provided however that θ4 and θ10 are not identically equal to zero, they determine a ruled surface uniquely up to projective transformations. (proof see [78] p.143)

As an exception to this theorem is the ruled surfaces, for which the two branches of the flecnode curve coincide, and those which have a straight line directrix.

Since the discriminant of the covariant C2 is equal to θ4 , the significance of the condition

θ4 = 0 becomes apparent. We may recapitulate as follows:

Theorem 2.1.2: The flecnode curve is determined by factorising the covariant C2. Its two intersections with the generators of the ruled surface are distinct if θ4 ≠ 0; they coincide if

θ4 = 0. If the integral curves cy and cz of a system of form (A) are the two branches of the flecnode curve, this system of differential equations is characterized by the conditions

uu12 =21 =0. (2.15)

Moreover we have:

Theorem 2.1.3: The necessary and sufficient condition for a ruled surface belonging to a single linear complex, which is not special, are

θ4 ≠ 0, θ9 =∆=0, θ10 ≠ 0 and Rank ∆ = 2. (2.16)

If θ10 = 0, while the other conditions remain the same, the complex is special. The surface belongs to a linear congruence with distinct directices if Rank ∆ = 1, while still θ4 ≠ 0. The directrices of the linear congruence coincide, if θ4 also vanishes. In this latter case the surface is, or is not a quadric, according as the equations

uu11 −=22 u12 =u21 =0 (2.17) are, or are not satisfied .

2.2 The ruled surface referred to its flecnode curves

By a suitable transformation on the dependent variable, the system (A) may be reduced to a new system of the same form: yp′′ ++z′ qy+qz=0 12 11 12 (F) zp′′ ++21 y′ q21 y+q22 z=0 2 The Method of WILCZYNSKI 9

for which pp11 ==22 0, pq12′ = 2,12 p21′ = 2q21. For this system (F), the two curves cy and cz are the two branches of the flecnode curve of Σ. When this transformation has been made, the fundamental invariants are:

2 θ42=−16()qq211 2 θθ10 = 4 pp12 21 3 (2.18) 2 θθ92=−2 4 ()pp11′′2p12p21 2 θθ4.1 =−894θ4′′ θ4′ +16()pp21 12 −2q11 −2q22 .

If θ4 ≠ 0 and θ10 ≠ 0, then the coefficients of (F) can be expressed in terms of the invariants by means of the formulae:

11θθ θθ 49ddxx− 49 θθ10 44∫∫θθ10 10 10 pe12 ==, p21 e, θθ44 p 3 p′′=+12 θ 2θ 24θθ −θθ′, 12 ( 4 9 4 10 4 10 ) 4θθ410 p 3 p′′=−21 θ 2θ +24θθ −θθ′, (2.19) 21 ( 4 9 4 10 4 10 ) 4θθ410 1 5 q =−16θθ+8θθ′′ −9θ′ 2 −8θ2 , 11 2 ( 10 4.1 4 4 4 4 ) 64θ4 1 5 q =−16θθ+8θθ′′ −9θθ′ 2 + 8.2 22 2 ( 10 4.1 4 4 4 4 ) 64θ4

To find the differential equations of the flecnode surfaces, we are going to use a tetrahedron of reference, two of whose vertices are the points Py , Pz , in which g yz is cut by cy , cz , and whose other two vertices are the points Pρ , Pσ , determined by substituting y= yk , zz= k ( k = 1, 2, 3, 4 ) into the expression (2.11), i.e., from the expressions

ρ =+2,yp′ 12 z σ =+2.zp′ 21 y (2.20)

The unit point is so chosen that any expression of the form uykk= α12++ααzk3ρk+α4σk,

k = 1, 2,3, 4, represents a point Pu whose coordinates are (α1,,αα23,α4). We are going to call it the fundamental tetrahedron of reference.

The defining systems of differential equations for the two sheets Φ1, Φ2 , of the flecnode surface are obtained by eliminating from equations (F) and (2.20) and others derived from them by differentiation, on the one hand z and σ , and their derivatives, and on the other hand, y and ρ and their derivatives. When this is done, surface Φ1 is referred to the curves cy and cρ , and surface Φ2 to the curves cz and cσ as directrix curves. For Φ1, cy is one branch of its flecnode curve but cρ is not the other branch. Similarly, cz is one branch of the flecnode curve of Φ2 but cσ is not the other branch. To find the canonical form of these two systems of differential equations, which uses for directrix curves on these two surfaces the two branches of their flecnode curves, simple transformations on the dependent variables are required. For the sheet Φ1,

p12η = qy, qyζρ=−γ12 , 2 The Method of WILCZYNSKI 10

and for the sheet Φ2 ,

p21ϕ = qz, qzψ =−σγ21 , where

qq=−11 q22 , γ12 =2,qp12 12 −q′ q γ 21 = 2.qp21 21 − q′ q The resulting systems are ηζ′′ ++ef′ η+fζ=0, 12 11 12 (F1) ζη′′ ++ef21 ′ 21η+f22ζ=0, where we put

qp12 2 ee12 =− , 21 = ⎣⎦⎡⎤2γγ12′ + 12 +2()q11 +q22 −p12 p21 , p12 q ⎛⎞∆ 1 11′′2 1 fq11 =−⎜⎟+++24γγ12 12 22, f12 = ()2qq12 −p12q, pp2 ⎝⎠12 12 (2.21) ⎛⎞∆ 11′ 2 1 f22 =−⎜⎟24γγ12 + 12 + 2 , ⎝⎠p12 p fq=+12 ⎡⎤γγ′′ 2 γ′ +11γ3 ++q−ppγ++qq′′−pq−pq. 21 q ⎣⎦12 12 12 2212 ()11 22 12 21 12 11 22 12 21 21 12

For the sheet Φ2 ,

ϕψ′′ ++hj12 ′ 21ϕ+j12ψ=0, (F2) ψ ′′ ++hj21ϕϕ′ 21 +j22ψ=0, where p q 21 ⎡⎤′ 2 hq12 =− ⎣⎦22γγ12 + 12 + ()11 +q22 −p12 p21 , h21 = , q p21 ⎛⎞∆ 11′ 2 2 j11 =−⎜⎟24γγ21 + 21 + 2 , ⎝⎠p21 1 ⎛⎞∆ ′ 11′ 2 2 j21 =− 2 ()2,qq21 −p21q jq22 =⎜−−−24γγ21 21 2 ⎟ (2.22) 2 p21 ⎝⎠p21 p jq=− 21 ⎡⎤γγ′′ +2.γ′ +11γ3 + +q− ppγ+q′ +q′ −pq−pq 12 q ⎣⎦21 21 21 2221 ()11 22 12 21 21 11 22 12 21 21 12 with 221 3 21 3 2 ∆=11pq222−2p12p1′′2+4p1′ 2, ∆=22pq111−2p21p2′′1+4p2′1. (2.23)

2.3 The ruled surface referred to its LIE-curves

The LIE-curve, may be determined by factorising a quadratic covariant, viz., C5. For system

(F), the covariant C5 reduces to

2 22 Cu51=−()1u22()p21y−p12z (2.24) 2 The Method of WILCZYNSKI 11 such that the transformation which replaces the two branches of the flecnode curve with the two branches of the LIE-curve is

yp=+21 yp12 z; zp=−21 yp12 z (2.25) and in this case the surface Σ will be referred to its LIE-curves as directrix curves, and the differential equations (F) will take the form yp′′ ++y′ pz′ +qy+qz=0, 11 12 11 12 (L) zp′′ ++21 y′ p22 z′ +q21 y+q22 z=0. where

⎛⎞qq12 21 ⎛⎞qq12 21 pp11 =−12 p21 ⎜⎟+,,p22 =−p12 p21 −⎜⎟+ ⎝⎠pp12 21 ⎝⎠pp12 21

⎛⎞qq12 21 pp12 ==21 ⎜⎟− , ⎝⎠pp12 21 (2.26) 11⎛⎞∆∆21 qq11 ==22 22⎜⎟22+ =()δδ2 +1 , ⎝⎠pp21 12

11⎛⎞∆∆21 qq12 ==21 22⎜⎟22− =()δδ2 −1 ⎝⎠pp21 12 where

221 3 221 3 ∆=11pq222−24p12p1′′2+ p1′ 2, ∆=22pq111−24p21p2′′′1+ p21.

2.4 The osculating quadric – The flecnode congruence

The osculating quadric QΣ along a generator g of a ruled surface Σ is defined as the limit of the regulus determined by g and two neighbouring generators of Σ as each of these approaches g, remaining on Σ. It contains two different sets of lines; the first one are the generators of QΣ which belong to the same set as g, the concordant set of lines. The asymptotic tangents of Σ along g constitute the second set of generators of QΣ , the transversal set of lines.

Now if we let the generator g yz of our ruled surface vary over the surface, then there is clearly a one-parameter family of osculating quadrics, and the totality of their concordant set of lines is a two-parameter family, or a congruence of lines, which WILCZYNSKI called the flecnode congruence C of the ruled surface. Actually, the flecnode congruence receives its name from its relation to the flecnode surfaces, which constitute its focal surfaces. Moreover, if we substitute y= yk , zz= k ( k =1, 2, 3, 4 ) into the expression (2.11), the two sets of four quantities ρk and σ k may be interpreted as the homogenous coordinates of two points Pρ and

Pσ ; let the line which connects them be gρσ . Now, as x changes, gρσ describes a ruled surface Σ′, whose generators belong to the congruence C, and we will call it according to WILCZYNSKI the derivative of Σ with respect to x. This surface is a developable if J = 0.

2 The Method of WILCZYNSKI 12

Thus we have the following theorem:

Theorem 2.4.1: The flecnode congruence contains two families of ∞1 developables, which coincide if and only if θ4 = 0, i.e., if and only if the two branches of the flecnode curve of Σ coincide. To determine any developable surface of the congruence, it is necessary and sufficient to find a solution of the equation

4,{}ξξxI2 ++2{,x}J=0, (2.27) where {ξ, x} denotes the Schwarzian∗) derivative of ξ with respect to x, and to take this solution ξ = ξ(x) as the independent variable of the defining system of differential equations. The derivative of Σ with respect to ξ will then be a developable surface, and all developables of the congruence may be obtained in this way. Moreover, any four developables of the same family intersect all of the asymptotic tangents of Σ in point-rows of the same cross ratio.

There exists in general two families of ∞2 non-developable ruled surfaces of the congruence C each of which has one branch of its flecnode curve on the flecnode surface of Σ. But may it happen that the derivative of Σ, as a non-developable ruled surface of the congruence , has both of the branches of its flecnode curve on the flecnode surface of Σ, one on each sheet? We have the following theorem as an answer on this question,

Theorem 2.4.2: If there exists a ruled surface of the flecnode congruence C, different from Σ, which has both of the branches of its flecnode curve on the flecnode surface of Σ, one on each sheet, then the surface Σ belongs to a linear complex. Conversely, if Σ belongs to a linear complex, but not to a linear congruence, there will be ∞2 ruled surfaces in its flecnode congruence which have the required property.

If Σ belongs to a linear congruence, this is true not merely for ∞2 but for all ruled surfaces of the flecnode congruence which, in this case, coincides with the linear congruence to which Σ belongs. Another question arises, whether among those ∞2 ruled surfaces of the congruence C there exists one (there can not in general be more than one) whose flecnode surface has one of its sheets in common with that of Σ.

Actually, we have,

Theorem 2.4.3: If the surface Σ is not of the second order, and if its flecnode surface has a sheet Φ′ which does not degenerate into a straight line, no other ruled surface of the congruence C has Φ′ also as a sheet of its flecnode surface. Or, in other words, the second sheet of the flecnode surface of Φ′ does not belong to the congruence C.

2 ∗) fx′′′() 3 ⎡⎤f′′(x) The Schwarzian derivative of a non constant meromorphic function f : ^→ ^ is defined by DSchwarzian ≡−⎢⎥. fx′′() 2⎣⎦f(x) It is historically the first and most fundamental projective differential invariant. (see [52])

2 The Method of WILCZYNSKI 13

If we trace any curve on a ruled surface Σ, there will correspond to it a perfectly definite curve on the derivative Σ′; the lines joining the corresponding points being the asymptotic tangents of Σ. The investigation of such correspondences gives rise to a number of important results. It may be shown that there exists a single infinity of ruled surfaces in the flecnode congruence two of whose asymptotic lines correspond in this way to the flecnode curve on Σ. In other words this may be expressed as follows,

Theorem 2.4.4: If Σ is a ruled surface with two distinct branches of its flecnode curve and not belonging to a linear congruence, there exists just a single one parameter set of ruled surfaces in the congruence C , whose intersections with the two sheets of the flecnode surface of Σ are asymptotic lines upon them. They are the derivatives of Σ, when the independent variable is chosen such as to make the seminvariant I vanish. Moreover, the point-rows, in which any four of these surfaces intersect the asymptotic tangents of Σ, all have the same anharmonic ratio.

The derivative Σ′, of Σ with respect to x, is cut thus by the flecnode surface of Σ along a pair of asymptotic lines if I = 0. But this relation turns out to be reciprocal, i.e., the surface Σ′cuts out an asymptotic curve upon each sheet of the flecnode surface of Σ. The asymptotic lines upon the two sheets of the flecnode surface correspond to each other. Since the flecnode surface is also the focal surface of the congruence C, we see that the flecnode congruence is a W-congruence.

The principal surface Σ1 of the flecnode congruence (cf. [78] p.216) my be constructed as follows:

We consider the flecnodes Py and Pz of g, the planes π y and π z osculating the flecnode curves at these points, and the points P1 and P2 upon the flecnode tangents f1 and f2 whose loci are the second branches of the flecnode curves on the two sheets Φ1 and Φ2 of the flecnode surface respectively. The plane π y intersects f2 in a certain point to which corresponds a point on f1 such that the line joining them is a generator of the osculating quadric. This latter point together with P1 constitute a pair, such that the harmonic conjugate of Py with respect to it is the point in which the principal surface intersects f1. The intersection with f2 is found in the same way. The curves in which this principal surface intersects the two sheets of the flecnode surface of Σ are their principal curves.

We might also say, that in this way there is determined, upon the generators of the osculating quadric, an involution whose double elements are g and the generator of the principal surface. For Σ1 the following can be easily verified:

– If θ15 = 0, θ4 ≠ 0, the principal surface Σ1 is the harmonic conjugate of Σ with respect to the two ruled surfaces of the flecnode congruence which cut out the second branches of

the flecnode curves on Φ1 and Φ2.

– If θ4.1 = 0, or θ64θθ−910=0, the principal surface Σ1 intersects Φ1 and Φ2 along 22 asymptotic curves, where θ6 =−24I (IJ) +5(K−I′ ) +4(K−2J′′ +II′′ ).

25 – If θθ4.1 −64 4 =0, the principal surface Σ1 is developable. 2 The Method of WILCZYNSKI 14

223 – If θθ15 −9 θ4 =0, the principal surface Σ1 intersects one of the sheets of the flecnode surface along the second branch of its flecnode curve. It thus intersects both sheets if θ =θ =0. 15 9

We have to notice that our construction of the principal surface becomes indeterminate if Σ has two straight line directrices.

The process of forming the derivative of a ruled surface with respect to an independent variable may be repeated. We thus obtain Σ′′ the second derivative of Σ with respect to x. This may coincide with Σ itself. We then have the following result,

Theorem 2.4.5: Every ruled surface which has a second derivative coinciding with Σ itself may be defined by a system of form (A) for which pik = 0, and for which the quantities qik are constants. If the independent variable is chosen such that the second derivative Σ′′ coincide with Σ, the first derivative Σ′ is a projective transformation of the original surface, and its asymptotic lines correspond to those of Σ. Such surfaces are a special case of a ruled surface Σ contained in a linear congruence (i.e. one, or both, branches of the flecnode curve are straight lines), which we will study later.

2.5 The derivative cubic

Let g′, a generator of the concordant set on QΣ , be the corresponding generator of Σ′, and let Q′ be the quadric which osculates Σ′ along g′. Since g′ is situated entirely upon Q′ as well as upon QΣ , the rest of the intersection of these two quadrics is, in general, a space cubic. This cubic is called the derivative cubic. Using the fundamental tetrahedron of reference PPyz Pρ Pσ , then the parametric equations of the cubic will take the following form

2 22 x = At t , x31=+()Bt 2,Ct1t2+Dt2t1=ψ t1 112 (2.28) 2 22 x21= At t2, x41=+()Bt 2,Ct1t2+Dt2t2=ψ t2 where

Au=−2(11u22 u11 u22 ),Bu= v , 12 22 ψ =+Bt 222.Ct t +Dt (2.29) 1 1122 Cu=−2 ()11vv22 u22 11 , Du=−v 21 11,

We obtain, in this way, associated with every ruled surface, surfaces containing a single infinity of space cubics. The derivative cubic degenerates only if the two branches of the flecnode curve of Σ coincide, if Σ has a straight-line directrix, or if the derivative surface Σ′ is one of the developables of the congruence C. The relations of the derivative cubic to the ruled surface are numerous and of considerable interest. We shall mention only the following theorem:

Theorem 2.5.1: Let the derivative surface Σ′ be one of the ∞1 surfaces of the congruence C which intersects the flecnode surface along an asymptotic curve. The derivative cubic will then intersect the generator of Σ in its LIE-points (cf. [78] p.207).

If the two branches of the flecnode curve of Σ coincide, there is a derivative conic to consider instead of a derivative cubic. 2 The Method of WILCZYNSKI 15

As a space cubic always determines a null-system, then associated with this null-system, we have a linear complex. We are going to call it the linear complex of the derivative cubic. For the relation between this complex and the osculating linear complex of the ruled surface, we have,

Theorem 2.5.2: The osculating linear complex and the complex of the derivative cubic are in involution if the first derivative ruled surface cuts out asymptotic curves on the flecnode surface of Σ, and the cubic passes through the LIE-points of g.

The coordinates of the plane, which corresponds to a point ( xx12,,0,0) of g in the null- system of the cubic are

2 2 4,()CB− Dx2 −−4,()CBDx1 −ABx12− 2,ACx 2.ACx12+ ADx This plane contains g if , and only if, CB2 − D= 0, i.e., if the derivative cubic is tangent to g. It will coincide with the plane tangent to Σ at this point, if further

−−ABx12,ACx2=ωx2 2,ACx12+ ADx =−ωx1 where ω is a proportionality factor, or

Abx12++(20AC ω)x =, (20AC + ω) x12+=ADx , whence follows ω =−AC or −3AC. We have therefore

x12::xC=− B=−D:C or x12::xC= B= D:C.

These points are harmonic conjugates with respect to the flecnodes. Therefore,

Theorem 2.5.3: If the derivative cubic is tangent to g, there are two points of g whose tangent planes are the planes corresponding to them in the null-system of the cubic. These points and the flecnodes form a harmonic group on g. They never coincide with the LIE- points unless the ruled surface has a straight line directrix.

Generally, it can be proved that

Theorem 2.5.4: There exist in general two points on g, harmonic conjugates with respect to the flecnodes, such that the planes, corresponding to them in the null-system of the derivative cubic, pass through a generator of the osculating quadric.

Let c be one branch of the flecnode curve of a ruled surface Σ, and let the developable surface formed by its tangents be called its primary developable. There exists another important developable surface containing c, which WILCZYNSKI called its secondary developable, as indicated in the following two theorems,

Theorem 2.5.5: If at every point of the flecnode curve of Σ there be drawn the generator of the surface, the flecnode tangent, the tangent of the flecnode curve, and finally the line which is harmonic conjugate to the latter with respect to the other two, the locus of these last lines is a developable surface, the secondary developable of the flecnode curve. 2 The Method of WILCZYNSKI 16

Theorem 2.5.6: We can find a one parameter set of ruled surfaces, each having one branch of its flecnode curve in common with that of Σ. This family of ∞1 surfaces may be described as an involution, in which any surface of the family and its flecnode surface form a pair. The primary and secondary developables of the branch of the flecnode surface considered, are the double surfaces of this involution. In fact, the generators of these surfaces, at every point of their common flecnode curve, form an involution in the usual sense (see [78] p.233).

2.6 The differential equations of curves situated on a ruled surface

To find the differential equations of the integral curves cy and cz of the ruled surface described by the system (A), we shall eliminate once z and then y, so as to obtain the linear differential equations of the fourth order which each of these functions must satisfy. The required differential equations take the form:

yp(4) ++46y(3) py′′ +4py′ +py=0, 1234 (2.30a,b) (4) (3) zq++4612zqz′′ +4q3z′ +q4z=0 where 11 pq1 =−( 12l12 p12m12 ), p2 =(s12l12 −r12m12 ), 46∆∆11 11 qq1 =−( 21l21 p21m21), q2 =(s21l21 −r21m21), 46∆∆22 pm=−1 ()prpr+l(ps−qr)−l∆, 3 4∆ []21 1211 1112 12 1112 1211 11 1 1 (2.31) 1 pm4=−[]12()p121s1 q11r12 +l12(q11s12−q121s1)−m111∆ , ∆1 1 qm3=−[]212()p1r22p22r21+l212(p2s21−q21r22)−l22∆2, 4∆2 1 qm3=−[]21( p212s2 q22r21) +l21(q22s21−q212s2) −m22∆2 . ∆2

We shall assume that ∆1 ≠ 0 and ∆≠2 0; i.e. neither cy nor cz is a plane curve.

The quantities ∆1, ∆2 , lij , mij and sij are defined as follows

∆=11p 2sq12−12r12, ∆22=p 1sq21−21r21

2 rp11 =+11 p12 p21 −p1′1 −q11, sp11 = 11q11 +−p12q21 q11′ ,

rp12 =+12 ()p11 p22 −p12′ −q12,sp12 =+11q12 p12q22 −q1′2,

rp21 =+21()p11 p22 −p2′1 −q21,sp21 =+21q11 p22q21 −q21′ , 2 rp=+pp−p′ −q, sp22 =+21q12 p22q22 −q22′ , 22 22 12 21 22 22 (2.32) ′ lp11 =− 11r11 −p21r12 +r11′ +s11, mr11 =−11q11 −r12q21 +s11, ′ lp12 =− 12r11 −p22r12 +r1′2 +s12, mr12 =−11q12 −r12q22 +s12, ′ lp21 =− 11r21 −p21r22 +r2′1 +s21, mr21 =−21q11 −r22q21 +s21, ′ lp22 =− 12r21 −p222rr2++2′2 s22, mr22 =−21q12 −r22q22 +s22. 2 The Method of WILCZYNSKI 17

The fundamental seminvariants and invariants of the integral curve cy are given by the following expressions:

2 3 Pp22=−p1′ −p1, Pp33=−p1′′ −33p1p2+p1,

22 4(3)2 Pp44=−43p1p3−p2+12p1p2−6p1−p1 +3P2, (2.33)

3 ′ ′′6 ′6 2 θ33=−P2 P2, θ44=−PP2.3+5 P2−25 P2

On replacing p, P and θ by q, Q and θ′ respectively, we obtain the corresponding expressions for the integral curve cz .

The tangents to the integral curves cy and cz will belong to linear complexes if, and only if, the invariants θ3 and θ3′ vanish identically. Similarly, if cy and cz are conics, then θ3 and θ3′ vanish identically. By making use of (2.31) and the first of (2.23), one can express θ3 and θ3′ in terms of the coefficients of (A) as follows p′ p′ 12 1 ′′ ′ 21 1 ′′ θ32=−()qq211−2 ()q22−q11, θ3=−()qq1122−2 ()qq11−22. (2.34) p12 p21

Thus the conditions θ3 =θ3′ = 0, reduce to

2 p12′ 2 p21′ qq11′′−=22 ()qq11 −22 , qq11′′−=22 ()qq11 −22 . (2.35) p12 p21

As the flecnode curve and the LIE-curve are of fundamental importance for the projective theory of ruled surfaces, it becomes essential to find out to what extent they may be arbitrarily assigned. To answer this question, let us consider a curve c be given by means of its differential equation

dd43yyd2ydy ++46pp12+4p3+p4y=0, (2.36) ddxx43dx2dx where p1,,… p4 are given functions of x. In the system of differential equations (A) defining the surface Σ, we must regard the coefficients pik and qik as unknown functions. We may, however, assume without exception that u12 = 0, so that cy is one of the branches of the flecnode curve on Σ , that p21 = 0 , so that cz is an asymptotic curve on Σ , and that pp11 =22 =0. Under these assumptions we form the differential equation (2.30a) of the curve cy . Since cy is to be identical with c , it must be possible to transform equation (2.30a) into (2.36) by a transformation of the form y= ϕ()xy, x = fx(). (2.37)

The functions ϕ and f are not independent. For, while the equations u12 = 0 and p21 = 0 are not disturbed by any transformation of this form, the conditions pp11 = 22 = 0 are. In fact, a transformation of the form (2.37) converts (A) into another system of the same form whose corresponding coefficients p11 and p22 will be ϕ′ f ′′ ϕ′ f ′′ p ++2 p ++2 11 ϕ f ′ 22 ϕ f ′ p = , p = . (2.38) 11 f ′ 22 f ′ 2 The Method of WILCZYNSKI 18

In order, therefore, that after this transformation p11 and p22 may again vanish, we must have

ϕ = C (2.39) f ′ where C is an arbitrary constant, which may be put equal to unity.

If then we apply the transformation (2.37) to (2.36), we shall get an equation

dd43yyd2ydy ++46pp12+4p3+p4y=0, ddxx43dx2dx which we must identify with (2.30a). Equating coefficients gives us a system of four equations with five unknown functions of x, viz.: f , pq12, 11, q21, q22 . One may, therefore, impose another condition and still obtain an infinity of ruled surfaces. The most general curve cz which is capable of being the second branch of the flecnode curve on a ruled surface, for which cy is the first branch, involves therefore, in its expression one arbitrary function of x. It cannot, therefore, be an arbitrary curve, as that would involve two arbitrary functions.

The same can be proved analytically for the LIE-curves, thus we state with WILCZYNSKI the following theorem,

Theorem 2.6.1: Any arbitrary space curve being given, can be considered as one branch of the flecnode curve (LIE-curve) of an infinite set of ruled surfaces into whose general expression there enters an arbitrary function. Two curves taken at random can not be connected, point to point, in such a way as to constitute the complete flecnode curve (LIE- curve) upon the ruled surface thus generated.

Pσ (0,0,0,1)

Pz ()0,1,0,0

Pρ (0,0,1,0) cy

Py (1,0,0,0)

Fig. 2.1

Let us consider a space curve cy , whose linear homogenous differential equation of the fourth order is

(4) (3) yp++4612ypy′′ +4p3y′ +p4y=0 (2.40) with the relative semi-covariants, besides y which is obviously itself a semi-covariant, 2 The Method of WILCZYNSKI 19

zy=+′ p1y,

ρ =+yp′′ 212y′ +py, (2.41) (3) σ =+yp3312y′′ +py′ +p3y.

z ρ σ The absolute semi-covariants are y , y and y . All other semi-covariants and seminvariants are functions of these and of the derivatives of P2 , P3 and P4 defined in (2.33). If we make a transformation of the independent variable ξ = ξ (x), we find that

zz=+1 3 η y, ξ′ ()2 1 ⎡⎤1 2 ρρ=+2 24ηzy+6 ()µ+9η, (2.42) ()ξ′ ⎣⎦ 1 ⎡⎤33231 ′ σσ=+3 22ηρ+()µ+ηzy+4()µ+43ηµ+η. ()ξ′ ⎣⎦ where we have put

ξ′′ η = , µ =−ηη′ 1 2. (2.43) ξ′ 2

If the functions y1,,… y4 constitute a fundamental system of (2.40), then we may interpret them as the homogenous co-ordinates of a point Py of the curve cy in ordinary space. If we put yy= k (k = 1,2,3,4)into the expression (2.41), we obtain three other points Pz , Pρ and Pσ which, as x varies, describe three curves cz , cρ and cσ , curves which are closely connected with cy . Pz is clearly a point on the tangent of cy constructed at Py ; Pρ is in the plane osculating cy at Py , while Pσ is outside of this plane (see Fig.2.1).

In order to study the curve cy in the vicinity of Py , it will be convenient to introduce the tetrahedron PPyz Pρ Pσ as tetrahedron of reference, with the further condition that, the unit point is chosen such that any expression of the form uykk= α12++ααzk3ρk+α4σk, k =1, 2,3, 4, represents a point Pu whose coordinates are (α1,,αα234,α).

We may without affecting the positions of the points Pz , Pρ and Pσ assume that (2.40) ∗) is written in the semi-canonical form , so that p1 = 0. For, in order to put (2.40) in the semi- canonical form, we need only to multiply y by a certain factor λ, which will then also appear multiplied into the semi-covariants z, ρ and σ.

Let us then assume p1 = 0. We shall have from (2.41) z= y′. If we differentiate (2.40) and eliminate y between the resulting equation and (2.40), we shall find

∗) It is always possible to reduce the equation (2.40) to a canonical form in which pp12= = 0 , i.e. P2 = 0. This will be called the LAGUERRE-FORSYTH canonical form.

It can be also proved that: ”the most general transformation which leaves the LAGUERRE-FORSYTH canonical form invariant, is α+x β Cy ξ= , η= , the totality of these transformations constitutes a four-parameter group. (cf. [78] p.25) n−1 γ+x δ ()γ+x δ

2 The Method of WILCZYNSKI 20

2(4) (3) 2 ()PP42+−36z ()P4′′+P2P2z+6P2(PP42+3) z′′ ++⎡⎤6PP′4 P+3P2 −6PP′+6PP′z′ (2.44) ⎣⎦()23()42 2()422 + ⎡⎤4PP′′++3P22P+3P−4PP+6PP′z=0 ⎣⎦()34 2(4 2)3()4 22

2 if PP42+≠30. 2 If PP42+3=0 we find (3) zP++642z′ P3z=0. (2.45)

Equation (2.44) determines the curve cz in the same way as (2.40) determines cy. But if 2 PP42+=30, z satisfies (2.45) showing that the curve cz is in this case a plane curve, i.e., is a plane section of the developable surface whose cuspidal edge is cy .

We shall need to consider the ruled surfaces generated by those edges of our tetrahedron which meet in Py . Of these we know one immediately, namely the developable which has cy as its edge of regression, and of which PPyz is a generator. The ruled surface Σ yρ generated by PPy ρ clearly has cy as an asymptotic curve; for, the plane PPyρ Pz is both osculating plane of cy at Py , and tangent plane of the surface at Py . If we assume that p1 = 0, we can find the system of differential equations representing this ruled surface as follows: From (2.41), we have

yp′′ =−ρ 2 y, (3) yp=−ρ′′22y−py′, (2.46) (4) yp=−ρ′′ 22y′′ −2.p′ y′ −p2′′ y

Substituting these values in (2.40), we obtain the equation

2 ρρ′′ +−(4pp322 ′ )y′ +(p4−pp2′′ −5 2)y+5p2=0 (2.47)

Hence, assuming p1 = 0 , then this ruled surface may be studied by means of the equations

yP′′ +−2 yρ =0, (2.48) 2 ρρ′′ +−(4PP322 ′ )y′ +(P4−PP2′′ −5 2)y+5P2=0.

Similarly, assuming p1 = 0, the differential equations representing the ruled surface Σ yσ generated by PPy σ are obtained as yp′′ ++y′ pσ ′ +qy+qσ =0, 11 12 11 12 (2.49) σσ′′ ++py21 ′ p22 ′ +q21y+q22σ=0, where

2 PP32−+′′1 P43P2−P3 P3 p11 ==, pq12 , 11 = , q12 =0, p22 =−, q22 =3P2 PP2233P2 P2 1 23 p21 =−⎡⎤33P3 +P3P2′′+2PP2 3 −3PP2 2′′+PP2 4 −6P2 , (2.50) P2 ⎣⎦ 1 22 qP21=−⎡⎤34P−66P2P3+P3P3′′+P2P4+P2P2′−P23P′′. P2 ⎣⎦ 2 The Method of WILCZYNSKI 21

Now, if (2.40) is written in the LAGUERRE-FORSYTH form, i.e. P2 = 0. In that case, the two equations (2.49) reduce to the single equation

′′11 ′ Py34++33σ (P −P3)y=0, (2.51) which proves that, in this case, the surface Σ yσ is a developable. Moreover, if τ =+λµy σ represents the point in which PPy σ intersects the edge of regression of the developable, then it can be easily proved that,

τ =+3Py3 σ (2.52) represents the edge of regression of the developable to which Σ yσ reduces when P2 = 0.

To find the equations of both the osculating cubic and the osculating conic of cy at Py , we are going to assume that the differential equation be written in the LAGUERRE-FORSYTH canonical form, and that Py corresponds to the value of x = a. The solution Y of our equation may then be developed by TAYLOR’s theorem into a series proceeding according to powers of (x − a) up to the fifth order, and for greater convenience in writing, we shall put a = 0. So we shall find

′′11′2(3)31(4)41(5)5 Yy=+yx+23yx+!yx+4!yx+5!yx (2.53) an expression which may be written in the form

Yy=++12yyzy3ρ +y4σ (2.54) where yy12,,y3,y4 are themselves such power-series, which will represent the curve cy up to th terms of the 5 order in the vicinity of the point Py , referred to the system of co-ordinates which has just been defined.

Since pP12==0, we shall find from (2.41) by successive differentiation that

yz′′==,,y′ρσy(3) =−Py+,y(4) =−Py−4Pz, 343 (2.55) (5) yP=− 44′′y−(4P+ P3)z−4P3ρ.

If therefore we put Y into the form indicated above, we shall have:

345 yP13=−120 20 x−5P4x−P4′x+… yx=−120 20Px45−(4P′ +P)x+… 2334 (2.56) 25 yx33=−60 4Px+… 3 yx4 =+20 …

We see at once that the following equations are exact up to terms no higher than the fifth order

32yy−=y2 0, 24 3 (2.57) 2 5(2yy13−−y2) 6P3y3y4=0.

These same equations must be satisfied by the co-ordinates of any point of the osculating cubic of the curve cy , since this cubic must have contact of the fifth order with cy at Py . 2 The Method of WILCZYNSKI 22

These equations (2.57) are, therefore, referred to this special tetrahedron of reference. In terms of a parameter t we may write

3 2 3 x13=+15 12Pt , x2 = 30t, x3 = 30t , x4 = 20t . (2.58)

It can be also proved that the expression

′ 3632 ()12PP32−+12 24P2τ +15ττy+20( 52P2+) z+30τρ+20σ (2.59) represents an arbitrary point of the osculating cubic when the tetrahedron of reference is not restricted to the condition P2 = 0 , because this expression remains invariant under the general 1 transformation ξ = ξ(x), and reduces to (2.58) for P2 = 0 , if τ = t . From this we can find the equation of the osculating conic as

22 −+40xx13 15x2+32P2x3=0 (2.60) which may be defined as a part of the intersection of the developable of the osculating cubic with the osculating plane.

To deduce the equation of the osculating linear complex, i.e. of that linear complex determined by five consecutive tangents of the curve, denote by Y and Z the expressions of

y and z in the vicinity of Py . Then we have up to terms of the fourth order

1134 141213 Y =−y()1,62Px34−4Px +z( x−6Px3) +2ρσx +6x

Z=−y 11Px23−Px −1Px′4+z⎡14−2Px3−1P′+P x4⎤ (2.61) ()2634244⎣ 3324()34⎦ 1142 +−ρσ()xP623x+x.

If we denote the coefficients of yz, , ρ, σ in these two expressions by y1,,… y4 and z1,,… z4 respectively, then the PLÜCKER line-coordinates of the tangent will be

ωik =yzi k −yk zi whence 1134… 1 2 ω12 =−1,28Px3 +P4x + ω23 =+2 x …, 1 4 1 3 ω13 =−xP12 3x+…, ω24 =3 x +…, (2.62) 1 2 … 1 4 ω14 =+2 x , ω34 = 12 x +… Therefore, the equation of the osculating linear complex , referred to the special tetrahedron of reference, is

ω14 −ω23 =0. (2.63)

In general, it can be proved that the equation representing the osculating linear complex when the tetrahedron of reference is general , takes the form

ω14 −−20P2ωω34 23 =. (2.64) 2 The Method of WILCZYNSKI 23

The equations of the null-system defined by the osculating linear complex are then

ωv =−x , ωv =−x +2,Px 14 3224 (2.65) ωv23= x , ωv41=−x 2Px23 or ω′xP=+2,vv ω′x = v , 1224 32 (2.66) ω′xP22=−2,vv1− 3ω′x41=−v .

We are going to find the position of the point Pρ on the generator PPy ρ . The curve cy , on the ruled surface Σ yρ generated by PPy ρ , is an asymptotic curve upon it, and the surface is characterized by the equations (2.48).

According to the general theory of ruled surfaces, the asymptotic tangents to Σ yρ at the points Py and Pρ are obtained by joining these points to 2z and 24σ −+Pz232Py respectively. Therefore, the asymptotic tangent to Σ yρ at any point (αα1,0, 3,0) of PPy ρ joins this point to

22Py33α +−(αα1 4P23)z+2α3σ.

The equation of the plane tangent to Σ yρ at (αα1,0, 3,0) is then

−+αα3xP212()−20α3x4=.

To the same point of PPy ρ there corresponds a plane in the osculating linear complex. According to (2.65), this is the plane

αα32xP+−()1 202α3 x4=. (2.67)

It is clear that these planes form an involution, and the double planes of this involution are the osculating plane x4 = 0, and a plane x2 = 0 which contains σ. The point which corresponds to this latter plane is

β=2Py2 +ρ. (2.68)

At the same time the osculating conic intersects PPy ρ in Py and in Pα where

α =4Py2 +5ρ. (2.69)

The cross ratio of the four points Py , Pρ , Pα , Pβ is then 5 ()αβ,,y ,ρ= 2 .

Hence if, upon the generator of Σ yρ , there be marked its intersections with the osculating conic, and the point Pβ whose tangent plane coincides with the plane corresponding to it in the osculating linear complex, the point Pρ is determined by the condition that the cross ratio 5 of these four points shall be equal to 2 .

We notice that if P2 = 0, then Pα and Pβ coincide with Pρ , and in that case Pρ can be defined as the point at which the plane tangent to the ruled surface and the plane corresponding to it in the osculating linear complex coincide. In this exceptional case it can be 2 The Method of WILCZYNSKI 24

also proved that the curves cy and cρ shall be harmonically divided by the branches of the flecnode curve of the ruled surface.

Now, we proceed to deduce the equation of the osculating linear complex of Σ yρ . Let us assume that pP12==0, and let Y and R denote the developments of y and ρ in the vicinity of an ordinary point x = a, and replace again ( x − a) by x in the developments. Then we shall find Yy=++yyzyρ +yσ , 123 4 (2.70) Ry=++ρ12ρρz3ρ+ρ4σ where 1123′′12′4… ρ13=−Px−26P4x − P4x +24(4P3−P4) x + 23114 ρ2=−24Px3− ()P34′′+P x − ()2P34′+P′x +… 612 (2.71) 2134′ ρ33=−1832Px −4()P3+P4x +… 1 4 … ρ43=−xP6 x+ and y1,,… y4 have been computed before in equations (2.56).

Denoting by ωρik =−yi k yk ρi , the PLÜCKER-coordinates of the line joining the two points, we find 2311′′′′4 ω12 =−Px3 −31(22P3 −P4 ) x − 2( P3 −P4 ) x +…, 1134′ ω13 =−1236Px3 −()P3 −P4 x +…, 1 4 ω14 =−xP3x+…, 6 (2.72) 1 4 ω23 =+xP6 3x+…, 2 ω24 =+x …, 1 3 ω34 =+3 x …

Now, let

abω12 ++ωω13 c14 +dω23 +eω24 +fω34 =0 be the equation of the osculating linear complex of the surface in question. Then, the coefficients of all powers of x up to and including x4 must be zero, if we substitute the above development for ωik into the left term. This gives us the following equations

bc=+0, d=0, c+d=0 −P3a+e=0, 11′ 1 −−33()20PP34a−P3b+3f=, (2.73) 11′′ ′ ′ 11 −−12 ()22PP34a−6 ()P3−P4b−6 P3c+6 P3d=0 and finally we find thus the equation of the linear complex osculating the ruled surface Σ yρ as 2 −−PP31ωω2 ()4′′−24P3′(14−ω23)−P3ω24+4(P4−2P3′) P3ω34=0. (2.74) 2 The Method of WILCZYNSKI 25

We notice that it coincides with the osculating linear complex of cy if, and only if, P3 = 0, i.e. if cy belongs to a linear complex. The coordinates vk of the plane, which corresponds to a point (x1,,xx23,x4) in the linear complex (2.74) are given by

ωv13=∗+42Px2+ ∗ +(P4′′−P3′) x4, 2 ωv23=−42Px1+∗−()P4′′−P3′x3+ 4P3x4, (2.75) ωv3=∗+()PP4′′−2432′x+ ∗ −P34(PP−234′)x, 2 ωv44=−()PP′′−243′x1− P3x2+4P3()PP4−23′x3+ ∗ , where ω is a proportionality factor.

Thus, if the curve cy belongs to a linear complex, i.e., θ3 = 0, and we put P2 = 0, then from (2.33) also P3 = 0. Hence, the ruled surface Σ yρ is a non-developable surface, and its equations take the form:

ρ′′ +Py4 =0, y′′ −=ρ 0. (2.76)

It belongs to the same linear complex as cy , and cy and cρ are the two branches of its LIE- curve, which is at the same time an asymptotic curve.

To find the equation which determines the curve cρ in the same way as (2.40) determines cy in this special case, we have to differentiate (2.40) twice ( taking into account that p1 = 0, PP23==0) then eliminate y and y′. Finally the equation determining cρ takes the form:

(4) 2 pp44′′(3) ⎛⎞ ρρ−+⎜⎟2−ρ′′ +p4ρ=0. (2.77) pp44⎝⎠

In this case PPz ρ generates a developable, since z′ = ρ, of which cz is the cuspidal edge. PPρ σ generates a developable, since ρ′ = σ , of which cρ is the cuspidal edge. Finally PPz σ generates a ruled surface whose equations are

P4′ σσ′′ −+′ Pz4 =0, z′′ −=σ 0, (2.78) P4 upon which cz and cσ are asymptotic lines. So we have the following theorem,

Theorem 2.6.2: If a given curve belongs to a linear complex we may, in an infinity of ways, choose the fundamental tetrahedron so that four of its edges give rise to developables, whose cuspidal edges are described by the four vertices. The other two edges of the tetrahedron will then give rise to ruled surfaces, upon which the vertices of the tetrahedron trace a pair of asymptotic curves. The latter coincide with the two branches of the special LIE-curves as asymptotic lines on the surface.

Now, let us see under what conditions can the given curve cy be one branch of the flecnode curve upon one of these surfaces. Let us study the ruled surface generated by PPy σ , on which cy can never be an asymptotic line. In fact, if we form the quantities uik of the theory of ruled surfaces for system (2.49), we find

P2′ u12 =− 2 . P2 2 The Method of WILCZYNSKI 26

But u12 = 0 is the condition that cy may be a branch of the flecnode curve on the surface generated by PPy σ . Suppose that the variable has been so chosen as to make P2′ = 0. The most general transformation which leaves this relation invariant satisfies the condition

55′ −−20ηµP2 63+ηµ= ′ 1 2 or, since µ =−η2 η, then

3 −−51ηη′′ 2P2 +15ηη′ −5η=0 (2.79) a differential equation of the second order for η. Moreover, two different solutions of this equation give rise to two distinct ruled surfaces. For, let η1 and η2 be two such solutions, and let σ1, σ 2 be the corresponding values of σ. Then, according to (2.42),

σσ=+1 ⎡⎤33ηρ+µ+η2z+1 µ′ +4ηµ+3η3y x = 1, 2 . (2.80) xx32⎣⎦()x2x4( xxxx) () ()ξx′

But, if the same ruled surface corresponds to η1 and η2 , the three points y, σ1 and σ 2 must be collinear. We must, therefore, be able to reduce

′′333 ()ξσ11−=(ξ2)σ22 ()η1−η2ρ+… to a multiple of y, But clearly, this is possible only if η2=η1. We see, therefore, that there 2 are ∞ of the ruled surfaces Σ yσ upon which cy is one branch of the flecnode curve.

If P2′ ≠ 0, our problem leads to the differential equation

d1η′′−=11ηη226 PP−2+2ηη′−η (2.81) dx ()25225( 2) which is of the second order and of the third degree. Corresponding to the ∞2 solutions of 2 this equation, we find ∞ positions for Pσ , viz.:

′′3 6 12 32 46()ξ ση=()55PP22− ++ηµ3ηy+(4µ+6η) z+6ηµ+4σ.

The locus of these points is a cubic surface

32 23 27Px24′ −+36Px23x4 90x2x3x4−90x1x4−40x3=0 (2.82) which contains PPyz, the tangent of cy , as a double line. It is, therefore, a ruled surface. It is a surface of the kind known as CAYLEY’s cubic scroll.

Suppose that a solution η of (2.81) be known. We may make a transformation of the ξ′′ independent variable, ξ = ξ(x) such that =η. For the resulting equation follows P′ = 0. If ξ′ 2 we again denote the independent variable by x , (2.81) becomes

′′ ′ 12 ′ 1 2 ηη−=η−5P2η+2η(η−2η), (2.83) where P2 is a constant, since P2′ = 0. But we may satisfy this equation by putting η = const., which gives the equation 2 The Method of WILCZYNSKI 27

3 12 ηη+=5 P2 0, whence

η = 0, ±−12 P . (2.84) 5 2

The root η = 0 gives the original solution, the other two are new. Thus, if one of the surfaces

Σ yσ is known, upon which cy is a branch of the flecnode curve, two others may be found by merely solving a quadratic equation.

2.7 CARPENTER’s surfaces with planar flecnode curves

A. F. CARPENTER [10] considered ruled surfaces whose flecnode curves consist of two plane branches. The necessary and sufficient conditions that cy and cz as flecnode curves shall be plane curves are:

221 3 ∆=11pq222−2p12p1′′2+4p1′ 2=0, (2.85) 221 3 ∆=22pq111−24p21p2′′1+ p2′ 1=0.

We have from (2.19) by differentiation

p12 p12′′=+()θθ4 4θ9 24θ4θ10 −θ4′θ10 , 4θθ410

p21 p21′′=−()θθ4 4θ9 +24θ4θ10 −θ4′θ10 , 4θθ410 p p′′ 12 (432 2′ 2 32′ 22 6 ′ 12 =−22θθ4 9 θ4θ10 +θ4 θ10 −θ4 θ4θ4θ9θ10 (2.86) 16θθ410 22 2 −+16θ4θθ4′′10θ10 4θ4 θ4θ9′θ10 +8θ4θ10θ10′′−16θθθ4 4′′10 ),

p21 32 2 2 22 p21′′′=−22(4θθ4 9 θ4θ10 +32θ4′θ10 +6θ4 θ4θ4′θ9θ10 16θθ410 22 2 −−16θ4θθ4′′10θ10 4θ4 θ4θ9′θ10 +8θ4θ10θ10′′−16θθθ4 4′′10 ).

By substituting from (2.19) and (2.86) in (2.85) we obtain ∆1 and ∆2 in terms of the fundamental invariants, in the form

2 2 p12δ1 p21δ2 ∆=1 22=0, ∆2 ==22 0 64θθ410 64θθ410 where

32 2223222 δ1=−16θ10 θ4.1θ10+40θ44θθ′′ 10−25θ4′ θ10+θ4θ9+20θ4θ1′0−16θθθ44′ 10θ1′0 22 −+16θθ410θ1′′0 4θ4 θ4()2θ4θ10−3θ49′θθ10−2θ49θ′θ10+3θ49θθ1′0 , (2.87) 32 2223222 δ2 =−16θ10 θ4.1θ10 +40θθθ4 4′′ 10 −25θ4′ θ10 +θ4θ9 +20θ4θ10′ −16θ44θθ′ 10θ1′0 22 −−16θθ4 10θ1′′0 4θ4 θ4 ()2θ4θ10 −3θ4′θ9θ10 −2θ4θ9′θ10 +3θ4θ9θ10′ . 2 The Method of WILCZYNSKI 28

Now p12 can not be zero for otherwise the first of equations (F) would reduce to a second order equation in y alone, that is, cy would be a straight line contrary to our hypothesis.

Similarly for p21. We conclude therefore that δ1 = 0 and δ 2 = 0 are the conditions for a flecnode curve with two distinct non-rectilinear plane branches.

The differential equation for cy is, in view of (2.19),

yp′′′ ++3312y′′ py′ +p3y=0, (2.88) where

p12′1 3p12′q11 1 p1 =− ,,pp2 =− ()12 p21 −q11 p3 =− −2 p12 p2′1 +q1′1. (2.89) 23p12 2p12

Let us suppose that both cy and cz are conics. By addition of the two conditions of (2.34), we have

⎛⎞pp12′′21 ⎜⎟−()qq22 −11 =0. ⎝⎠pp12 21

But if qq22 −11 =0, then by (2.18), θ4 = 0, that is the two branches of the flecnode curve are coincident, contrary to our hypothesis. It follows that pp12′ 12 = p21′ p21 and therefore that

22 d ⎛⎞pp12′′1 ⎛⎞12 p12′′3 ⎛⎞p12′ ⎜⎟−=⎜⎟ −⎜⎟ d2xp⎝⎠12 ⎝⎠p12 p12 2⎝⎠p12 (2.90) 22 d ⎛⎞pp21′′1 ⎛⎞21 p21′′3 ⎛⎞p2′1 =−⎜⎟⎜⎟=−⎜⎟. d2xp⎝⎠21 ⎝⎠p21 p21 2⎝⎠p21

Now from (2.85) we have

2 2 p12′′ 3 ⎛p12′ ⎞ pp21′′′3 ⎛⎞21 2,q22 =−⎜⎟ 2q11 =−⎜⎟ (2.91) p12 2 ⎝⎠p12 pp21 2 ⎝⎠21 and this, with (2.90) gives again qq22 − 11 = 0. We conclude therefore that

Theorem 2.7.1: There are no ruled surfaces whose flecnode curves are composed of two distinct, non-degenerate, conics.

We can also prove that if either branch of flecnode curve of a ruled surface is a conic then the principal surface of the flecnode congruence intersects one of the sheets of the flecnode surface along the second branch of its flecnode curve. Moreover this theorem holds when only one branch of the flecnode curve is plane.

It is possible by a suitable chosen transformation of the independent variable to make uu11 −22 =1. Let us then assume system (F) is such that

pp11 ==22 u12 =u21 =0, uu11 − 22 =1 (2.92) and at the same time that cy is a conic while cz is any plane curve not a conic.

2 The Method of WILCZYNSKI 29

From equations (2.18) we have

θ4 = 1, (2.93) and since cy is a conic (i.e. θ3 = 0 ) then

θ91+2θ′0=0. (2.94)

By means of (2.93) and (2.94) the first of equations (2.87) gives

2 δ1 =+θθ10 ⎣⎦⎡⎤82()10 1−θ4.1 +(θ9 +2θ1′0 )(θ9 +10θ10′′) −8θ10 (θ9 +2θ1′′0 )

2 =+θθ10 ⎣⎦⎡⎤82()10 1−θ4.1 .

But in our case δ1 = 0 and θ10 ≠ 0, so that we find

θ4.1 =+82()θ10 1. (2.95)

Remembering also that δ 2 = 0, then δ1−δ 2 may be reduced by means of (2.93) and (2.94) to

22 23θθ10 10′′ −+θ1′0 θ10 =0. (2.96)

The general solution of (2.96) is

x 4Ae2 θ10 = , (2.97) x 2 AA12()− e and this result substituted in (2.94) and (2.95) gives

8A eAxx+ e ⎡ x ⎤ 22() ⎢ 8Ae2 ⎥ θ9 =− 3 , θ4.1 = 812 + . (2.98) AA ex ⎢ x ⎥ 12()− ⎣ AA12()− e ⎦

We now have our four fundamental invariants expressed explicitly in terms of the independent variable and two arbitrary constants. Consequently it is possible to calculate the coefficients of system (F). By means of (2.93) and (2.19), we find pA12 ==3 const. From equations (2.18) then

x θ10 4Ae2 p21 == 2 . (2.99) p12 x AA13()A2− e

From (2.19) we find

xx 2Ae22()A + e q12 = 0, q21 = . (2.100) x 3 AA13()A2− e

From the first of equations (2.85), since p12 is a constant, q22 = 0 , and since uu11 −=22 41()q22 −q11 =,

1 q11 =−4 . 2 The Method of WILCZYNSKI 30

The coefficients of system (F) have now been calculated. They involve three arbitrary constants of integration, but two of these are not essential; since the transformation y η = , ζ = z, ξ =−x log A2 (2.101) A3 leaves conditions (2.92) unchanged and serves to remove A2 and A3 . The transformed system of equations is

1 ηζ′′ +−′ 4 η=0, ξξ 2Aeξ Ae ()1+ e (2.102) ζη′′ ++23′ η=0. ()11−−eeξξ()

Since each value of A gives a system of equations defining a class of mutually projective ruled surfaces, we conclude that

Theorem 2.7.2: There exists a single infinity of classes of mutually projective ruled surfaces having the property that their flecnode curves consist of two distinct plane branches, one of which is a conic.

We are going to give a geometrical construction of a ruled surface Σ whose flecnode curve consists of two plane branches, as an answer of the question of the extent to which a ruled surface is determined by the condition that the two branches of its flecnode curve shall be distinct, non-rectilinear, plane curves.

Let c be any given plane curve. On c let any set of points P1,..., P5,... be taken, corresponding for instance to values of the parameter ξ differing by ∆ξ . Through each of the points P1, P2 , P3 draw an arbitrary line and let Q1, Q2 , Q3 , be the points of intersection of these three lines g1, g2 , g3 , with an arbitrary plane π . Through P1 and Q1 respectively draw the lines f1′ and f1′′ which intersect g1, g2 , g3.

The plane determined by f1′ and P4 will intersect

f1′′. This point of intersection, together with P4 will determine a line g4 which intersects both f1′ and Q2 Q3 Q f1′′, and cuts the plane π in a point Q4. Through P2 1 Q4 π and Q2 respectively draw the lines f2′ and f2′′ which intersect g2 , g3 , g4. The plane determined by f2′ and P5 will intersect f2′′. This point of f1′′ intersection together with P5 will determine a line g which intersects both f ′ and f ′′, and cuts the 5 2 2 c plane π in a point Q5. Proceed thus, locating each time a new point in the plane π. The sets of lines f ′ P P P4 i P1 2 3 and fi′′ are such that each line of either set intersects four successive lines of the set gi . When ∆ξ f1′ approaches zero the set of lines gi will constitute a g3 g4 g1 ruled surface Σ, while the sets of lines fi′ and fi′′ g2 will also constitute ruled surfaces F′ and F′′ such that each line of either surface will be a four-point Fig. 2.2 tangent to Σ, those of F′ having as locus of their 2 The Method of WILCZYNSKI 31 points of tangency the curve c and those of F′′ having as locus of their points of tangency the locus of the points Qi . Σ will therefore be a ruled surface whose flecnode curve consists of two plane branches.

2.8 Ruled surfaces whose LIE-curves have planar branches

As the pair of LIE-curves of a ruled surface is closely connected with its flecnode pair, we are going to study what happens for the LIE-pair if we assume that cy and cz are plane curves. From (2.26), then

qq11 ==22 q12 =q21 =0 hence the system (L) will take the form

⎡⎤qq qq ′′ ⎛⎞12 21 ′ ⎛12 21 ⎞′ yp+−⎢⎥12 p21 ⎜⎟+y+⎜−⎟z=0, ⎣⎦⎝⎠pp12 21 ⎝p12 p21 ⎠ (LP) ⎛⎞qq12 21 ⎡⎤⎛q12 q21 ⎞ zy′′ +−⎜⎟′ −⎢⎥p12 p21 −⎜+⎟z′ =0. ⎝⎠pp12 21 ⎣⎦⎝p12 p21 ⎠

Thus we can find the differential equations for the LIE-pair cy and cz in the form

yp′′′ ++11⎡⎤pp−p′ y′′ +ppp−pp′ +p′ p−p3 y′ =0 pp⎣⎦()11 22 12 12 ( 11 12 22 11 12 11 12 12 ) 12 12 (2.103) 11 3 zp′′′ ++⎣⎦⎡⎤()11 p22 p12 −p12′ z′′ +()p11 p12 p22 −p22 p12′ +p22′ p12 −p12 z′ =0 pp12 12 assuming that p12 ≠ 0, otherwise it would be p12 = kp21 and it follows from (2.85) that

qq11 −22 =0, i.e., θ4 = 0. As the equations (2.103) are of the third order only, then we conclude that,

Theorem 2.8.1: If the two branches of the flecnode curve of a ruled surface are distinct (over ), non-rectilinear, plane curves, then the two branches of the LIE-curve have this same property. Moreover, the four planes of these curves form a harmonic pencil of planes in which the planes of the two branches of the flecnode curve are separated by the planes of the two branches of the LIE-curve. The converse of this theorem also holds.

Now let us consider the planes osculating the four curves cy , cz , cy , cz , at the points Py ,

Pz , Py , Pz . Actually, it can be proved that,

Theorem 2.8.2: The planes osculating the two branches of the flecnode curve and the planes osculating the two branches of the LIE-curve at the four points in which these curves cut a line element of their supporting ruled surface, will form a pencil if, and only if,

22 Dp≡∆21 1 −p12∆2 =0. (2.104)

2 The Method of WILCZYNSKI 32

CARPENTER [12] gave an extension of the preceding theorems to include the involute curves as follows

Theorem 2.8.3: The planes osculating the flecnode curve, the LIE-curve and the involute curve at the six points in which these curves cut any ruling g of Σ, will belong to a pencil, if and only if

22 Dp≡∆21 1 −p12∆2 =0.

The axis m of this axial pencil generates a ruled surface Ψ. If the tangent to cy at Py cuts

m in one of its flecnode points, the tangent to cz at Pz will cut m in its other flecnode point, the tangents to cy , cz , at Py , Pz , will cut m in its LIE-points, and Ψ will be the space dual of Σ. Conversely, if Ψ is the space dual of Σ , then the remaining conditions follow.

Moreover, it can be proved that (CARPENTER [10]):

Theorem 2.8.4: In order that the line of intersection of the planes osculating the flecnode curve of a given ruled surface at the two points in which it is cut by a generator of the surface may envelope a plane curve, it is necessary and sufficient that one branch of the flecnode curve be a plane curve.

2.9 Ruled surfaces with coincident plane branches of the flecnode curve

If we choose the curve with which both branches of the flecnode curve coincide for one directrix curve, cy , and any asymptotic curve of Σ for the other directrix curve, so in this case the system (A) will take the form 1 yp′′ ++12 z′ q22 y+p12′ z=0, 2 (FA)

zq′′ ++21 yq22 z=0.

The following theorem can be easily proved,

Theorem 2.9.1: If the two branches of the flecnode curve of a ruled surface Σ coincide in a plane curve, then there exists a one-parameter family of mutually projective ruled surfaces each having the two branches of its flecnode curve coinciding with that of Σ, and to this family belong Σ and its flecnode surface. In order that the two plane branches of the flecnode curve of a ruled surface may coincide it is necessary and sufficient that the secondary developable of one of these curves be a cone (cf.[13]).

If cy and cz coincide in a plane curve, the osculating planes coincide with the plane of this curve and the secondary developable becomes a cone the vertex of which is the common pole of the plane of this curve taken with respect to the osculating quadrics of Σ. We make use of this pole and polar relation in a construction of this ruled surface:

Let c be any given plane curve. On c let points P1,..., P5,... be taken corresponding to values of the independent variable differing by ∆ξ. Through P1 and any point P not in the plane of

c, pass the line PP1. This line, together with the tangent t1 to c at P1, determines a plane 2 The Method of WILCZYNSKI 33

PP11t . In this plane draw any line g1 through P1. In the same way through P2 and P3 draw lines g2 and g3 lying respectively in the planes PP2t2 and PP33t . Through P1 draw the line

f1 which cuts g2 and g3. It will intersect the plane PP44t in a point which with P4 will determine a line g4. Through P2 draw the line f2 which cuts g3 and g4. It will cut the plane

PP5t5 in a point which with P5 will determine a line g5. So proceed, obtaining alternately new lines f and g . The two sets of lines g and f are such that each line of either set intersects four consecutive lines of the other set. Now let ∆ξ approach zero. As the points

Pi−1, Pi , Pi+1 approach coincidence, the quadric determined by three lines gi−1, gi , gi+1, approaches a limit quadric Qi . Moreover, with respect to this quadric P and the plane of c have the pole and polar relation since P is the point common to the planes tangent to Qi at

Pi−1, Pi , Pi+1. At the same time the two sets of lines gi and fi will approach two ruled surfaces Σ and Φ each of which is one sheet of the flecnode surface of the other, and c will be one branch of the flecnode curve for each surface. The quadrics Qi will osculate Σ along the generators gi and the plane of c will have with respect to all these osculating quadrics the common pole P. Since the locus of the poles of the plane of c with respect to the Qi is the edge of regression of the secondary developable of c, this developable is a cone and hence Σ, and therefore Φ, have the two branches of their common flecnode curve coincident and plane.

2.10 Ruled surfaces with rectilinear branches of the flecnode curve

As we explained before that, the necessary and sufficient conditions for a ruled surface with two distinct straight line directrices are that Rank ∆ = 1, and θ4 ≠ 0. In this case the ruled surface belongs to a linear congruence (see the introduction) and the system (A) will take the form yq′′ +=y0, 11 (R) zq′′ +22 z=0, where q11 and q22 are non-vanishing constants, and qq11 − 22 ≠ 0, since we have assumed

J ≠ 0 and θ4 ≠ 0.

The integral curves of (R) are the straight line directrices of Σ. Let the edge xx34==0 of the fundamental tetrahedron of reference coincides with cy , while the edge xx12==0 coincides with cz . We may then put

rx1 rx2 ye= , y= e, y3 = 0, y4 = 0, 1 2 (2.105) rx3 rx4 z1 = 0, z2 = 0, ze3 = , ze4 = , where

⎧⎫r1 ⎧⎫r3 ⎨⎬=± −q11 , ⎨⎬=± −q22 . (2.106) ⎩⎭r2 ⎩⎭r4 If we put

xkk=+αβyzk, where α and β are arbitrary functions of x, we shall obtain ( x1,,… x4) as the coordinates of an arbitrary point of the surface. 2 The Method of WILCZYNSKI 34

We find

rx1 rx2 rx3 rx4 x1 = αe , x2 = αe , x3 = βe , x4 = βe .

The relation between x1,,… x4, obtained by eliminating α, β, x from these equations, will be the equation of the ruled surface Σ. It is

22−−qq22 11 2−q22 2−q11 xx14−=xx23 0, or more briefly

λµ λµ xx14−=xx23 0 (2.107) where λ and µ are constants.

If the directrices of the congruence coincide, the equation representing the ruled surface Σ can be proved to be of the form

x3 2qx()1x4−=x23x x34xlog (2.108) x4 where q is a constant.

Let Σ be a ruled surface with straight line directrix, and let us take this line as one of the reference curves cy . Let cz be an arbitrary curve. We may write y =1, yx= , y = 0, y = 0, 1 2 3 4 (2.109) zf11= ()x, zf22= ()x, zf33= ()x, z4 =1.

The system (A), for which these are the fundamental solutions, is found to be y′′ = 0,

⎛⎞ff13′ ′′ f2′ f3′′ f3′′ ⎛f1′f3′′ ⎞ (2.110) zx′′ +−⎜⎟f12′′ f′′ −x + y′ −z′ +⎜−f1′′⎟y=0, ⎝⎠ff33′′ f3′⎝f3′⎠ whence by (2.6)

1 uu11 ==0, 12 0, u22 =−2 { f3, x}, (2.111) ⎛⎞ff13′′′ f3′′ uf21=−2,{}3x(f2′ xf1′)+2(x′ +2)⎜⎟f1′′−−3()xf12′′−f′′ +2()xf12′′′−f′′′ , ⎝⎠ff33′′ where { f3 , x} indicates the Schwarzian derivative of f3 with respect to x. The flecnodes are given by

η = y, ζ = −−uy21 u22 z. (2.112)

The second flecnode point ζ is given by ζ =−uu− f, ζ = −uf, 121221 3223 (2.113) ζ 22=−ux1−u22f2, ζ 42=−u 2.

2 The Method of WILCZYNSKI 35

Thus, we obtain the following result,

Theorem 2.10.1: The equations of the most general curve which can serve as the second branch of the flecnode curve of a ruled surface with a straight-line directrix can be obtained without any integration.

Let p21 = 0 . Then system (F) becomes

yp′′ ++z′ qy+1 p′ z=0, 12 11 2 12 (2.114)

zq′′ +=22 z0 so that cz is a straight line. Moreover, by (2.18), θ9 = 0 and θ10 = 0. If we think of the non- vanishing invariants θ4 and θ4.1 as given functions of x, it is possible to determine q11 and

q22 from (2.18). But p12 remains as an arbitrary function, that is, the invariants do not suffice to characterize the ruled surface. However, if we suppose cy to be a plane curve, then from the first of equations (2.85), p12 may be determined except for two constants of integration.

We have seen that a ruled surface can not have two non-degenerate conics for the two branches of its flecnode curve. But these two branches may consist of a straight line and a conic as we shall see.

Under the conditions pp11 ==21 p22 =u12 =u21 =0, the value of the invariant of weight three of cy is given by (2.34). For θ3 = 0 we get on integrating

p12 =−cq( 22 q11 ) or, if we suppose the independent variable to be so chosen that

uu11 −=22 41()q22 −q11 =,

1 1 then p12 = 4 c. The first of equations (2.85) then gives q22 = 0 so that q11 =−4 and (2.114) becomes

11 yc′′ +−44z′ y=0, z′′ =0. (2.115)

Equations (2.115) are indeed a special case of equations (2.102).

We proceed to obtain the Cartesian equations of a ruled surface which has a straight line and a conic for the two branches of its flecnode curve. For convenience let c = 1. Then a fundamental system of solutions of (2.115) is

11xx−−1x 1x 22 2 2 y12==e,1ye+,y3=e−1,ye4=, (2.116)

zz12==1, x, z3=−x, z4=1.

An arbitrary point on the line g yz is given by

1 x −1 x xe=+λµ2 , x =+λ(1ex2 )+µ , 1 2 (2.117) 1 x −1 x 2 2 x3 =−λ(1ex)−µ , xe4 =+λµ. 2 The Method of WILCZYNSKI 36

We have

1 x 2 Xx1=+1x23+x−x4=2,λµe X3=x1−x2−x34+x=2,

−1 x 2 Xx2=− 1+x234+x+x=2,λλe X4=x1+x2−x3−x4=2(+µx), (2.118)

11xx− 2 XX12λλ22X4λX1X2λX1x ==ee, , =+x, 22=, =e, XX33µµX3µX 3 µ X2 whence

X XX X 4=± 12+log 1, (2.119) X 33XX2 which becomes

ξ =−(ζηlogξ)2 (2.120) where XX21= ξ, XX31=η and XX41= ζ are interpreted as the Cartesian coordinates of a point in the three space. The introduction of X i in place of xi amounts merely to a projective transformation and this does not change the character of the curves involved. It is easy to show that for the above surface the flecnode curve is made up of the parabola ζ 2 = ξ in which it is cut by the ξζ − plane, and the line at infinity common to the set of planes perpendicular to the ξ − axis. The rulings of the surface are the straight lines

ξ = k, ζη=±log kk. (2.121)

2.11 The primary and secondary flecnode cubics and LIE-cubics

The quadric determined by any ruling g of a general ruled surface Σ, the line gF of intersection of the planes osculating the two branches of the flecnode curve of Σ at their points of intersecting with g, and the line which corresponds to this line by means of the linear complex which osculates Σ along g, intersects the quadric osculating Σ along g, in g and a non-degenerate space cubic, which passes through the LIE-points of g. CARPENTER [13] called this cubic the primary flecnode cubic CF. If we use the system (F) and choose the fundamental tetrahedron of reference PPyzPρ Pσ explained before, where for the system (F), equations (2.11) take the form

ρii=+2,yp′ 12 zi σ ii=+2,zp′ 21 yi (i =1,…, 4 ) (2.122) then the parametric equations of this cubic may be written in the form

222 222 ξ11=−2,tp()2q21t p21q12 ξ21=−2,( p 2qt21p21q12) (CF) 2 2 ξ31=−pp221t()p12t p21, ξ41=−pp221()p12t p21.

Similarly we can find a primary LIE-cubic CL, if we replace in the previous section the word „flecnode“ wherever it occurs, with the word “LIE”. The parametric equations of this cubic are found to be 2 The Method of WILCZYNSKI 37

222 2 ξ1=−2,Pt ()p122q 1t p21q12 +p122q 1Dt ( p12t −p21) =Pξξ1+D 3

222 2 ξ2=−2,Pp()122q1t p21q12 +p122q1Dp( 12t−p21) =Pξ2+Dξ4 (CL)

2 2 ξ31=−p 2pP21t()p12t p21=Pξ3, ξ41=−p 2pP21( p12t p21) =Pξ4. where,

p21 p12 Pp=−12q21 p21q12 , D =∆1−∆2. (2.123) p12 p21

If we think of the parameters in (CF), (CL), as being identical, there is thereby established between the points of these two curves a 1-1 correspondence.

Let us now consider the plane osculating (CF) at any one of its points, then the point which corresponds to this plane by means of the osculating linear complex of Σ along g is found to be

222 22 2 η12=+2(p 13pq1221t p21q12), η21=+23pt2( p12q21t p21q12), (CF´) η =+pp223,pt p 22 31221()12 21 η41=+pp221t()p12t 3p21

These four equations constitute the parametric equations of a new space cubic whose points are in (1,1) correspondence with the points of (CF). CARPENTER called it the secondary flecnode cubic CF´. The lines joining corresponding pairs of points of (CF) and (CF´) determine a ruled surface Ω upon which these two cubics are directrix curves. For this surface Ω, we have the following theorem, (cf. [13])

Theorem 2.11.1: The (1,1) point correspondence established between the primary and secondary flecnode cubics associated with each element g of a ruled surface Σ by means of the linear complex osculating Σ along g, determines a second ruled surface Ω on which these two cubics are asymptotic curves. The surface Ω belongs to a linear congruence with distinct directrices which are in fact the two branches of the flecnode curve of Ω , and which cut every ruling of Ω in points harmonically separated by the points in which this ruling is cut by the primary and secondary flecnode cubics.

Similarly we can find the secondary LIE-cubic, whose parametric equations are

222 2 2 η12=+23p 1Pp()12q21t p21q12+p12p21Dp(312t+p21) =Pηη1+D3,

22 2 3 2 η21=+23pP2t()p12q21t p21q12+p12p21Dt( p12t+3p21) =Pη2+Dη4, (CL´)

22 22 η31=+p 2pP21()3,p12t p21=Pη3 η41=+p 2pP21t( p12t 3.p21) =Pη4

A comparison of equations (CF), (CL), (CF´), (CL´) shows that the primary and secondary LIE-cubics are projectively related to the primary and secondary flecnode cubics. It follows that the ruled surface Ω determined by the primary and secondary LIE-cubics is projectively related to the ruled surface Ω, and the properties established for Ω are also properties of Ω.

2 The Method of WILCZYNSKI 38

Thus we have

Theorem 2.11.2: The primary and secondary LIE-cubics associated with each ruling g of a ruled surface Σ , as well as the ruled surface Ω which they determine, are projectively related to the primary and secondary flecnode cubics and the ruled surface Ω determined by them.

The correspondence established between the points of (CF) and (CL), (CF) and (CF´), (CL) and (CL´) also holds for the pair of cubics (CF´) and (CL´), and hence these two cubics are directrix curves of a ruled surface Ψ. For this surface it can be proved the following theorem,

Theorem 2.11.3: The ruled surface Ψ determined by the secondary flecnode cubic and the secondary LIE-cubic associated with each ruling g of a ruled surface Σ, belongs to a linear congruence whose directrices coincide in g and at the same time constitute the complete flecnode curve of Ψ.

Moreover, there are two and only two rulings of Ψ which lie upon the osculating quadric of the ruled surface Σ along g and these correspond to the values 0 and ∞ of the parameter t.

They are indeed the flecnode tangents of Σ at Py and Pz .

2.12 The flecnode- LIE tetrahedron

If the osculating planes of the two branches of the flecnode curve and those of the two branches of the LIE-curve do not form a pencil, then they form a tetrahedron, which N. B. MACLEAN [45] called the flecnode-LIE-tetrahedron of the integrating ruled surface Σ. As the independent variable x varies, the edges of this tetrahedron will generate six new ruled surfaces, while the vertices will describe space curves, each lying on three of the new ruled surfaces. MACLEAN studied these six new ruled surfaces, which are covariantly associated with the original ruled surface Σ, and has found the differential equations of form (A) defining them. He paid a special attention to the consideration of the surfaces ΣF , which generated by the line of intersection of the osculating planes to the flecnode curves (denoted by gF ), and ΣL , which generated by the line of intersection of the osculating planes of the

LIE-curves (denoted by gL ). He studied their relations to one another and to the original ruled surface Σ.

For the surface ΣF , the differential equations of form (A) can be found as follows: Let us put q q ry=λ +y′, sz= µ + z′, where λ = 21 , µ = 12 (2.124) p21 p12 to denote two points Pr , Ps on the tangents of the flecnode branches cy , cz of Σ at the points

Py , Pz , where these tangents meets the generator gF respectively. These two points describe two curves cr , cs on ΣF , which we are going to take as directrix curves of ΣF . The required differential equations are finally found to be: (cf. [45])

rc′′ ++11r′ c12s′ +d11r+d12s=0, (AΣ F ) sc′′ ++21r′ c22s′ +d21r+d22s=0 2 The Method of WILCZYNSKI 39 where d1 d1 c11 = log , cp12 =12 , c21 =p21, c22 = log , dx δ2 dx δ1

dq=−2λ′ +λd logδ, dp=+′ λ p−pd logδ , (2.125) 11 11 dx 2 12 12 12 12 dx 2

dp=+′ µ p−pd logδ , dq=−2lµ′ +µδd og 21 21 21 21 dx 1 22 22 dx 1 and δ1, δ 2 from (2.87).

Similarly we can find the differential equations of form (A) for the surface Σ L by putting ry=λ +y′, sz=λ +z′, (2.126)

qq where λ ==12 21 (cf. (2.26)). p12 p21 By differentiating and eliminating yz,,y′,z′, we find the required differential equations of

ΣL as

rm′′ ++11r′ m12s′ +n11r+n12s=0, (AΣ L ) sm′′ ++21r′ m22s′ +n21r+n22s=0 where

mp=−d,log q, mp12 = 12 11 11 dx 1 d mp22 =−22 log q2 , mp21 = 21, dx

⎛⎞p − λ ′ ′ ′ 11 ⎛⎞p12 nq11 =−11 λ +q1 ⎜⎟, nq12 =+12 q1 ⎜⎟, q1 q ⎝⎠ ⎝1 ⎠ (2.127) ′′ ⎛⎞p22 − λ ⎛p21 ⎞ nq=−λ′ +q , nq21 =+21 q2 , 22 22 2 ⎜⎟ ⎜⎟q ⎝⎠q2 ⎝⎠2

22 q1 = λ′′−+qp11 λλ11 −, q2 =λ−q22 +λp22 −λ.

MACLEAN studied both surfaces and deduced:

(1) The ruled surface ΣF will be a developable if and only if either ∆1 = 0 or ∆=2 0, i.e., if,

and only if, either cy or cz is a plane curve.

(2) The ruled surface ΣL will be a developable if and only if either q1 or q2 is zero.

(3) The tangents to the flecnode curve at Py , Pz of g, and the tangents to the LIE-curves at

Py , Pz of g, intersect not only gF and gL respectively, but also both pairs intersect both

gF and gL . 32 3 (4) If θθ49+16θ10=0, the ruled surface Σ has the following characteristic property: The planes, which osculate the LIE-curves at the two points of its intersection with any generator g, intersect in a line which is tangent to the osculating quadric along g. 2 The Method of WILCZYNSKI 40

Let us consider an arbitrary point P of the generator g of Σ, which is given by

α y+ β z, where the ratio α : β is arbitrary. The plane of P and gF determines a tangent t to Σ at P. Thus, we obtain a one-parameter family of curves on Σ such that the tangents of them along g intersect gF . The plane determined by P and gF is given by the equation

xyαβ++zq21 yp21 y′q12 z+p12 z′=0. (2.128)

As the independent variable changes, P describes a curve on Σ and d α y+ β z which is a dx () point on the tangent t to this curve at P, will lie on the plane (2.128), provided

d αβyz++αβyzqy+py′′qz+pz=0. (2.129) dx () 21 21 12 12

On expanding, simplifying, and dividing out by Dy= ′′zyz≠ 0, this condition reduces to

α′ β′ 1 ⎛⎞p21′ p12′ −=⎜⎟− . (2.130) αβ2 ⎝⎠p21 p12

The integration of this equation gives

p α = c 21 (2.131) β p12 where c is an arbitrary constant.

It may be noted that for c =±1, the LIE-points, Py , Pz of g are obtained. So we get the following result,

Theorem 2.12.1: The one-parameter family of curves on Σ, characterized by the property that the tangents to them along the generator g intersect gF , are those determined by the condition of forming a constant cross-ratio with the LIE-curves of Σ. This one-parameter family has the property that the tangents to the curves taken in points of g , intersect not only

gF but also gL .

Similarly, a point P′ of gF and g determine a tangent t′ to ΣF at P′. Thus, in a reciprocal manner, we obtain another one-parameter family of curves on ΣF such that the tangents to them along gF intersect the generator g of Σ. The differential equation representing this family is found to be of RICCATI form, and from the characteristic property of any four solutions of a RICCATI equation, we deduce that,

Theorem 2.12.2: Any four points of the generator gF , which are such that the tangents to the curves they describe (at the points where these curves meet gF ) intersect the generator g, have the property that their anharmonic ratio is constant.

It may be noticed that the tangents along g to the one-parameter family of curves on Σ constitute the rulings of the transversal set of lines of the quadric whose concordant set of lines is formed by the lines g, gF and gL . This quadric is called by MACLEAN [45] the flecnode-LIE Quadric of the given integrating ruled surface Σ. To find the equation 2 The Method of WILCZYNSKI 41 representing this quadric, let us choose the fundamental tetrahedron of reference determined by PPyz Pρ Pσ , where ρ and σ are determined from (2.21). Using this system, the required equation of the Flecnode-LIE Quadric is found to be

⎧2⎛⎞qq12 21 2⎫ xx14−−x2x3 ⎨p12x3−2⎜⎟− x3x4−p214x ⎬=0. (2.132) ⎩⎭⎝⎠pp12 21

This quadric is characterized by the following property:

The null-points of the planes, which osculate the LIE-curves at the two points of their intersection with any generator, as determined by the osculating linear complex, lie on the flecnode-LIE quadric of Σ.

CARPENTER [16] generalized MACLEAN´s system of one-parameter curves on Σ, which we have just mentioned. Let us use the system (F) and choose the previous tetrahedron of reference. The curves generated by the points η = αβyz+ , ζ = αy− βz, where α, β are arbitrary functions of the independent variable x, cut each generator g of Σ in points harmonically separated by the flecnode points of that generator. If the tangents to the four curves cy , cz , cη , cζ at points which they cut a generator g of Σ are to lie on a quadric, then the matrix of the coordinates of the general points on these tangents must be of rank two. Using this condition, the equation representing this quadric is found to be

22 xx14−−x2x3 p12x3+p214x −20(αα′′−β β) x3x4=. (2.133)

If now α = cx11θ (), β = c22θ (x), where θ1, θ2 are arbitrary functions and c1, c2 constants, then

α′′αβ−=βθ11′θ−θ2′θ2 (2.134) and it follows that,

Theorem 2.12.3: There exists on Σ, for each choice of the ratio θ1θ2 a one-parameter family of curves forming an involution such that the double elements are the two branches of the flecnode curve, and such that the tangents to the curves of each family at points of intersection with a generator of Σ constitute one regulus of a quadric which is itself uniquely determined by the choice of θ12θ .

12 12 If θ1= p21 and θ2= p12, quadric (2.133) becomes the flecnode-LIE quadric defined by

MACLEAN. The curves cη , cζ are now generated by the points

12 12 12 12 η =+cp121y c2p12z, ζ =−cp121y c2p12z. (2.135)

When c1 =1, c2 =1, cη and cζ are the LIE-curves of Σ. On the other hand, when c1 =1, c2 = i, the points determined by η, ζ are the involute points of g, and cη , cζ are the involute curves of Σ. Moreover, it can be proved that,

Theorem 2.12.4: There exists on Σ a one-parameter involutory family of curves whose double curves are the two branches of the flecnode curve and which are such that any four curves of the family cut all generators of Σ in constant cross ratio. The tangents to these 2 The Method of WILCZYNSKI 42 curves along any generator constitute one regulus of a quadric whose equation may be written in the form

22 xx14−−x2x3 p12x3+p214x =0. (2.136) and on this quadric lie the asymptotic tangents of Σ at the LIE-points of g.

CARPENTER [16] studied a particular type of ruled surface which is characterized as follows,

Theorem 2.12.5: There exists a one-parameter family of ruled surfaces defined by a system of differential equations

ye′′ −+−−xxz′ qy+1 ez=0, 11 2 (A´) xx1 ze′′ ++y′ 2 ey+q22 z=0, where

qc11 =≠1 c2 =q22 , 2qq11 ++222 1=0 (2.137) for each of which the derivative surface as well as each sheet of the flecnode surface is projectively equivalent to the given surface, and on which there lies a one-parameter involution family of curves whose double elements are the two branches of the flecnode curve, all curves of the family being projectively equivalent anharmonic curves.

2.13 The complex-permute quadric – The flecnode-reciprocal quadric

In another work of CARPENTER [15] about the point-line correspondences associated with a general ruled surface, he found some other properties of the LIE-curve. To explain this properties, let us denote the pair of flecnode tangents to g by f1 and f2 , and let Φ1 and Φ2 be the corresponding two sheets of the flecnode surface. Further, let QΣ , QΦ1 and QΦ2 be the quadrics which osculate Σ along g, Φ1 along f1 and Φ2 along f2 respectively. The equations representing these quadrics referred to the fundamental tetrahedron PPyz Pρ Pσ are found to be

xx14−=x2x3 0 (QΣ)

2 px12 ()1x4 −−x2 x3 20qx4 = (QΦ1 )

2 px21 ()1x4 −−x2 x3 20qx3 = (QΦ2 ) where qq=−11 q22.

It may be verified that QΣ and QΦ1 have for their complete intersection the lines g and

f1 each counted twice, that QΣ and QΦ2 have similarly g and f2 for their complete intersection, and that QΦ1 and QΦ2 have for their complete intersection g counted twice together with two lines, one through each of the LIE-points of g.

Now let us consider any point P in space and let π1 and π 2 are the polar planes of P with respect to QΦ1 and QΦ2 respectively. Now, the plane π1 has a pole with respect to the quadric QΦ2 , say a point P12 , and similarly the plane π 2 has a pole with respect to QΦ1 say a 2 The Method of WILCZYNSKI 43

point P21. There is thus made to correspond to each point of space a line PP12 ∨21 ≡l1. On the other hand the two planes π1 and π 2 intersect in a line l21≡ π ∩π 2, so that there is thus made to correspond to P a second line l2. These lines are in general distinct. If we denote these two point-line correspondences by τ1 and τ 2 respectively. It is found that both lines l1 and l2 intersect g, and the line l1 goes also through P. Moreover, both l1 and l2 belong to a special linear congruence, which is the common congruence of two linear complexes. The first of these is the linear complex K which osculates Σ along g, and the second is composed of all the lines of space which intersect g. The relations of the correspondences τ1 and τ 2 to this linear congruence are established in the following theorem, (cf. [15])

Theorem 2.13.1: Let π be any plane on g, M its point of tangency with QΣ and M ′ the point which corresponds to it in the null-system of K. There exists a unique plane π ′ also on

g for which M ′ is its point of tangency with QΣ and M the point which corresponds to it in the null-system of K . To all the points of π there correspond by τ1 the lines of π through

M ′, and by τ 2 the lines of π ′ through M. Reciprocally, to all the points of π ′ there correspond by τ1 the lines of π ′ through M , and by τ 2 the lines of π through M ′.

Let us choose a general point on a line of the concordant set of lines of QΣ . To this point corresponds by τ1 a line whose locus is found to be a quadric ruled surface, and whose equation is

px12 1x3 −=p21x2 x4 0. (2.138)

This quadric has in common with QΣ the line g , the two rulings of QΣ (different from g ) through the LIE-points of g, and that ruling of QΣ to which the quadric in question corresponds. Similarly, to the same point corresponds by τ 2 a line whose locus is another quadric and whose equation is found to be

px12 1x3 −−p21x2 x4 4qx3 x4 =0. (2.139)

From these two equations, we notice that these two quadrics must be tangent to each other along g, and we thus have,

Theorem 2.13.2: To the lines of the concordant set of the osculating quadric QΣ there correspond by τ1 (τ 2 ) the quadrics of the family determined by the pair of planes tangent to

QΣ at the LIE-points of g and that quadric whose tangent lines along g correspond by

τ1 (τ 2 ) to the points of the planes on g.

For a point on the line gF of intersection of the two planes osculating the two branches of the flecnode curve, corresponds by τ1 and τ 2 , as before, two lines. The locus of the first of these is found to be a quadric, whose equation

2222 p12 p21 ()px12 1x3 − p21x2 x4 −−20()p12q21x3 p21qx12 4 =. (2.140)

It is called the complex-permute quadric, and is determined by g, the line gF , and the polar reciprocal of this line gF with respect to the osculating linear complex K. It is easy to show that of the two reguli of this quadric, the one whose lines cut g belongs to K, while the 2 The Method of WILCZYNSKI 44 other has the property that the polar reciprocal with respect to K, of any of its lines, is another of its lines. For the second line, its locus is found to be another quadric with the equation

22 px12 1x3 −−p21x2 x4 2qx12 3 −4qx3 x4 +2q21x4 =0. (2.141)

It is called the flecnode-reciprocal quadric. It is the locus of a general point on the line joining two general points on the two polar reciprocals of gF with respect to QΦ1 and QΦ2 respectively. Thus we can say that,

Theorem 2.13.3: To the line of intersection gF of the planes osculating the two branches of the flecnode curve at their points of intersection with g there corresponds by τ1 the complex- permute quadric and by τ 2 the flecnode-reciprocal quadric.

In every instance thus far to the points of a straight line have corresponded the lines of a quadric, irrespective of which of the two correspondences τ1 and τ 2 was employed. Moreover, in every case these quadrics have had g as one of their lines. Thus we can say

Theorem 2.13.4: To each straight line of space the parameter on which is independent of the parameter on Σ , there corresponds by either τ1 or τ 2 , so long as the Σ -parameter x is held constant x = x0 , a quadric, and all such quadrics intersect in that ruling of Σ which is determined by the value x = x0.

Let us now think of a space curve cx as being put into correspondence with the ruled surface Σ by choosing arbitrarily the variable x for the parameter along the curve. The points of the curve will thus correspond to the lines of Σ. To each point Px of cx there corresponds by τ1 a line which passes through Px and cuts that ruling of Σ which corresponds to Px parametrically, in a point Pτ1 , and similarly corresponds to Px by τ 2 a line which cuts that ruling of Σ which corresponds to Px parametrically, in a point Pτ 2 . As Px moves along cx ,

Pτ1 traces a curve cτ1 on Σ, which we will call it the first image of cx on Σ, and in the same time Pτ 2 traces another curve cτ 2 on Σ, the second image of cx on Σ. For these two curves cτ1 and cτ 2 , we have the following two theorems,

Theorem 2.13.5: All those space curves cx to which correspond by τ1 developables whose cuspidal edges are the first images of cx on Σ, lie upon the LIE-developables of Σ (i.e. the developables whose cuspidal edges are the two branches of the LIE-curve of Σ ) and these developables are self-corresponding under τ1 . Any curve on either developable satisfies this condition, its first image on Σ being one or other branch of the LIE-curve of Σ.

Theorem 2.13.6: All those space curves cx to which correspond by τ 2 developables whose cuspidal edges are the second images of cx on Σ, lie upon two ruled surfaces tangent to Σ along the two branches of the LIE-curve. Any curve upon either of these ruled surfaces satisfies this condition. Its second image on Σ is one (or other) branch of the LIE-curve and the ruled surface which corresponds to it by τ 2 is the developable of that branch.

Moreover, the two branches of the LIE-curve are the only two curves on Σ that can serve as coincident first and second images of a curve cx .

2 The Method of WILCZYNSKI 45

2.14 Ruled surfaces whose flecnode curves belong to linear complexes

J. W. HEDLEY [23] deduced the differential equations of the two branches of the flecnode curve cy and cz from the system (F) and put them in the following simplified forms:

∆∆′′⎛⎞pp′′2 ∆ (4) 11(3) 12 3 12 1′′ yy−+⎜⎟−p12 p21 +q11 −33q22 −22+4+ y ∆∆1 ⎝⎠pp12 12 2 p12 1

53 ∆1′ +− p12′′pp21 − 12 p21 +2q11′+()p12 p21 −q11 y′ {}22 ∆ 1 2 (2.142) ⎧ 2235 pq12′ 11 +−⎨ pq12 22 − p12′′− p12 p2′1 −3 2 +qq11′′ − 3 11q22 ⎩ 44 p12 ⎛⎞4qp⎛⎞′2q∆′ ⎫ ++1 11 ∆+1 pp′′−q+3 12 11 1 y=0 ⎜⎟2 11⎜⎟2222111p ∆ ⎬ ⎝⎠p12 ⎝⎠12 1 ⎭ and a similar equation (2.143) for z. Thus we can compute the invariants of weight three for both (2.142) and (2.143), and let us denote them by θ3y and θ3z where pq′ 31′′1′3′12 22 θ3y =− pp1221− pp1221+ q11+ q22+3 8844p12 3 31pp12′′12 +−33∆11+2∆′ 4 pp12 12 4 p12 2 (2.144) ⎛⎞1199p12′ ∆1′ +−⎜⎟pp12 21 q11 −q22 −2 ⎝⎠88816p12 ∆1 23 93pp12′′∆∆11′′12 ′15∆1′9∆11′∆∆′′ 1 1′′′ +−23−+2 − , 16 pp12 ∆∆118 12 ∆1 32 16 ∆1 8 ∆1

and similar expression for θ3z .

∗) Hence if θ3y = 0, then the flecnode curve cy is a complex curve , and if θ3z = 0, the same is true for cz , but the linear complexes to cy , cz may be distinct. Moreover, with a suitable transformation, we can make the invariant θ4 be reduced to unity. This causes no loss of generality but amounts to a convenient selection of the parametric representation of the surface. This transformation causes (2.18) and (2.19) to become:

θθ4 ==1, 9 ±2 ()p21 pp12′′−12 p21 , θ10 =p12 p21, (2.145) θ4.1 =−16()pp21 12 2q11 −2q22 .

θθ ± 1199dx ∓ dx 44∫∫θθ10 10 pe12 ==θθ10 ; p21 10 e

p12 p12 p12′′=±()θθ9 +2;10 p2′1 =(∓θθ9 +210′) (2.146) 4θ10 4θ10 11 qq11 =−64 ()16θθ10 4.1 ∓ 8 ; 22 =−64 ()16θθ10 4.1 ±8 .

∗) i.e. its tangent belong to a linear complex 2 The Method of WILCZYNSKI 46

Depending on these calculations, HEDLEY studied in details the system (F), and the different forms which it takes under different conditions. He deduced some theorems and propositions, which we will summarize as follows (proof see [23], [24]):

1 (1) If (F) has constant coefficients, all different from zero, and if qq22 =11 ±4 , then the differential equations of the flecnode curves become yP(4) ++y′′ Qy=0, 11 (4) zP++22z′′ Qz=0,

where P1, P2 , Q1 and Q2 are constants; Q1 and Q2 are different from zero.

1 (2) If p12 , p21, q11, q22 , occurring in (F), are different from zero, and if qq22 =11 ±4 , and if the differential equations derived from (F) become yP(4) ++y′′ Qy=0, 11 (4) zP++22z′′ Qz=0, then (F) has constant coefficients.

(3) If θ4 ≡1, θ9 ≡ 0, θ10 = c1 (a constant, not zero); θ4.1 = c2 (a constant ); then q11, q22 , p12 1 and p21 are all constant; p12 ≠0, p21 ≠0 and qq22 = 11 ± 4 ; and conversely.

(4) If θ4 ≠ 0, θ9 ≡ 0, and if in (F) p12 ≠ 0, p21 ≠ 0 , then (F) becomes

1 yp′′ ++z′ q11 y+2 p′z=0, (F1) 1 zp′′ ++y′ 2 p′y+q22 z=0

where p ≠ 0 and q22 ≠ q11; and conversely. We notice that, since θ9 = 0 is the condition that a ruled surface belongs to a linear complex, then one can tell that a ruled surface belongs to a linear complex by either calculating the invariants or observing the coefficients of the system (F).

1 (5) If in (F), p12 , p21, q11, q22 are constants all different from zero and if qq22 =11 ±4 , then the ruled surface belongs to a linear complex, the flecnode curves belong to linear complexes; they are anharmonic curves and lie on quadric surfaces.

(6) If θ3y = 0, θ3z = 0, θ4 =1, θ9 = 0, θ4.1 = c1, ∆11= a , ∆2= a2, where a1, a2 , and c1 are

constants, a1 ≠ 0, a2 ≠ 0, and if p12 , p21, q11, q22 are all different from zero, then the flecnode curves belong to linear complexes and lie on quadric surfaces and the ruled surface belongs to a linear complex.

(7) If θ3y = 0, θ3z = 0, θ4 =1, ∆=1 a, ∆2 = a and p21 = c, where a and c are constants,

different from zero, then either p12 is a constant, different from zero, or qp−−21dp=−1 cqp. 22 ∫ 12 12 2 22 12

1 (8) If p12 is not a constant, ∆=11a , qa22 = 2 , qq11 = 22 ± 4 (where a1 and a2 are constants,

different from zero) in the homogenous linear differential equations (2.114) and if cy belongs to a linear complex, then 1 2 ⎛⎞a1 1/2 pa12 =+⎜⎟sin{}2 2 ()xa3 , ⎝⎠3a2 and conversely. 2 The Method of WILCZYNSKI 47

(9) If p21 ≠ 0, p12 ≠ a constant, θ4 ≡1, ∆1= a1, where a1 is a constant, different from zero

and cz is a conic, then if cy belongs to a linear complex, it follows that dp 12 =+x a , ∫ 1 3 41232 ()32ap11∓ 2+ a2p12 and conversely. (10) If the flecnode curves belong to linear complexes and if the ruled surface belongs to a

linear complex and if p12 ≠ 0, p21 ≠ 0, θ4 =1, ∆1= a1 and ∆2= a2, where a1 and a2 are constants, different from zero, then the four planes osculating the flecnode and LIE-curves always form a non-degenerate tetrahedron.

pp21 12 Moreover, according to HEDLEY if θ4 ≠ 0, θ10 ≠ 0, ∆1 ≠ 0, ∆2 ≠ 0 and ∆−12∆ =0, pp12 21 then θ9 ≠ 0. Conversely, if θ4 ≠ 0, θ10 ≠ 0, ∆1 ≠ 0, ∆2 ≠ 0 and θ9 = 0, then

pp21 12 ∆−1∆2≠0. So, from that and from (2.104), we see that, pp12 21

Theorem 2.14.1: If the planes osculating the flecnode curve and the LIE-curve of a ruled surface at the four points, in which they intersect any ruling, form a harmonic pencil, then the ruled surface does not belong to a linear complex, and if the ruled surface belongs to a linear complex then these four planes do not form a harmonic pencil.

2.15 The flecnode sequel

In his dissertation J. W. LASLEY [42] called one sheet of the flecnode surface the first flecnode transform of Σ and denoted it by Φ(1) . The second sheet is the minus first flecnode transform of Σ, denoted it by Φ(1− ), for the reason that the first flecnode transform of Φ(1− ) is Σ itself. The first transform of Φ(1) is the second transform of Σ and is denoted by Φ(2). Continuing in this way one obtains a sequel of surfaces called the flecnode sequel. LASLEY studied the questions naturally arising as to the cases in which this sequel either terminates or returns again into itself and he stated the following results: (1) A necessary and sufficient condition that the flecnode sequel terminates with its first

transform is θ10 = 0, i.e., the given ruled surface has a straight-line directrix. (2) The necessary and sufficient condition that the flecnode sequel terminates with its second transform is that

33 222322 θ30 ≡−32θ15 θθ10 20 −2θ4 θ9θ15 +2θ4 θθ10 14 −θ4θ9 +θθ10 4.1 =0 (2.147)

where θ30 as an invariant is deduced by the relations explained by WILCZYNSKI ([78] p.119). Moreover, the equations of both branches of the flecnode curve on Σ may be determined without any integration if the flecnode sequel terminates with its second transform. (3) A necessary and sufficient condition for the generators of the second and minus second transforms to intersect is that

22 θθ18 =−9 θ4θ10 =0. (2.148) 2 The Method of WILCZYNSKI 48

(4) There exists a single infinity of classes of mutually projective ruled surfaces the generators of whose second and minus second flecnode transforms intersect, which have the additional property that the flecnode curve consists of two distinct branches, one of which is a conic. (5) There are no ruled surfaces whose second and minus second flecnode generators intersect, which have the property that their flecnode curve consists of two distinct plane branches, one of which is a conic. (6) In case the generators of the second and minus second flecnode transforms intersect, the minus first flecnode surface belongs to a special linear complex if, and only if, the first flecnode surface belongs to a special linear complex. In this case the second and minus second flecnode surfaces are straight lines having a point in common.

LASLEY studied also the case of periodicity of the flecnode sequel and he concluded that

Theorem 2.15.1: The flecnode sequel is periodic, of period two (i.e. the first flecnode transform of Φ(1) coincides with Σ ), when, and only when, the flecnode curve meets every generator in two coincident points, or is indeterminate. Moreover, the flecnode sequel can not be periodic, of period three, or of period four.

2.16 The metric properties of the flecnode curves

S. W. REAVES [54] studied in his dissertation the metric properties of the flecnodes using the method of WILCZYNSKI. He studied ruled surfaces in Euclidean space whose branches of the flecnode curve are the line of striction and the line at infinity on the surface. Actually, G. TZITZEICA [67] has pulled attention to ruled surfaces with (non-degenerate) director cone∗) which have the two properties that the line of striction and the line at infinity on the surface are the two branches of the flecnode curve, and which have the additional property that the director cone is a cone of revolution. These three properties, he says, characterize the ordinary ruled helicoid, generated by a straight line subject to a helicoidal movement around an axis to which it is not perpendicular.

Let the Cartesian equations of the ruled surface Σ be written in the form

x =+x0 tl, yy=+0 tm, zz= 0 + tn (2.149) where x00,,yz0 denote the coordinates of a point O on the line of striction of Σ, expressed in terms of the arc length s measured along the line of striction from one of its points; lm, ,n are functions of s and denote the direction cosines of the generator g through O; and t is the distance from O to an arbitrary point on g. We shall assume that the positive directions on the coordinate axes form a right-handed system.

Let g be a tangent to the surface Σ drawn through O and perpendicular to g. The line g generates a surface Σ, known as the conjugate surface to Σ. The two surfaces Σ, Σ have the same curve for line of striction and at each point of this curve they have the same surface normal n .

∗) The director-cone of a ruled surface is the cone formed by drawing through a point lines parallel to the generators. 2 The Method of WILCZYNSKI 49

To express the coefficients pik , qik of the system (A), let us choose a movable system of axes which for each value of the independent variable s consists of the trirctangular system O − ggn, where O is the point on the line of striction given by that value of s, and gg, , n are respectively the corresponding generator of Σ, the generator of Σ, and the surface normal through O . If we introduce as homogenous coordinates x1 = x, x2 = y, x3 = z, x4 =1, and in (2.149) replace t by a function of s, say ts(), we may regard

x10=+xtl, x20=+ytm, x3 = zt+ n, x4 =1 (2.150) as the parametric equations, in homogenous form, of an arbitrary curve on the surface Σ . Let us choose two special curves on Σ as directrix curves cy and cz by giving t first the value zero and then the value infinity. The equations of these curves are

cy : y1= x0, y2= y0, y3= z0, y4 =1, which is the line of striction, and

cz : z1 = l, z2 = m, z3 = n, z4 = 0, which is the infinitely distant line of Σ. Substituting these values in (A), we have eight equations from which to find pik , qik . On solving these we find the following results

Dp11 =−x0′′ l l′ , Dp12 = x0′ x0′′ l ,

Dp =−l l′′l ′, Dp22 =−x0′ l l′′ , 21 (2.151) Dq12 =−x0′′x0′l′, Dq22 = x0′ l′′l ′,

qq11 ==21 0 Dx= 0′ ll′ .

Finally, we find the required expressions as c′ c c p11 =− , p12 =(kk−), p21 =− , c k kk k′c′ cc′− cc′ 1 ⎛cc22⎞ p22 =−, q12 = , q22 =⎜+⎟, (2.152) kc c k⎝⎠k k

2 qq==0, D =−c , 11 21 k where

2 mn 2 22 c==∑∑1,−()lx0′ cl= cosθ = ∑ x0′, cc+ =1, (2.153) yz00′′ and k, k are the parameters of distribution along g, g, and θ is the angle between the positive directions on the generator g and the tangent t to the line of striction.

2 The Method of WILCZYNSKI 50

Using these formulae, REAVES gave a proof of TZITZEICA’s statement and gave some other characterizations of the flecnode curves and the surfaces Σ, Σ, which we will try to summerize as follows: (1) A necessary and sufficient condition for one branch of the flecnode curve on a ruled surface Σ to be at infinity is that kk 2tan2θ shall have the same value for every generator g of Σ. (2) A necessary and sufficient condition that the flecnode curve on Σ consist of the line of striction and the infinitely distant curve of the surface is that kk 2tan2θ and 2k+ k shall both be constants. (3) The ruled surfaces which have for flecnode curve the line of striction and the infinitely distant curve of the surface and whose director cone is of revolution, are characterized by the properties that k, k and θ are constants.

If we denote by Φ∞ the corresponding sheet of the flecnode surface for the one branch of the flecnode curve at infinity, and by P and P∞ respectively, the central (striction) points on the generators g and g∞ of the surfaces Σ and Φ∞ , then the two surfaces Σ and Φ∞ , each being the flecnode surface of the other with the curve of contact of corresponding generators at infinity on each, bear towards each other an entirely reciprocal relation. Thus, it can be proved that,

Theorem 2.16.1: When Σ and Φ∞ have the common branch of their flecnode curves at infinity, the point mid-way between the central points P and P∞ of corresponding generators

g and g∞ is also the centre C of the quadric QΣ which osculates Σ along g.

On account of the reciprocal relation existing between Σ and Φ∞ we may infer that C is also the centre of the quadric QΦ∞ which osculates Φ∞ along g∞ . For the locus of the centre C of the osculating quadric, we have the following different cases: (1) C lies on the edge of regression on the asymptotic developable if and only if one branch of the flecnode curve is at infinity. (2) C is a fixed point if and only if both branches of the flecnode curve are at infinity.

Moreover we have,

Theorem 2.16.2: If the centre of the osculating quadric of a ruled surface is constrained to move on a straight line, then it must be a fixed point. Finally, when both flecnodes are at infinity, the lines of striction of the two surfaces Σ and Φ∞ are symmetric to each other with respect to a fixed point which coincides with the centre of the common osculating quadric of

Σ and Φ∞ .

3 The Method of CECH

To understand the analytic basis of this method, we have to notice that CECH applied the method of FUBINI, which we summarized in the appendix, to study the projective differential geometry of ruled surfaces in the Euclidean space. In CECH’s theory of ruled surfaces [21], [22], the coefficients of the system (A) of WILCZYNSKI are decomposed into elements, from which the invariants of a ruled surface Σ is to be obvious.

3.1 The fundamental equations

Assuming a ruled surface Σ whose directrices are the two curves cy and cz. Designating by yz() the four homogenous coordinates of a moving point on ccyz(), where these quantities are functions of a parameter v, then Σ can be described by the equation

x(,uxv)==+yuz. (3.1)

The curves v = const. are the generators of Σ; all other curves on Σ are obtained by putting u = ϕ(v ). The surface Σ is a developable if the point

dx =+yz′′ϕ +ϕ′z dv is situated on the generator (,yz), i.e. the points yz,,y′,z′ are then linear dependant. The necessary and sufficient condition for a developable surface is therefore α = 0, where we have put α = (yz,,y′,z′) (3.2)

The plane tangent to Σ at a point x is

()x,,xxu v =+η0uζ0, (3.3) where

η0 = (yz,,y′), ζ 0 = (yzz,,′). (3.3a)

Actually, the geometric position of this plan does not depend on u if α = 0 and just in that case. Hence we have, the tangent plane of a developable surface is fixed along each generator. We are going to exclude the developable surfaces of our study, i.e. always α ≠ 0.

The planes η0, ζ 0 contain the line (,yz), hence we have evidently a relation of the form (,ηζ00)= λ(x,y). Precisely, we have

(,η00ζα)= (x,y) (3.4) that is to say λ = α. So that the tangent plane of Σ at a point x is

ξ =+λη(0uζ 0). (3.5)

One can choose λ such that at every point x, the asymptotic directions would be given by ()xxd± (ξ dξ ). (3.6) 3 The Method of CECH 52

A system of the asymptotics is formed by the generators v = const., so we have

( xx,u) =±(ξ,ξu) (3.7) that is to say

2 ()yz,,=±λ (η00ζ ). (3.8)

−1 Therefore, from (3.4), we conclude that λ=±α2 . To fix the idea, let us choose the sign in such a way that ξ =+ηζu (3.9) with

η = 1 (yz,,y′), ζ = 1 (yzz,,′). (3.9a) α α

Equation (3.4) can be then written as ()ηζ,= ω(yz,), ω = ±1 (3.10) the sign of ω is defined by ω = sgn(yzy′′z ). (3.11)

Depending on the previous assumptions, CECH applied FUBINI’s method for surfaces in ordinary space to develop his theory of ruled surfaces. According to CECH’s theory, the system (A) of WILCZYNSKI takes the form:

yb′′ =−(2θθ′ )y′ +2az′ +(−b′ +b′ +ac−b2 −B−j)y+(a′ −aθ′ +A)z, (Ac) zc′′ =−2(y′ + θθ′ +2b)z′ +(−c′ +c′ −C)y+(b′ −bθ′ +ac−b2 +B− j)z. where the coefficients a, b, c and θ are defined as:

a ==ω ()yzy′y′′ ,,b ωω()yzy′z′′ −11θ′ =()yzy′y′′ +θ′ 22aa22222a2 12 12 12 (3.12) ω ′′′ cy==2 ()zyz,lθ oga12 . 2a12

2 with ()yzy′′z = ωa12 , ω = sgn (yzy′′z ).

3 The Method of CECH 53

The quantities A, B, C and j are deduced by CECH and described in the form

A =−ω ⎡⎤yzy′y′′′ +33yy′z′y′′ +θ′ yzy′y′′ , 2 ⎣⎦()()() 2a12

ω ′ ′′′ ′ ′′′ ′ ′ ′′ By=−2 [3()zyz−(yzzy)+(yyzz) 4a12 ++33(zy′z′y′′ )θθ′ ()yzy′z′′ +3′ ()yzz′y′′ ], (3.13) C =−ω ⎡⎤yzz′′z ′′ +33zy′′′z z ′ +θ′ yzz′′z ′ , 2 ⎣⎦()()() 2a12

ω ′ ′′′ ′ ′′′ ′ ′ ′′ ′ ′ ′′ j =−2 [()yzz y ()yzy z −(yy z z )+()zy z y 4a12 ′′ ′′ ′ ′ ′′ ′ ′ ′′ ′′ ′′ ′ ′ ′′ ′′ 1 ′′ ′ 2 ++23(yzy z )θθ()yzz y −()yzy z −ω(yzy z )(y z y z )]+2 ()θ−θ.

where a12 is from FUBINI’s theory of surfaces (cf. [21] p. 52).

3.2 The projective linear element of a ruled surface

CECH defined the projective linear element of a general non-ruled surface by F3:F2, where

1 22 F2 =−Sdxdξ, Fx3 =−2 Sd()dξξd dx (3.14) and the symbol S means summation on the coordinates of the point or the plane under consideration, i.e.,

S,x22=+xy2+z2+t2 S,ξ 22=+ξη2+ζ 2+τ 2 S,ξ xx= ξη+++yζzτt… etc. assuming that the homogenous coordinates of a point are (,x yz,,t), and of a plane are (,ξ ηζ, ,τ).

The linear projective element is intrinsic (independent of the choice of the curvilinear coordinates u, v ) and does not change by the change of the homogenous coordinates x, ξ.

The form F3 (similarly F2 ) is almost intrinsic (only its sign can be changed if one changes u, v ). If one multiplies x, ξ by a factor σ , then the form F3 (similarly F2 ) is multiplied by 2 σ . Moreover, the linear projective element vanishes on the DARBOUX∗) lines, such that

F3 = 0 is the equation of these lines in an arbitrary curvilinear coordinates.

For a ruled surface, the projective linear element takes the form F 2 3 =+AB2 u+Cu2 dv (3.15) () 2 F2 2dua++()2bu+cudv

∗) The curves of DARBOUX on a surface Ψ in ordinary space are the three one-parameter families of curves such that at each point A of Ψ the three tangents of these curves are in the directions in which may coincide with the three triple-point tangents of the curve of intersection of Ψ with a quadric having a second-order contact with Ψ at A. (cf. LANE [40] p.76) 3 The Method of CECH 54

where the equation F2 = 0 defines the asymptotic lines on the surface. Thus, a point y+ uz describes an asymptotic curve cyu+ z on the surface Σ if, and only if, u, as a parameter, is the solution of the RICCATI differential equation: uf′ +()u=0, (3.16) where

f ()ua=+2bu+cu2. 3.17)

We notice that there are generally two points on each generator at which F3 = 0. These are the flecnode points of the generator considered. The point y+ uz describes then the flecnode curve cyu+ z on the surface if, and only if, u is the root of the quadratic “flecnode form” F()u , where

F()uA=+2Bu+Cu2. (3.18)

The geometric significance of the flecnode points can simply be explained as follows:

For a ruled surface Σ the osculating quadric QΣ coincides with LIE-quadric HΣ , which is consequently fixed along a generator g of the surface. Now, for an arbitrary point x on a general surface Ψ, the LIE-quadric HΨ has a contact of second order with Ψ in x, and the curve of intersection of Ψ and HΨ has therefore a triple point; the tangents of the intersection at x will be the DARBOUX-tangents F3 = 0. So, it results that the order of contact of Ψ with HΨ is higher than two if, and only if, the DARBOUX tangents at x are undetermined. Hence: the flecnode points on a ruled surface Σ are the points where Σ has a contact of the third order (at least) with the osculating quadric QΣ. The flecnode points can also be defined as the inflexion points on the asymptotic lines. Moreover, they can also be defined as the points on a generator g at which the asymptotic tangents are the directrices of the osculating linear congruence of Σ along g (cf. [22] p.130).

CECH considered the following theorem as the fundamental theorem of ruled surfaces,

Theorem 3.2.1: If the projective linear element is given, then it will be enough to know j, in order to determine the ruled surface completely (proof cf. [21] p.187).

Let us designate by x the homogenous coordinates of a moving point on an arbitrary non-ruled surface Ψ, and by ξ the homogenous coordinates of the tangent plane to Ψ. Assuming that Ψ is non developable, we can choose the factors of the coordinates such that the two asymptotic tangents of the surface being given by ()xxdd± (ξξ) (3.19) for an arbitrary infinitesimal displacement d. Let Ψ′ be another non developable surface, and y(η) the coordinates of its points (tangents planes), and ()yydd± (ηη) (3.20) 3 The Method of CECH 55 its asymptotic tangents. Supposing that the two surfaces touch each other along a non asymptotic curve c, then without losing the generality, we can suppose that y= x along c. As the two surfaces touch each other along c, then we will have the relation η = σξ at each point of c. It is easy to obtain a geometric significance of the factor σ. In fact, at each point x of c, the asymptotic tangents of Ψ are given by (3.19), where the differentiations are made in the direction of c. Similarly, for the asymptotic tangents of Ψ′ we have the expression (3.20), which is reduced to

( xxdd) ±σξ2 ( ξ) (3.21) in virtue of the relations yx= , η = σξ which are valid along the curve c. Thus,

Theorem 3.2.2: At each point of c, the couple of asymptotic tangents of Ψ, as well as those of Ψ′, belong to an involution J whose double elements are the tangents (dx x) at c and the conjugate tangent (ξξd). The two double elements and the two couples of asymptotic tangents form four couples of the involution J. The anharmonic ratio of these four couples is equal to σ 4.

Now let c be a DARBOUX-curve relative to Ψ; then along c we have the relation Sd(xd2ξξ−d d2x)=0. (3.22)

The differential equation of DARBOUX-curves of the surface Ψ′ is Sy(dd2ηη−d d2y)=0. (3.23)

Then along c we have

⎡⎤2222 2 Sd⎣⎦xd()σξ −=d()σξ dxxσS(ddξ −dξ dx) +dσS(2dxdξ −ξdx) (3.24) =−σξSd()xxd22dξd +3dσSdxdξ.

From (3.22) this becomes 3dσSdxdξ. (3.25)

But Sdxdξ ≠ 0, since the curve c can not be an asymptotic, thus:

Theorem 3.2.3: If the curve of contact c (non rectilinear) of two surfaces Ψ, Ψ′ is a DARBOUX-curve on Ψ, in order that to be c also a DARBOUX-curve on Ψ′, it is necessary and sufficient that the anharmonic ratio of the asymptotic tangents of the two surfaces to be constant along c. That happens in particular when the contact of Ψ, and Ψ′ is of second order along c.

If the surface Ψ is a ruled surface Σ, then at an arbitrary point of Σ the three tangents of DARBOUX coincide with the generator through this point; an exception are the flecnode points where these tangents are indeterminate. Thus, on a ruled surface, a non asymptotic curve is a curve of DARBOUX if, and only if, it is a flecnode curve. 3 The Method of CECH 56

As an application on a special case, let us consider one of the two surfaces, assume Ψ′, be a ruled surface Σ′, so we have the following theorem:

Theorem 3.2.4: If a curve c traced on a surface Ψ is a DARBOUX-curve on Ψ, then it is a flecnode curve of the ruled surface generated by the asymptotic tangents of Ψ along c, and reciprocally.

Moreover, let c be a given curve generated by a point x()s , and given for each value of s the tangent plane ξ (s) of c, but different from its osculating plane. There exist ∞1 ruled surfaces having the given curve c as a flecnode curve, and touch at each point x(s) of c the given plane ξ ()s . These surfaces can be determined by quadrature (cf. Theorem 2.6.1). In fact, it can be said that the required surface is generated by the line

()xxd±σξ2 (dξ), (3.26) on choosing σ such that

⎡22⎤ Sd⎣xxd()σξ −d d()σξ ⎦=0. (3.27)

From (3.24), then the required surface will be given by ()xxd+ t(ξξd) (3.28) with

Sx()dd22ξξ−d dx −∫ tc= e Sxddξ (3.29) and c is an arbitrary constant.

It is interesting to remark that the ruled surfaces under consideration can be obtained without quadrature, if c is a plane curve. In this case we can suppose that xx= ()12,,xx3,0. Let (ξξ12,,ξ3) be the coordinates of the tangent of c. The factors can be chosen such that the following relation is valid

3 22 ∑()ddxxiiξξ−=did i0. (3.30) i=1

Now, the plane ξ contains the line (ξξ12,,ξ3) of the plane x4 = 0, so that ξξ= ()12,,ξξ3,ϕ. From (3.29) and (3.30), the ruled surfaces which having the given plane curve c as a flecnode curve are generated by the line ()xxdd+ C(ξξ), (3.31) xx==()12, x, x3,0 , ξξ(1,ξ2,ξ3,ϕ) where C is a constant and ϕ is an arbitrary function. For example, if c is a conic section, one can suppose that 3 The Method of CECH 57

xs= ()1, , s2 , 0 , hence ξ =−()ss2 ,2,1,ϕ and one can easily find that the ruled surfaces having the given conic section as a flecnode curve are generated by the line (x, y) where

ya=+⎣⎡2,()ϕϕ′′ s()+a, 2sϕ, −2s⎦⎤ (3.32) with a is a constant and ϕ is an arbitrary function. (cf. § 2.7) Moreover, let us consider an arbitrary non-ruled surface Ψ and let us put

1 22 F2 =−Sdxdξ, Fx3 =−2 Sd()dξξd dx such that the projective linear element of Ψ is F3:F2. From (3.24) we have

⎡22⎤ Sd⎣xd()σξ −=dxd(σξ )⎦2σ F3−3dσ F2. (3.33)

A curve c of Ψ is then a flecnode curve on the ruled surface generated by the line

()xxd±σ 2 (ξξd) (3.34) if, and only if, F dσ = 23 . (3.35) σ 3 F2

Hence we have the following theorem,

Theorem 3.2.5: Let c be a non asymptotic curve on a surface Ψ; one can distribute the tangents of Ψ along c in a family F of single infinity of ruled surfaces in the manner that c be a flecnode curve on each surface of the family. At each point of c, let us consider the ordinary involution of the tangents of Ψ, whose double elements are the tangent of c and its conjugate. Let ϕ be the anharmonic ratio of four couples according to that involution: the two double elements, the couple of asymptotic tangents of Ψ, and those which contain the generator of a ruled surface Σ of the family F. The increment of logϕ along c is given by the integral F 8 ∫ 3 (3.36) 3 F2 and that whatever the way one chooses the surface Σ in the family F (cf. [22] p.139).

3.3 The pangeodesic

The pangeodesics on a surface (cf.[41]p.203)are those curves whose osculating planes, at each point A of the surface, are able to serve as the osculating planes of curves each of which has at the point A a stationary (four-point) osculating plane and also a stationary (four-plane) ray- 3 The Method of CECH 58 point. According to CECH [22], the pangeodesics on a surface are those curves which annulled the variation of the integral of the projective linear element F3:F2. They are characterized by the following geometric property: If A is a point of a pangeodesic curve on a surface Ψ, and if α is its osculating plane at A, and let A, A′, A′′, A′′′ are four infinitely close points (but not on the pangeodesic) on the intersection of α and the surface Ψ, then the tangent planes of Ψ at A, A′, A′′, A′′′ pass through the same point. The envelope of the osculating planes of the pangeodesics on Ψ at A is a cone, which is called the cone of SERGRE. This cone has its vertex at A and has the tangent plane of the surface at A for quintuple plane; the lines of contact being the two asymptotic tangents and the three tangents of DARBOUX at A. To find the conditions for the flecnode curve to be a pangeodesic on a ruled surface, we have

Theorem 3.3.1: A flecnode curve on a ruled surface Σ is pangeodesic if, and only if, this flecnode is a rectilinear directrix of Σ, or iff it is a double flecnode curve. In particular, If the ruled surface have a non rectilinear plane flecnode curve c, then c is a double flecnode curve on Σ if, and only if, the tangent planes of Σ along c pass through a fixed point.

3.4 The intrinsic fundamental forms

3.4.1 The quadratic intrinsic form f (t)

The quantities abc,,,A,B,C,j which previously defined, are evidently invariant for a unimodular substitution. It is necessary to find the expressions which being more independent of the particular choice of the parameter and invariant for a multiplicative substitution. Assuming a point x =+yuz where y and z are independent of the parameter v. The more general transformation which does not change this hypothesis is

λ2+ µ2u u = , v = V , x =+ρλ( 11µu) x (3.37) λµ11+ u

2 where λ11,,µλ2,µ2,V ,ρ are functions of v and (λµ12− λ2µ1)=1. We are going to call every expression, which is not varied by this transformation, as intrinsic. For an intrinsic expression under the behaviour of a multiplicative substitution, this transformation may be reduced to

u= u, v= v, x = ρx. (3.38)

The transformations (3.37) have two cases, the first case for which λ12µλ−2µ1=1, whereas for the second case λ12µλ−2µ1=−1. An expression which is not changed by the first case, and changes on contrary the sign if λ12µλ− 2µ1=−1 is called improperly intrinsic.

From (3.37)

λµ − λµ 12 21 v vv′ dduu=+2 (…)d, dd= V ()λµ11+ u hence 3 The Method of CECH 59

22 1 (2AB++uCu)dv 2 d(ua++2bu+cu2 )dv

⎡⎤222 2 Au()λµ11++2dB()λµ11+u()λ2+µ2u+C()λ2+µ2uv = V ′ ⎣⎦ 2()λµ12− λ2µ1 d(u + …)dv therefore

AB++2 uCu2 = 2 (3.39) V′ ⎡ 22⎤ =+Au()λµ11+2.B()λµ1+1u()λ22+µu+C()λ22+µu λµ12− λ2µ1⎣ ⎦

From FUBINI’s theory of surfaces

2 aV12 =′ρλ(1µ2 −λ2µ1 )a12 (3.40) hence

1 AB++2 uCu2 a 2 () 12 (3.41) =+11⎡⎤Auλµ22+2Bλµ+uλ+µu+Cλ+µu 42⎣⎦()11 ()11()22 ()22 ρλ()12µ− λ2µ1 a12 therefore

2 12()λµ11+ u 12 ()A ++2Bu Cu =42.(A +2Bu +Cu ) (3.42) a12 ρλ()12µ− λ2µ1 a12

Since

λ + µ u xy=+uz=+y 22z= 1 x λµ11+ u ρλ()11+ µu

=+1 ()yuz ρλ()11+ µu then from (3.37)

yy=ρλ()1+λ2z, z=ρµ( 1y+µ2z) (3.43)

Putting g= (yz), then g= ρ 2g, the substitution (3.37) of the first kind does not change then the factor of the coordinates of the generator. t Let us now put u = 2 in (3.42), in order to introduce homogenous coordinates on the t1 generator, and simplifying , we get:

1224 12 2 2()At1++2Bt1t22Ct =ρλ()1µ2−λ2µ12( At1+2Bt1t2+Ct2) (3.44) aa12 12 where we put the condition 3 The Method of CECH 60

ty12+=tz ρ ()t1y+t2z and therefore

tt11=+λ1µ1t2, tt22=+λ1µ2t2. (3.45)

In particular, setting ρ = 1 and λ12µλ− 2µ1=±1, we get the quadratic form

1 22⎛⎞t1 2 (AtB11++2tt2Ct2)=f⎜⎟=f(t) (3.46) a12 ⎝⎠t2 which is improperly intrinsic, that is to say, by changing the parameter v, and using (3.45) 2 in which (λµ12−λ2µ1)=1, it does not changes or changes the sign according to λ12µλ−2µ1=1 or λ12µλ−2µ1=−1.

3.4.2 The bilinear intrinsic form f (t,τ)

The curved asymptotic lines are described by the differential equation

du ++ab2u+cu2 =0 (3.47) dv which is of RICCATI form so its integral takes the form λ + µα u = 22, α = arbitrary constant. λ11+ µα

Setting λ + µ u u = 22 λ11+ µ u then the point y+ uz describes an asymptotic line if u is constant. Let us try to find the corresponding transformation of the homogenous coordinates t1, t2.

Putting ut= 2:t1, then (3.47) becomes

22 t21′′t −+t1t2 at1 +20bt1t2+ct2 =, which, by introducing a new parameter σ , can be written in the form

tb1′ =+()σ t1+ct2, ta21′ =− t+(−b+σ )t2.

Assuming ρ = 1, then it must be according to our convention λ12µλ−=2µ1 ±1 =const., σ = 0, then

tb11′ =+tct2, ta21′ =− t−bt2. (3.48)

Thus, if the ruled surface Σ referred to its asymptotic lines, then abc= ==0 . If Σ is not referred to its asymptotic lines, then the parameter may be transformed by placing λ + µ u u = 22 λ11+ µ u 3 The Method of CECH 61 and more specified

tt11=+λ 1µ1t2, tt22=+λ 1µ2t2 (3.49) without to change v and the factor of x, where λ1,λ2 and µ1, µ2 are two solutions of the linear system (3.48) in such a way that

λ12µλ−2µ1=1.

From (3.43), it will be then

yy=+λ1λ2z, zy=+µ1µ2z. (3.50)

Let us consider v, t1, t2 as independent variables; the differentiations of t1, t2 with respect to v are given by (3.48). Let

ty12+=+tz t1y t2z and by differentiation with respect to v, then

ty12 +=tz t1y′+t2z′ (3.51) where we have set yy =+′ by−az, zz =+′ cy−bz. (3.52)

Introducing y and z, then equations (Ac) take the following form yb′ =− y+az+y, zc′ =− y+bz+z yB′=−()+ jy+Az+(θ′−b)y+az, (3.53) zC′′=− y+()B− jz−cy +(θ +b)z.

As we consider v, t1, t2 as independent variables, then by differentiating the identity (3.51), and from (3.53) we obtain

tB12⎣⎦⎡⎤−+()jy+Az+θθ′′y +t⎣⎡−Cy+(B−j) z+z⎦⎤

′′ ′′ ⎡⎤′ ′ ⎡′ ′ ⎤ =+ty121tz =t⎣⎦−()B+j y+Az+θθy +t2⎣−Cy+()B−j z+z⎦ and therefore from (3.51) we have

tB12⎣⎦⎡⎤−+()jy+Az+t⎣⎡−Cy+(B−j) z⎦⎤

⎡⎤⎡⎤ =−+tB12⎣⎦()jy+Az+t⎣−Cy+()B−jz⎦.

Indicating by τ1, τ 2 and τ1, τ 2 as alternative values for t1, t2 and t1, t2 respectively; then the bilinear form

S()ty12 +−tt {τ1⎣⎡()B jηζ−A ⎦⎣⎤⎡+τCη−(B+j)ζ⎦⎤} does not change by the substitution 3 The Method of CECH 62

tt11=+λ 1µ1t2, τ11=+λτ1µ1τ2, λ12µλ− 2µ1=1. tt22=+λµ12t2, τ 22=+λτ1µ2τ2,

Thus this bilinear form becomes

1 ⎡⎤ ft(),τ =+2 ⎣At1ττ1 B()t12+t21τ+Ct2τ2+j(t1τ2−t21τ)⎦ (3.54) a12 which is improperly intrinsic. To explain the geometric significance of the projectivity defined over every generator by the bilinear relation ft(,τ )= 0, let us consider a ruled surface Σ generated by p= (yz), and another ruled surface Σ′ generated by qy= (′′z). Then, py′′=−()z(zy′)

p′′ =−()yz′′ (zy′′ )+2.(y′z′ )

Substituting from (Ac), then p′′=−22cy()y′+(θθ′+ b)(yz′)−(′′′−2b)(zy)−2a(zz)+2(ac−b2 − j)()yz+2(y′z′). assuming that Σ′ is referred to its asymptotic lines, then we will find the following relation 2qp=−′′ θ′ p′ +2jp (3.55) which is also valid, if Σ′ is not referred to its asymptotic lines. Now let us consider the line q as an image of the factor of p, then we can easily proof the following theorem, which explains the required significance of the projectivity ft(,τ )= 0:

Theorem 3.4.2.1: Let us consider the ruled surface Σ′ generated by q, then for a fixed value of the parameter v, if we construct the tangent plane of Σ in the point (ty1+ t2z); and if in the point of intersection of this plane with q, we construct the plane tangent to Σ, then the intersection of this last plane with p is the point (τ1y+τ 2z) corresponding to ()ty1+ t2z in the projectivity ft(,τ )= 0.

The line q is called the principal generator of the osculating quadric, which generates a ruled surface called by WILCZYNSKI the principal ruled surface of the flecnode congruence (cf. § 2.4). j Finally, it can be proved that the quantity 2 is intrinsic and if we apply the transformation (3.38) we get a12

⎛⎞1 ′′ ρ′ jj=+ρ⎜⎟+θ′ (3.56) ⎝⎠ρ ρ assuming that the surface is referred to its asymptotic lines, i.e. abc= ==0.

3 The Method of CECH 63

3.5 The flecnode congruence – The normalization of coordinates

As we explained before, the totality of the concordant set of lines of the osculating quadrics form the flecnode congruence of the ruled surface. Now let us choose the factor ρ from (3.37) in such a way that q describes a developable surface of the flecnode congruence ; the necessary and sufficient condition for that is the projectivity ft(,τ )= 0 degenerates, that is to say it satisfies the condition AC −+()B j ()B −j =AC −B22+j =0. From (3.56) and (3.39) this condition takes the form

′′ ′ ⎛⎞11⎛⎞ 2 ⎜⎟−+θ′⎜⎟ jB±−AC=0. (3.57) ⎝⎠ρρ⎝⎠

The double points of the projectivity ft(,τ )= 0 are the flecnode points. j To find from the intrinsic quadratic form f (t) and from the intrinsic quantity 2 an a12 expression which is also an invariant, it is necessary to examine the effect of the multiplicative substitution x = ρx. Applying this transformation and using (3.42) and (3.46) one finds that ⎡ ⎤ 1 j 11⎛⎞′′ ρ′ f ()t= f()t, =+⎢ j ρ⎜⎟+θ′ ⎥. (3.58) ρ 4 aa22 ρ ρ 12 12 ⎣⎢ ⎝⎠ ⎦⎥

The variables t1, t2 of the quadratic form f (t) are not completely determined; they can be replaced by t1, t2 where

tt11=+λ 1µ1t2, tt22=+λ 1µ2t2, λ12µλ− 2µ1=±1; if λ12µλ−2µ1=−1, then it is necessary to change the sign of AB, ,C. The discriminate B2 − AC of f (t) is therefore intrinsic; the effect of the multiplicative substitution x = ρx is B28−=AC ρ (B2−AC ) (3.59) The sign ε =sgn(B2 −AC) (3.60) of B2 − AC is therefore intrinsic and invariant. For ε =1, the flecnode curves are real and distinct; for ε =−1, the two flecnode curves are imaginary and for ε = 0 the two flecnodes are coincident. We are going to suppose that ε 2 =1; the parameter will be normalized by supposing the intrinsic and invariant ε =−(B2 AC). (3.61) In that case the corresponding coordinates of p are called the normal coordinates. If p is not normalised, then the normal coordinates are (cf. [21] p.207)

±−4 B2 AC p. 3 The Method of CECH 64

The sign of the normal coordinate is not determined; the choice of it is equivalent to the choice of the positive direction of the generator. We can choose the independent variable v in a substantially determined way, assuming ( in normal coordinates) a12 = ±1, that is

1 Sdpp⋅=d yzdydz=ω dv 2 , ω = ±1. (3.62) 2 ()

The parameter v determined from (3.62) except for the sign and one additive constant will be called the projective arc length. For non normal coordinates

ω ddv 22=−B AC ()yz ydz .

3.6 The fundamental system of invariants

3.6.1 Ruled surfaces with distinct flecnode curves

In the following we are going to assume that the coordinates are normalised and the projective arc length is the independent variable, besides that the ruled surface Σ is referred to its asymptotic lines. The variables tt1,2, are not yet completely determined; for fixing the positive direction of the generator, it is valid to set

tt11=+λ 1µ1t2, with λµ12− λ2µ1=1 (3.63) tt22=+λµ12t2, where λ1, λ2and µ1, µ2 are constants. If instead we invert the positive direction of the generator, we must put

tt11=+λ 1µ1t2, with λ12µλ− 2µ1=−1 (3.63a) tt22=+λµ12t2, and change the sign of A, B and C. The quantity j in (3.13), corresponding to our assumptions, is intrinsic and invariant. The quadratic form

22 f ()tA=+t112Btt2+Ct2 (3.64) with the discriminate BA2 −C=ε is improperly intrinsic (it changes the sign, if the positive direction of the generator is inverted), and invariant; and so also are the forms

f ′′()tA=+t222B′tt+C′t, 1122 (3.65) 22 f ′′()tA=+′′t112B′′tt2+C′′t2, where the dashes indicate differentiation with respect to the projective arc length v , but we have to notice that f ′(t) changes the sign also if the positive direction of the projective arc length is inverted. Let abc===0, then Σ is uniquely determined, up to a collineation, if f (t) and j are given as functions of v. Differentiating (3.61) we get AC′′+−CA 20BB′=, the quadratic forms f (t) and f ′(t) are therefore conjugate; the two points which are defined by ft′( ) = 0 on each generator are harmonic conjugate with the flecnode pair; they are the 3 The Method of CECH 65 harmonic points (Involute points) of the generator; they describe the harmonic curves on the surface. The resultant of the forms f (t) and f ′(t)

42()B22−−AC (B′ A′′C )−()AC′+CA′−BB′2 which is reduced to 4εh, where

hB=−′′2 AC′ (3.66) is intrinsic and invariant. We notice that h = 0 if, and only if, the ruled surface Σ has a rectilinear directrix. The quantity

ABC kA= ′′BC′ (3.67) AB′′ ′′ C′′ is improperly intrinsic and invariant, but changes the sign also if the positive direction of v is inverted. k = 0 if, and only if, the ruled surface belongs to a linear complex; in particular if h = 0, then also k = 0 . This can be verified as follows

ABCC−2BA 22kA2 =−′′BC′C′ B′A′= AB′′ ′′ C′′ C′′ −2B′′ A′′

−−2(BA2 C) AC′ +CA′ −2BB′ AC′′ +CA′′ −2BB′′ =+AC′CA′−22BB′ −(B′2 −A′′C )A′′C ′+C′A′′−2B′B′′ AC′′ +−CA′′ 22BB′′ AC′ ′′ +CA′ ′′ −BB′ ′′ −2(B′′ 2 −A′′C′′)

−20ε 2h =−02hh−′ ==0, if h 0. 22hh−−′′(B′−AC′′ ′′)

We demonstrate the first fundamental theorem,

Theorem 3.6.1.1: Let h ≠ 0, i.e. Σ has two distinct flecnode curves, and does not have a rectilinear directrix, then the surface Σ is determined, except for a collineation, if ε = ±1, h ≠ 0, k and j are given as functions of the projective arc length v. In order that Σ be existed, it must be h > 0 if ε =−1. It can be also proved that, if h = 0, one of the roots t12: t of ft()= 0 is constant, and one of the flecnode lines is a rectilinear directrix.

Let us define the sign σ of the quantity

D = A′′ (3.68) A′ which is also intrinsic and invariant, then we have the second fundamental theorem, 3 The Method of CECH 66

Theorem 3.6.1.2: A ruled surface Σ with one rectilinear directrix and with the other flecnode curve non rectilinear is determined, except for a collineation, if σ =±1, D and j are given as functions of the projective arc length.

We have to notice that the ruled surface Σ has two distinct rectilinear directrices, real or imaginary, if all the coefficients of f (t) are constants. We have thus the following theorem,

Theorem 3.6.1.3: A ruled surface Σ with two distinct rectilinear directrices, and which belongs to a non special linear congruence is determined, up to a collineation, if ε =±1 and j is a given function of the projective arc length.

Now, let us find the locus of the poles corresponding to a generator, which corresponds to a fixed value of the parameter v, with respect to the osculating conics of the asymptotic lines. The pole of a generator with respect to the osculating conic of the asymptotic line generated by a point x =+yuz is found to be

2 4.()yu′′++z AB′′++2 uC′u()y+uz AB++2 uCu2

Putting ut= 1:t2, then the pole of the generator p with respect to the osculating conic of the asymptotic ty1+ t2z is

4(f tt)()12y′′′++tz f(t)(t1y+t2z). (3.69)

Hence, for a fixed value of v, the required locus of the poles is a space cubic c. The generators of the concordant system of lines of the osculating quadric QΣ are bisecants of c; in particular, a generator g of Σ intersects c in the flecnode points. The principal generator

q of QΣ intersects c in a pair of harmonic points with respect to the flecnodes. The generators of the transversal system of lines of QΣ , which pass through the points of intersection of q and c, meet g in the its harmonic points.

The generator g and the cubic c are the base of a bundle of quadrics, to which belong two cones whose vertices are respectively in the flecnode points; and remarkable is the quadric of the bundle which is harmonic conjugate to QΣ with respect to the two cones of the bundle, that is therefore the locus of the straight lines that connecting the poles of the generator g with respect to the osculating conics of the asymptotic lines that passing through various points x of g to the points harmonic conjugate to x with respect to the flecnode points; such a quadric will be indicated by W1. Assuming a general point of space

x =+λµyz+λ11y′′+µ z, (3.70) and regarding λ, µ, λ1, µ1 as coordinates of x in a system of reference depending on v, then it can be verified that the equation representing W1 takes the form

22 4⎣⎦⎡⎤ABλλ11++()λµ λ1µ +−Cµµ1( A′′λ1+2Bλ1λ2+C′λ2) =0. (3.71)

3 The Method of CECH 67

Correlatively, the plane polar to g with respect to the osculating cone of the asymptotic ty1+ t2z is

4(ft)()t12η′′++tζηf′(t)(t1+t2ζ′). (3.72)

The bundle of planes with axis g and in the same time the planes which obtained in (3.72) by various values of t1, t2 are the base of a set of quadrics. We are going to indicate by W2 the quadric of this set which is harmonic conjugate of QΣ with respect to two conics of the set.

The quadrics W1 and W2 are reciprocal polar with respect to QΣ , as well as with respect to the osculating linear complex of Σ . If Σ belongs to a linear congruence, then W1 and W2 coincide together. In general W1 and W2 touch each other along g, and have in common two generators of the transversal system through the harmonic points.

Theorem 3.6.1.4: The two generators of W2 passing through a flecnode point of g (one of which is g itself) are harmonic conjugate with respect to the flecnode tangent and the tangent of the flecnode curve ( cf. Theorem 2.5.5).

Analogue to q, we are going to define the principal generator of W1, which is the generator q1 of the concordant system (to which also belongs g ) of W1 that intersects c in two points harmonic conjugate to those generators of the transversal system of W1 which passing through the flecnodes of g. It can be simply verified that:

2 q1 =−()A′′C B′ ()yz +44(A′C −BB′)( yz′) −( AC′−BB′)( zy′) (3.73) +−4()BCC′′B(yy′)+4()AB′−A′B(zz′)+16()AC−B2 ()y′z′.

Correlatively, we define the principal generator q2 of W2 , whose coordinates are:

2 q2 =−()A′′C B′ ()yz +44(AC′−BB′)( yz′) −( A′C −BB′)( zy′) (3.74) −−44()BCC′′B(yy′)−()AB′−A′B(zz′)+16()AC−B2 ()y′z′, thus

qq12++23hp+2εq=0.

So one concludes that the two pairs of lines gq,;q1,q2 belong to a linear set and are harmonically separated.

If h = 0, than q, q1 and q2 have a common point on the directrix of Σ.

3.6.2 Ruled surfaces with coincident flecnode curves

In this case BA2 −C=0; the discriminate of f (t) is null, and

⎛⎞t1 22 f ⎜⎟=−σα()12ttα21 =σ(αt), σ = ±1, (3.75) ⎝⎠t2 where the sign σ depends on the selected positive direction on the generator of Σ . We are going to exclude the case in which the unique flecnode curve α1y+α2z is rectilinear, since in 3 The Method of CECH 68 that case, the ruled surface belongs to a special linear congruence. Assuming that Σ is referred to its asymptotic lines and supposing that α1:α2 is not constant, so that the determinant

α12αα′′−=2α1 (αα′) will be ≷ 0. For the case ()αα′ = 0 for a particular generator; it is a singular case, which will be excluded. Putting δ = sgn ()αα′ , δ =±1, (3.76) then δ is intrinsic (does not depend on the positive direction on the generator) and invariant, but depends on the positive direction of the parameter v. To choose the arbitrary factor of f (t) in a determined way, it is sufficient to suppose that ()ααδ′ = . (3.77)

The coordinates of a generator g corresponding to (3.76) and to σ =1, are called the normal coordinates. If the coordinates of g are normal, then the projective arc length of Σ, as an independent variable, is determined by

1 2 2 Sdpp⋅=d ()yzdydz=ωdv , ω = ±1, δ = ±1. (3.78)

Supposing that g has normal coordinates; Σ referred to its asymptotic lines, and that v is the projective arc length, so that

f ()t= (αt)2 , ()αα′ =1, then by differentiating we have ()αα′′ = 0, so that

α11′′ = lα , α22′′ = lα (3.79) where l is evidently intrinsic and invariant. Thus the third fundamental theorem is,

Theorem 3.6.2.1: A ruled surface Σ not belonging to a linear congruence, and whose flecnode curves are coincident, is determined, up to a collineation, if the invariants l and j are given as functions of the projective arc length.

Now let us consider the case in which α12:α = const., and Σ still referred to its asymptotics. One can determine the factor of f (t) and therefore that one of g up to a constant factor by placing the condition that α1 and α2 are constants; in fact, one can choose dv according to (3.78). Let c be an arbitrary constant, one can also substitute dv and j by cdv, cj−2 . Supposing j ≷ 0, and let

ej= sgn , wj= ∫ dv then we have the fourth fundamental theorem, 3 The Method of CECH 69

Theorem 3.6.2.2: A ruled surface Σ, which belongs to a special linear congruence and for dlog j which j ≷ 0, is determined, up to a collineation, if e = ±1 and is a given function in dw w, where w and e from the previous relations.

If instead j = 0, then there exist no intrinsic differentials and invariants and no finite intrinsic and invariant expressions: the variable v can not be fixed except for a substitution of the type av + b with a,b are arbitrary constants. The ruled surface Σ, if j = 0, admits therefore ∞3 collineations in itself, and is thus the CAYLEY’s ruled cubic surface. Applying that by substituting in system (Ac) by A =1, BC==0, θ′ = 0, j = 0, abc= ==0 then (Ac) become y′′ = z, z′′ = 0 and by integrating

1123 z = ()0,0,1,vv, y = ()1, vv, 26, v

1123 x =+yuz=()1, vv, 26+u, v+uv that is, indicating the four coordinates of x by 1, x, y, z, then

1 2 1 3 x = v, yu=2 v +, zu=+6 vv hence 1 3 zx=−y3 x.

To interprete the normal coordinates geometrically, let us consider the locus of the poles of p with respect to the osculating conics of the asymptotic lines for a fixed value of the parameter v. It is described by (3.69), but the factor (αt) can here be omitted; then remains

2.()αα12tt−+21 (t1y′′t2z)+(αα12′tt−2′1)(t1y+t2z) (3.80)

Therefore: The locus of the poles of a generator with respect to the osculating conics of the asymptotics is a conic. Indicating by W1 the plane of this conic, then this plane is described by

α12′′ηα+−ζ 2.()α1η′+α2ζ′ (3.81)

Correlatively, the planes polar to the generator p with respect to the osculating conics of the asymptotics are given by

2.()α12tt−+αη21 (t1 ′′t2ζ)+()α12′tt−αη2′1 (t1 +t2ζ) (3.82)

Those are the planes tangent to QΣ and passing through a point W2 described by

αα12′′yz+−2()α12yz+α (3.83) 3 The Method of CECH 70

where this point is the pole of W1 with respect to QΣ . If (αα′) = 0, the conic degenerates in a directrix and in a line qy= (′′z) corresponding to the chosen factor of p for which the coefficients of f (t) are constants. If instead (αα′) ≷ 0, then the conic is proper. Let us suppose that ()αα′ =1, and let us call the corresponding line q, as before, the principal generator of QΣ . The point of the conic (3.80) lying on the principal generator of QΣ will be situated in the plane osculating the unique flecnode curve. In fact, this point is given by

α12′′yz+α . On the other hand, equations (Ac), for the considered assumptions, are then reduced to

2 2 y′′ =−()α12ααjy+2z, zy′′ =−αα11−( α2+ j) z

αα12yz′′ +=′′ −j()α1y+α2z and therefore ′′ ()α12yz+=αα−j(12y+αz)+2(α1′y′′+α2z′)+α12′′y+α′′z taking into account (3.80), then

′′ 2.()αα1′′yz+=2′′ (α12y+αz)+(j−l)(α12y+αz)

The principal generator of QΣ meets therefore the intersection of the plane W1 with the osculating plane of the flecnode curve. Correlatively, the principal generator of QΣ meets the line connecting the point W2 with the cuspidal point of the developable circumscribing Σ along the unique flecnode curve.

Finally, to find a general expression for h and k for non asymptotic coordinates, i.e. for an arbitrary parameter v, we are going to use the substitution (3.49) and considering v, t1, t2 as independent variables. We derive the identity: f ()t= f(t) and it is found directly

f ′()tf= (t), f ′′ ()t= f(t) where

22 22 f ()t =+At112Bt t2+Ct2, f ()t =At11+2Bt t2+Ct2, A=+A′′2()Ab −Ba , B =B +Ac −Ca, C =C′+2(Bc −Cb) AA =+ ′ 2()A b−Ba ′′ ′ ′ ⎡⎤′ 22′ =A +42()A b −B a +⎣⎦A()b +2b −ac −++B()a 2ab Ca , (3.84) BB =+ ′ Ac−Ca =+BA′′ 2()′c−C′a++Ac()′′24bc−Bac−C(a−2ab), CC =+ ′ 2()Bc−C b ′′ ′ ′ ⎡⎤22′ ′ =+CB42()c−Cb+⎣⎦Ac+B(c−2bc)−C()b−2b+ac. 3 The Method of CECH 71

Then it is enough to substitute f(t) and f (t) for f ′(t) and f ′′(t) respectively, to find the general expressions of h and k for an arbitrary independent variable v. Using the same principle, one can generalize all the previous formulae assuming the independent variable v in an arbitrary way.

3.7 The osculating linear complex

In the following we are going to assume that Σ is referred to its asymptotic lines; equations (3.84) permit easily to extend the formulae for the case of arbitrary coordinates. The fundamental form (Ac) will be reduced to yy′′ =−θ′ ′ ()B++jyAz, (3.85) zz′′ =+θ′ ′ ()B−jz−Cy.

Putting p = (yz) and qy= (′′z), one finds therefore the following relations, in which the neglected terms are linear combinations of p, p′, pq′′ =+22θ′ p′ −jp, qC′′=+(yy)B⎣⎦⎡⎤()y′z−()yz′+A(zz′)+2,θ′q−jp′ (3.86) qC′′ =+′ (yy′ )B′ ⎣⎦⎡⎤()y′z−()yz′ +A′ (zz′ )+…, qC′′′ =+′′ (yy′ )B′′ ⎣⎦⎡⎤()y′z−()yz′ +A′′ (zz′ )+…

Let r be a one directrix of the osculating linear congruence determined by four successive generators, then it must be SSrp ==rp′′SSrq ==rq 0.

The first three conditions give that

22 r =+()t12y t z,t1y′ +t2z′′=t1(yy )−t1t2⎣⎦⎡⎤( y′z) −( yz′) +t2( zz′)

2 so that, since ()yzy′′z = ωa12 , then

22 2 2 S2rq′ =−ωωa12 ()At1 + Bt1t2 +Ct2 =− a12 f (t).

Thus, one finds that the directrices of the osculating linear congruence are the flecnode tangents as it is already explained before. We see also that if AB= ==C0, then Σ is a quadric. Moreover, if AB==C=0 for a particular value of the parameter v, then it is said that the osculating quadric QΣ hyperosculates Σ along the generator corresponding to v. The condition in order that q′′ be a linear combination of p, p′, q, q′, or for which Σ belongs to a linear congruence is found directly to be

AB′′==C′ (3.87) ABC i.e. the roots of f = 0 are constants (which is already known at least for the case BA2 − C≷ 0 ). The condition in order that q′′′ be a linear combination of the previous ones, or that Σ belongs to a linear complex is k = 0, as we have already enunciated. 3 The Method of CECH 72

The lines of the osculating linear complex K are linear combination of p, pq′, , q′and

q′′. To it belong all the generators of the concordant system of QΣ , which are linear combination of p, p′, q. To fix the position of K, it is sufficient therefore to find the two generators of the transversal system of QΣ that belonging to it. Let

22 rt=+()12ytz, t1y′′+t2z=t1(yy′)−t1t2⎣⎦⎡⎤( y′z) −( yz′) +t2( zz′) be one of those two lines. In order that r be a linear combination of p, pq′, , q′ and q′′, it is necessary that r be a linear combination of q and q′′, so that we find the condition

2 tC1 C′ At12++Bt Bt12Ct tt12 −−B B′ = =0. (3.88) 2 At′′12++Bt Bt′12C′t tA2 A′

The first member of this equation is the Jaccobian of the forms f (t) and f ′()t ; its discriminant value is

44()ABA′′−−B(BC′′BC)−(AC′−A′C)2 =( AC−B22)( A′C′−B′) −()AC′+CA′−2BB′2.

In order that to be identically null, it must identically either

BA2 −=C0 or else

2 AC′′−=B′22d AC−B . ( dv )

There are two classes of ruled surfaces for which all the osculating linear complexes K are special: the first class are those surfaces whose flecnode curves are coincident; where as the second class are those surfaces whose directrices are rectilinear. For the second class, assuming normal coordinates, then h = 0 is the required condition as we explained before.

In the general case, (3.88) defines two points on each generator, the LIE-points of the generator, through them pass two generators of the transversal system of QΣ which belong to the osculating linear complex K.

The osculating linear complex along an asymptotic line of a ruled surface contains the osculating linear congruence of that surface. If all the asymptotic lines of a ruled surface belong to a linear complex, i.e. complex curves, then the surface belongs to a linear congruence. The opposite can also be simply verified.

3.8 Further studies on a hyperbolic ruled surface with non rectilinear directrices

Given a ruled surface Σ with two distinct flecnode curves in normal coordinates, referred to its asymptotic lines and v is the projective arc length. Supposing that h ≷ 0 (if ε = −1, h > 0 ), we are going to distinguish between two cases:

3 The Method of CECH 73

3.8.1 The first case: ε =1

The form f (t) will decompose in two linear real factors f ()t= 2()αt(βt) (3.89) where in the following we assume

()αtt=−α12 α2t1, ()βt =……,

For the significance of ε, we have ()αβ 2 =1 and evidently it is valid to assume that ()αβ =1. (3.90)

The linear forms (αt) and (βt) are not completely determined by (3.89) and (3.90); let ρ be an arbitrary function in v, then we can set the substitutions (αt), (βt) respectively for ρ ()αt , ρ −1 (βt), and therefore ()αα′ , (ββ′) respectively for ρ 2 (αα′), ρ −2 (ββ′). Let δ = sgn ()αα′′(ββ ). (3.91)

We notice that δ can not be vanished, as in that case it will be h = 0. We are going to exclude the particular values of v for which h = 0 and therefore δ = 0. We can thus fix the forms (αt) and (βt), except for a simultaneous change in sign, by setting the condition ()αα′′==δ (ββ )n ≷ 0. (3.92)

Differentiating (3.91), one finds ()αβ′′==(βα )m (3.93) where n and m , but not the sign δ , are evidently invariants by the substitution (3.63).

It is obvious that, if δ =±1, n, m are given as functions of the projective arc length v, then the quadratic form f (t) exist and well be determined up to such substitution. Actually, from (3.90) we notice that (α′t) and (β′t) are linear combinations of (αt) and (βt), and from (3.92) and (3.93) we deduce particularly ()αα′tm=− (t)+n(βt) (3.94) ()β′tn=−δα(t)+m(βt).

It is sufficient then to take ()α1, β1 and (α2, β2) as two solutions of the differential system α′ =−mnαβ+ , (3.95) β′ =−δαnm+ β associated from (3.90); it is possible that (αβ ) = const. would be a consequence of equations (3.95). From (3.95) we deduce at once that

ft′()=−2n⎡δαt22+βt⎤, h= 4δ n2 , km=16δ n2. (3.96) ⎣() ()⎦ 3 The Method of CECH 74

Given h and k, then δ , m and n2 are therefore determined, but not the sign of n. In fact such a change of sign inverts the positive direction on the generator, that is for a substitution of the form (3.63a).

If it is required to determine only the ruled surface Σ and not its asymptotics, then it is not necessary to integrate the system (3.95). In fact setting

tt1 = ()α , tt2 = (β ) then it is valid for ti the equations

tt11′′==()α bt+ct2, (3.97) t21′′==()β t −at −bt2.

Comparing with (3.94), we have

an= δ , bm=− , cn= . (3.98)

Hence, we have the following result: If the ruled surface Σ have two distinct non rectilinear flecnode curves, then we can normalise (except for a change of sign) the factors of the coordinates of the flecnode points in such a way that: an= δ , bm=− , cn= , δ = ±1 A = 0, B =1, C = 0, where m and n being associated with the invariants h and k from hn= 4,δ 2 km=16δ n, and the projective arc length of Σ is the independent variable Equations (3.53) take the form

ym′ =+yδ nz+y, ym′ =+yδ nz −(1,+j) y (3.99) zn′ =− y−mz+z, zn′ =− y −mz+()1.−jz

If δ =1, the harmonic curves are real and generated by the points y± z, if instead δ =−1, the LIE-curves are real and generated by the points y± z. The fundamental equations (Ac) take the form ym′′ =+22y′ δδnz′ +()m′ +n22−m−1−jy+δn′z, (3.100) zn′′ =−22y′ − mz′ −n′y+()−m′ +δ n22−m+1− jz.

We can deduce a simpler demonstration of CARPENTER’s theorem (cf. Theorem 2.8.1):

If the two flecnode curves of a ruled surface are plane curves, then the LIE-curves and the harmonic curves are also plane curves, and all their planes belong to the same bundle.

For, let yi and zi (i =1, 2, 3, 4 ) indicate the four coordinates of y and z. Choosing conveniently the system of reference, we can assume that y3 = 0, z4 = 0. Then (3.99) gives 3 The Method of CECH 75

20nz′′+=n z , 33 20ny44′′+=n y so that zy34: ==λ const. The curve described by the point x = y+ cz ()c = const. is therefore in the fixed plane

xc3−=λ x40.

3.8.2 The second case: ε =−1, h > 0

Let the quadratic form f (t) be defined, and the positive direction on the generator be chosen in such a way that f (t) is positive. We can therefore set

f ()tt=+()αβ22(t). (3.101)

For the significance of ε we have ()αβ 2 =1 and it can be also assumed that ()αβ =1.

Differentiating we deduce ()αβ′==(βα′)m. (3.102)

The linear forms (αt) and (βt) are still not completely determined; let ρ be an arbitrary function of the projective arc length v, they can be respectively substituted by cos ρα()tt+ sin ρ(β),

−+sin ρ ()αρttcos (β) and therefore the expressions (αα′) and (ββ′) respectively by cos22ρ ()αα′′++2mcos ρ sin ρ sin ρ (ββ ) +ρ′,

sin22ρ ()αα′′−+2mcos ρ sin ρ cos ρ (ββ )+ρ′, such that ()αα′− (ββ′) leads to be substituted by

⎣⎦⎡⎤()ααβ′′−+(β)cos 2ρ2msin 2ρ. It follows that we can fix (αt) and (βt), except for a simultaneous change in sign, by setting the condition m > 0 (it couldn’t be that m = 0 because that leads to h = 0 which is a contradiction) and ()αα′′==(ββ )n. (3.103)

As in the previous case, we deduce from (3.102), (3.98) and (3.103) the forms ()α′tm=− (αβt)+n(t), (3.104) ()β′tn=− (αβt)+m(t) 3 The Method of CECH 76 from which we deduce

f ′()tm=−2 ⎡⎤αt22− βt (3.105) ⎣⎦() ()

hm= 4,2 km= 32 2n, (3.106) it results that, for m > 0, then m, n are uniquely determined from h > 0, k. To construct from h > 0 and k, or similarly, from m > 0 and n, the quadratic form f ()t , it is sufficient to choose for (αβ1, 1) and (αβ2, 2) two solutions of the differential system α′ =−mnαβ+ ,

β′ =−nmαβ+ associated with (3.101). If we are not interested in the asymptotics of Σ, it is not necessary to integrate the previous equations. In fact the substitution

()αtt= 1, ()βtt= 2 is valid for (3.97) and comparing with (3.104) gives

an= , bm=− , cn= .

From (3.100) and (3.105) we have then

A =1, B = 0, C =1

At12++Bt Bt12Ct = 4.mt12t At′′12++Bt Bt′1Ct′2

Therefore: If a real ruled surface Σ without rectilinear directrices has two imaginary flecnode curves, we can normalise (except for a simultaneous change in sign) the factors of the coordinates y and z of the LIE-points in such a way that: a= n, bm=− <0, cn=

A =1, B = 0, C =1.

Where m and n are associated with the invariants h and k from (3.106) and the projective arc length is taken as the independent variable. The harmonic curves are described by the points y± z, the flecnode curves by the imaginary points y± iz. Equations (3.53) in this case take the form ym′ =+ynz+y, yj′ =− y+z+my+nz, (3.107) zn′ =− y−mz+z, zy′ =− −jz−ny −mz.

3 The Method of CECH 77

3.9 The flecnode surfaces

Let us consider a ruled surface Σ with two distinct real flecnode curves and non rectilinear directrices. Taking the projective arc length of Σ as independent variable and fixing conveniently the factors of the coordinates of y and z of the flecnode points of the generator, we can write equations (3.99) as:

ym′ =+yδ nz+y, ym′ =+yδ nz −(1,+j) y (3.108) zn′ =− y−mz+z, zn′ =− y −mz+()1.−jz

For every value of the parameter v, there exists the geometric significance of the tetrahedron of reference formed by the points y, z, y, z and we notice that ( yz) is the generator of Σ,

(yy )and ( zz) are the flecnode tangents, and ( yz) is finally the principal generator of QΣ . Let the flecnode surfaces of Σ are the ruled surfaces Φ1 and Φ2 generated respectively by the lines (yy ) and (zz). Being (yzy′′z )= ω, (3.109) we can determine the generator of Φ1 from the points y yy= , z = (3.110) 1 1 n it will be

(yz11yz11′′)=−ω. (3.111)

By simple calculations one can find from (3.108) the differential equations of form (Ac) representing Φ1

yy′′ =+nm′′′ 21nz′ +m′ −n+m22−δ n++jy+n′z, 11nn1( ) 11

jj⎡⎤1+ ′ 2 nm′′⎢⎥⎛⎞ ⎛⎞mn22n′′n′ zy11′′ =−22′ − z1′ + −⎜⎟− y1+⎜⎟m′ − +m−δ n− + 2 −1+ jz1. nn ⎝⎠n n ⎝⎠n nn ⎣⎦⎢⎥

Indicating the quantities related to Φ1 by the index 1, then comparing with (Ac) we have j a= n, b =−1 n′ , c = , 1 1 2 n 1 n A = 0, B =−1, C =−nm′ +2, (3.112) 1 1 1 n 2 2 j =−13nn′′ ′ −m′ +mn′ −m22+δ n. 1 24nnn2

We notice that, A1 = 0 means that the point y generates also a flecnode curve on Φ1. Such a property is already shown by WILCZYNSKI, who noticed more generally that there exists ∞1 ruled surfaces which touch Φ1 along the flecnode curve generated by y , and have also this curve of contact as a flecnode curve (cf. § 2.6).

2 From (3.112) we find that BA11−C1=1 and considering (3.111), then we deduce that a flecnode surface of Σ corresponds to Σ with a similarity of the projective arc length. 3 The Method of CECH 78

Analogue, setting

y= z, z = z , yzyz′′ = −ω, (3.113) 2 2 n ()22 22 the quantities related to the other flecnode surface Φ2 are j a= n, b =−1 n′ , c = , 2 2 2 n 2 n A = 0, B =1, C =+nm′ 2, (3.114) 2 2 2 n 2 2 j =−13nn′′ ′ +m′ −mn′ −m22+δ n. 2 24nnn2

The comparison of the first line from (3.112) and (3.114) shows that, if the point y1+ uz1 of

Φ1 corresponds to the point y2+ uz2 of Φ2 , then the asymptotic curves of Φ1 and Φ2 are correspondent. That explains the already known property of the flecnode congruence that it is a W-congruence.

For brevity, we are going to say yz() is the first flecnode point of the generator (yy ) of Φ1

[( zz) of Φ2 ]. From the second line in (3.112) we find that the second flecnode curve of Φ1 is generated by the point

−+1 n′ my+y, (3.115a) ( 2 n ) similarly, the second flecnode curve of Φ2 is generated by the point

−−1 n′ mz+z. (3.115b) ( 2 n )

Whence, the following theorem of WILCZYNSKI,

Theorem 3.9.1: The line connecting the two second flecnode points of Φ1 and Φ2 , corresponding to the same value of the parameter v, belongs to QΣ if, and only if, m = 0, i.e. iff Σ belongs to a linear complex.

Let from (3.108)

yn′′ +−()21my′ +()−m′ +m22=δδn++jy=2n⎡z +1 n′ −mz⎤ , ⎣⎢ ( 2 n ) ⎦⎥ then the point of intersection of the plane ( yy′y′′ ) with the line ( zz) is

zm +−1 n′ z, (3.115c) ( 2 n ) similarly, the point of intersection of the plane ( zz′z′′ ) with the line ( yy ) is

ym ++1 n′ y. (3.115d) ( 2 n )

Comparing the expressions (3.115) we deduce the following theorem,

3 The Method of CECH 79

Theorem 3.9.2: The generator of the concordant system of QΣ passing through the intersection point of the osculating plane of the first flecnode curve of Φ1 with the generator of Φ2 and the generator of QΣ passing through the second flecnode point of Φ1 are harmonic conjugate with respect to the generator of Σ and the principal generator of QΣ . Of course all these generators correspond to the same value of the parameter v.

WILCZYNSKI considered this theorem as a geometrical definition of the principal generator of

QΣ (cf. § 2.4).This theorem gives a geometric construction of the second flecnode point of

Φ1 (and of course also of Φ2 ). By means of (3.85) we deduce from (3.112) and (3.114):

2 j An==2, B nn′′−2m, C=−′+n′+2m′−2 , 11nn1n3 nn

2 An =−24′′+ mn,B =2nn′′ −2′ −4m+4j, 11n n2 3 j′′nj mj C =−nn′′′ +32′n′′ − n′ +2m′′ −2+2−4 1 nn23n4nnn2n

2 j An=−2, B =−nn′′−2m, C= ′−n′+2m′+2 , 22nn2n3 nn

2 An =+24′′mn,B =−2nn′′ +2′ −4m−4j, 22n n2 3 jn′′jmj C =−nn′′′ 32′n′′ +n′ +2m′′ +2−2−4. 2 nn23n4nnn2n

It follows that the expressions for the invariants h and k of Φ1 and Φ2 are respectively

2 hm=−24nn′′ ′ −mn′ +42 −4m′ +4j, 1 nnn2 2 nj′ km=−2 nm′′′ +12 n′′ +12 m′n′ −24 mn′ +4 ′′′−24mm+16m3 +16mjj−8 +4 ′, 1 nn n n n (3.116) 2 hm=−24nn′′ ′ +mn′ +42 +4m′ +4j, 2 nnn2 2 nj′ km=+2 nm′′′ 12 n′′ +12 m′n′ +24 mn′ +4 ′′′+24mm+16m3 +16mjj+8 +4 ′. 2 nn n n n Now, as an application on the formulae (3.116), we have the following question: is it possible that the two flecnode surfaces Φ1 and Φ2 of a ruled surface Σ to be correspond to each other such that the two flecnode tangents of the same generator correspond to each other in a fixed homology? The conditions for this problem are

h1= h2, k1= k2, jj12= .

Now from (3.112), (3.114) and (3.116) we have

′ hh−=−8 mn′ , jj−=−2nm 12 n () 12 ( n ) 3 The Method of CECH 80

m so that mn⋅ and n are constants. Two cases are thus possible: the first one is that: m and n are constants; while the second case is that: m = 0.

In the first case kk12−=−8, j′ so that also j = const. Conversely, if m, n and j are constants, then all the conditions of the problem are satisfied. In this case the differential equations (3.108) will not be changed if we put v + const. in place of v, i.e. Σ admits a continuous group of ∞1 homologies in itself.

In the second case kk12+ = 0, such that kk12= = 0 and all the surfaces Σ , Φ1 and Φ2 belong to linear complexes. Conversely, if Σ and Φ1 belong to the linear complexes K and

K1 respectively, then the null polarity accompanying K1 does not change Φ1, where as the null polarity accompanying K leads to Φ2 , such that the product of the two null polarities is a homology (bi-axial) which transforms Φ1 in Φ2. If m = 0, then nj′ kk−=−−4168.n′′′ − j′ 12 nn Therefore n can be taken arbitrary provided that it must be different from zero, and j can be determined from the linear differential equation

jj′ =−2,nn′ −1 ′′′ nn2 whose integration gives

1 ′′′2 jnnn=−+2 ()2 const. . (3.117) 4n This class of ruled surfaces which we obtained are interesting because each ruled surface of this class can be derived simply from a line has constant curvature of a space of three dimension of constant curvature.

We consider first the linear complexes K, K1, K2 to which belong the ruled surfaces Σ , Φ1,

Φ2 respectively, those complexes belong to a bundle, because as we have noticed, K1 and K2 are polar with respect to K. The base linear congruence Δ of the bundle can be general or special. For brevity, we will limit our study on the case Δ is general. The curves c, c1, c2 are 2 respectively the images of the ruled surfaces Σ , Φ1, Φ2 in a 1-1 correspondence on M 4 , (4) (4) (4) ∗) (4) (4) (4) contained in three spaces S , S1 , S2 of four dimensions . S , S1 , S2 intersect in a  (3) 2 space S of three dimensions which intersects M 4 in a quadric Q2 , which is the image of (4) (4) (4) 2 the non-special congruence Δ. The spaces S , S1 , S2 intersect M 4 respectively in three 3 3 3 quadrics of three dimensions Q , Q1 , Q2 ; the images of the complexes K , K1, K2. Let P ,

P1, P2 being respectively points of the curves c , c1 , c2 corresponding to an arbitrary chosen 2 value of the parameter v. The line PP12∨ is polar, with respect to M 4 , to the osculating (3) space of three dimensions S of the curve c in P , where P1 and P2 are the images of the directrices of the linear congruence osculating Σ along the generator corresponding to the chosen value of the parameter v. Now, S (3) be evidently found in the hyperplane S (4) so that 3 3 3 the line PP12∨ pass through a fixed point O. Let Q be the projection of Q1 and Q2 from

∗) For the system of notations, see chapter 7. 3 The Method of CECH 81 the point O on the hyperplane S (4). A simple considerations or simple calculations show that the two quadrics Q3 and Q3 of the space S (4) are touched in all the points of the quadric Q2. (4) Let c be the projection of c1 from the point O in the space S that is therefore a curve laying on Q3 and let P be the point of c corresponding to the value of v chosen before. The (3) 2 (3) line P1∨ P2 is the polar of S with respect to M 4 , the point P is the pole of S with respect to Q3 , and therefore the hyperplanes S (3) corresponding to various values of v touch the quadric Q3∗ , the polar of Q3 with respect to Q3 , which touches also Q3 in all the points of Q2. Introducing now in the space S (4) a metric Euclidean regarding S (3) as a hyperplane at the infinity, and Q2 as the absolute quadric: the quadrics Q3 and Q3∗ appear in such a metric as concentric hyperspheres. The curve c be on the hypersphere Q3 and its osculating hyperplanes (and therefore also its osculating planes) touch the concentric hypersphere Q3∗. Such osculating planes intersect Q3 in the osculating circles of c which have therefore constant radius and the curve c is a line with a constant curvature traced on the space with constant curvature (hypersphere) Q3. Analogue we can treat the case if ∆ is special, which leads to deduce Σ from a line with constant curvature of the ordinary Euclidean space.

3.10 WILCZYNSKI’s principal surface of the flecnode congruence∗)

Let us consider a ruled surface Σ, whose differential equations are given by (3.100), and for which the equations (3.53) are valid. The lines ( yy ), ( zz) are the flecnode tangents, and ( yz) is the generator of the principal surface Σ1 of the flecnode congruence of Σ. It follows that

⎛⎞jj′ ⎛j′ 221+ j⎞ ym′′ =+⎜⎟22y′ −δδnz′ +⎜m′ −m−m−n −1−j⎟y ⎝⎠11+−jj⎝1+j1−j⎠ ⎛⎞j′ 1 +−δ ⎜⎟nn′ −2nm z, ⎝⎠11+−jj (3.118) jj⎛⎞′′⎛1 j⎞ zn′′ =−22y′ −⎜⎟m+ z′ −⎜n′ +2nm+n⎟y 11+−jj⎝⎠⎝1+j1−j⎠

⎛⎞jj′ 221− −+⎜ mm′ +m+δ n −1+j⎟ z. ⎝⎠11−+jj and ()yzy′′z =−()j 2 1,ω where ω =()yzy′′z =±1. (3.119)

Thus, it follows that, Σ1 is a developable, if j2 =1,

25 (by WILCZYNSKI, this condition takes the form θθ4.1 − 64 4 = 0, cf. §2.4). We are going to exclude this case, and let us put

4 2 4 2 yy =1 j−1, zz =1 j−1 (3.120)

∗) cf. § 2.4 3 The Method of CECH 82 then, it will be

(yz11yz11′′)= ω. (3.121)

Substituting (3.120) in (3.118), then one finds

jn′ 2δ j ′′ ⎛⎞′ ′ ym11=−⎜⎟2 2 y− z1 ⎝⎠j −1 1− j ⎛ mj′′12++j j 2 jj′′−mjj′ ′ 22 1 +−⎜ mm−−δ n−1−j− 2 ⎝ 11+−jj2j −1 31jj22′′jj2 ⎞ ++2 ⎟ y1 422 jj2 −+11()⎟ ()j −1 () ⎠ ⎛⎞nj′′nj2 j +−δ nz′ −2nm − , ⎜⎟11++jj2 1 (3.122) ⎝⎠()jj−−11()

2nj ⎛⎞jn′′⎛⎞j nj2 j′ zy′′ =− ′ −−2mz′′−n+ +2mn + y 1 1+ j 11⎜⎟j2 −1 ⎜⎟11−+jj2 1 ⎝⎠⎝⎠()jj−+11() ⎛ mj′′12−+j j 2 jj′′+mjj′ ′ 22 1 +−⎜ mm− − −δ n+1−j− 2 ⎝ 11−+jj2j −1 31jj22′′jj2 ⎞ +−2 ⎟ z1. 422 jj2 −−11()⎟ ()j −1 () ⎠

Comparing this system with (Ac), one finds that δ nj j′ nj ab=− ,,=1 −mc= , 12−+jjj 2 −1 1 nj′′nj +−n′j 2mn nj2 j′ njj′ δ An=−′ + − + , 11+−jj()jj2 −−11()()1− j 2 (3.123) 43δ nj22− j′′ jj′ 22B =− 22+ , j −1 ()j2 −1 nj′′nj +−n′j 2mn nj2 j′ njj′ Cn=+′ − + + . 11−+jj()jj2 −+11()()1+ j 2

The focal surfaces of the flecnode congruence, i.e. the flecnode surfaces of Σ, intersect the principal surface Σ1 in the flecnode curves, if AC= = 0, or ⎛⎞nj 2 j′ 2 ′ δ AC−=− 2⎜23mn−nj+ 2⎟=0, (3.124a) 11−jj⎝⎠− and ⎛⎞njj′ δ AC 2 ′ +=2⎜n +32⎟=0. (3.124b) 11+−jj11−jj⎝⎠−

It follows that 3 m = 0, nK=⋅()j2 −1 2 (K= integration const.), 3 The Method of CECH 83

Thus we have,

Theorem 3.10.1: If the principal curves (cf. 2.4) , i.e. the intersection between the principal surface and the flecnode surfaces, are the flecnode curves on the principal surface Σ1 of the flecnode congruence of Σ, then the surface Σ belongs to a linear complex (cf. KLAPKA [33]).

The principal surface Σ1 is a quadric if, and only if, B = 0, i.e. from (3.123), iff 4δ nj22− j′′ 3jj′ 2−+22=0. (3.124c) j −1 ()j 2 −1

Let us put dj dp j′ ==p and jp′′ =⋅ dv dj then (3.124c) takes the form

dpj22 223 .3pp−⋅2 =2()j−1+4δ Kj()j−1, dj j −1 therefore, it follows

223 ⎧⎫jj+12 22 pj=−()1l⎨⎬n + 2 +4δ Kj+K1 , ⎩⎭j −1 1− j where K1 is the constant of integration. Thus, one can say,

Theorem 3.10.2: There exists ∞2 projective different ruled surfaces Σ, for which the principal surfaces Σ1 of the flecnode congruence are quadrics. Each surface of this system belongs to a linear complex (m = 0) and moreover, it is valid that

3 nK=⋅()j2 −1,2 dj v = , (3.125) ∫ 3 jj+12 jK22−+1l2 n +4δ j2+K () j −1 1− j 2 1

where K and K1 are the constants of integration.

∗) 3.11 Some remarks on the ruled surface ΣF

The ruled surface ΣF is generated by the line gF of intersection of the osculating planes of the flecnode curves of the surface Σ. As we explained before, ΣF is a developable if, and only if, at least one of the flecnode curves of Σ is a plane curve. KLAPKA [34] gave a proof this theorem using the formulae of CECH.

∗) cf. § 2.12 3 The Method of CECH 84

Let ccy1+c2z be a given curve on Σ , then this curve will be a plane curve if the ratio c12: c satisfies that

43 22 3 4 ϕ ()cc12, =+a0c1 a1c1c2+a2c1c2+a3c1c2+a4c2 (3.126) =+cy cz,,cy+cz′′cy+cz ′,cy+cz′′′=0 ()12()12(12)()12 where

2222 an0 =−23n′′ n′ +4n()2n−2m′n+mn′ +4n(m′ +δ n−m−1−j), 2222 an4 =−23n′′ n′ −4mnn′ +4n()m′ +δ n−m−1−j, ′ 3mm1⎛⎞aa40− δ aa14=+()−a0−2()a0+a4−⎜⎟, (3.127) ( 2nn) 2⎝⎠n

2 ′ nm−−33n′′nm ⎛⎞m m⎛⎞aa40− aa24=−()a0+⎜⎟δ +2,()a0+a4+⎜⎟ 2nn32⎝⎠ nn⎝⎠ ′ 3 mm1 ⎛⎞aa40− aa34=−()−a0−2 ()a0+a4 − ⎜⎟. ( 2 nn) 2 ⎝⎠n

Thus, cy is a plane curve if, and only if, a0 = 0, and similarly, cz is a plane curve if, and only if, a4 = 0.

The generator gF joins the points yn=+′′y2ny, (3.128) zn=+′′z2,nz where, from (3.100), y is the point of intersection of the tangent ( yy, ′) of the flecnode curve cy with the osculating plane ζ = (zz,,′ z′′ ), and similarly, z is the point of intersection of the tangent (zz, ′) of cz with the plane η = ( yy,,′ y′′ ). Thus, the necessary and sufficient condition in order that ΣF be a developable is ω ==()yz,,y′′,z 0 i.e. ω =+()ny′ 2,ny′ nz′ +2nz′, n′′y+3ny′ ′ +2ny′′, n′′ +3n′z′ +2nz′′ =0.

From (3.100), one finds then

ω =aa04(y,,z y′′,z)=0 (3.129) where (yz,,y′′,z)≠ 0, since Σ is not a developable.

To find the differential equations representing the surface ΣF , one can differentiate (3.128), then eliminate y′′ and z′′ using (3.100).

3 The Method of CECH 85

Finally, one finds

⎛⎞a′ ym′′ =+⎜⎟4 22y′ +δ nz′ ⎝⎠a4

1 ⎛⎞222aa44′′ ++2 ⎜⎟an4 31′′+2mnn+8nm′+6nn′′−8nm−6nn′y′ 4n ⎝⎠aa44

⎛⎞a4′ +−δ ⎜⎟52nn′ z, (3.130) ⎝⎠a4

⎛⎞aa′′⎛ ⎞ zn′′ =−22y′ +⎜⎟00− mz′ +⎜2n−5n′ ⎟y ⎝⎠aa00⎝ ⎠

1 ⎛⎞222a0′ a0′ ++2 ⎜⎟an0 31′′−2mnn−8m′n+6nn′′+8mn−6nn′z. 4n ⎝⎠a0 a0

To simplify the calculations, let us put

y z y1 = , z1 = (3.131) 4 4 aa04 aa04 where the denominator is real and positive. Thus, one finds easily from (3.129) that

()yz11,,yz11′′, =±(y,z,y′,z′)=±1, and equations (3.130) take then the form

⎡⎤a′ a′ ′′ 1 ⎛⎞4 0 ′ ′ ym11=+⎢⎥22⎜⎟− y+δ nz1 ⎣⎦2 ⎝⎠aa40 ⎡ aa′ ′ ++1 ⎛⎞an3 ′′22+12mnn+8nm′+6nn′′−8n2m44−6nn′ ⎢ 2 ⎜⎟4 ⎣4n ⎝⎠aa44 ′′2 ′′ ()aa04 a′ ()aa04 ()aa04 ⎤ ++13⎛⎞4 2my+−1 ⎜⎟22 ⎥ 1 41aa04 ⎝⎠a4 6aa04 4a0a4 ⎦

⎡⎤ ′ aa ′ ⎢⎥a4 1 ()04 +δ 5n′ −+2,nn z1 (3.132) ⎣⎦⎢⎥aa402 a4

⎡ ′ ⎤ ⎡⎤aa′′a′ ()aa04 ′′ ′ 11⎛⎞004 ′ ⎢ ′ ⎥ zn11=−22y+⎢⎥⎜⎟− − mz1+ −5n+2n− n y1 ⎣⎦22⎝⎠aa04 ⎣⎢ a0 a0a4⎦⎥

⎡ aa′ ′ ++1 ⎛⎞an3 ′′22−12mnn−8m′n+6nn′′+8mn200−6nn′ ⎢ 2 ⎜⎟0 ⎣4n ⎝⎠aa00

′ ′′2 ′ 1 ()aa04 ⎛⎞a0′ 31()aa04 ()aa04 ⎤ + ⎜⎟−+2.mz22 − ⎥ 1 4 aa04 ⎝⎠aa0016 aa04 4 a4⎦

3 The Method of CECH 86

Comparing these equations with (Ac) of CECH, one finds for ΣF :

′ 11⎛⎞aa00⎛′ 3a4′ ⎞ an==δδ,lb ⎜⎟n−m,c=n,A=4n′ +n⎜−⎟, 42⎝⎠aa40⎝ 2a4⎠

⎛⎞13a4′ a0′ Cn=+4,′ n⎜⎟− ⎝⎠22aa40 a′ a′ Bm=−12′ −2mn′ −3 m⎛⎞0 +4 na2 ⎜⎟a ⎝⎠04 (3.133) 2 2 ′ 31n′ ⎛⎞aa00′′aa44′′⎛⎞1⎛⎞a0′a4′ −−⎜⎟+⎜⎟22−−⎜⎟−, 28na⎝⎠04a ⎝⎠aa044⎝⎠a04a

2 35n′ ⎛⎞aa00′′aa44′′⎛⎞ jj= ++⎜⎟−⎜⎟+ 41na⎝⎠04a 6⎝⎠a04a ′′ 2 11()aa04 ⎛⎞aa00′′aa44′′1⎛⎞ +−⎜⎟−−m⎜⎟−. 41a0a4 6⎝⎠aa04 2⎝⎠aa04

The ruled surface ΣF is a quadric, if AB==C=0 (3.134) hence, one finds from (3.133) that

⎛⎞a′ a′ δ AC−=2,n⎜⎟0 −4 ⎝⎠aa04 from which ( n = 0 is excluded),

aa04:= λ :µ (3.135) where λ and µ are constants ≠ 0. Taking this relation into account, then equations (3.134) a′ Cn=−40′ n0 =, a 0 a′ An=−40′ n4 =, a4 give 4 4 a0 = λn, a4 = µn. (3.136)

From B = 0, it results that ()λµ−−nn3 88m′+mn′=0. (3.137)

Thus, the conditions in order that ΣF be a quadric are expressed by the equations (3.136) and m (3.137). Let us put n = α, then (3.137) takes the form

8α′ =−(λµ)n. (3.138) 3 The Method of CECH 87

We are going to exclude the case λ −=µ 0, as it involves n = 0, then it remains

n = 8α′ , m = 8αα′ . (3.139) λ − µ λ − µ From the relation

4 aa04−=()λµ−n, that is to say 8(nm−+′′nmn)=(λµ−)n3 thus, one finds from (3.139) 16α′′3 =−()λµα.

One can exclude α′ = 0 because of the assumption n ≠ 0, then the last relation gives

16α′2 =−λµ hence λ − µ α′ = 4 from which then

λ − µ α = v. 4 where the constant of integration can be neglected. Substituting in (3.139), then one finds the two invariants

n = 2 , m = 1 v. λ − µ 2

Finally, the relation of the third invariant j takes the form

8δ −−λµ j = , 2()λ − µ where δ =±1. We notice that the invariants n and j do not depend on v, i.e. they are constants. Assuming all the invariants are given, then the canonical flecnodal system of equations representing this class of the surface Σ takes the form

4δ ⎛⎞v 2 µ yy′′ =+v ′ z′ +⎜⎟−+ y, λµ− ⎝⎠4 λµ− (3.140) 4 ⎛⎞v 2 λ zy′′ =− ′ −v z′ +⎜⎟− + z λµ− ⎝⎠4 λ − µ where λ ≠ 0, µ ≠ 0, λ ≠ µ , δ =±1. One notices that each surface Σ of this class is determined by two constants λ, µ and a sign δ = ±1. 3 The Method of CECH 88

Moreover, equations (3.133) give

a = 2δ , b =−1 v, c = 2 . (3.141) λ − µ 2 λ − µ Finally, expressing j from (3.133), then (3.132) take the form 4δ λ4λ yy11′′ =+vv′ z1′ + y1,.z1′′ =− y1′ −z1′ + z1 (3.142) λµ−−λ −−µλµ λµ

From (3.133), (3.135), one finds that a= a, b= b, cc= , these relations give the following geometric property of this class of the ruled surfaces Σ,

Theorem 3.11.1: Let Σ be a ruled surface, and let ΣF be the corresponding ruled surface generated by the line of intersection of the osculating planes of the flecnode curves of Σ. Let

ΣF be a quadric, and let also x be an arbitrary point describing an asymptotic line on Σ. Then, the tangent plane of Σ at the point x of a generator g of Σ intersects the corresponding generator gF of ΣF in a point x which describes a line of the complementary regulus of Σ . F

According to WILCZYNSKI, the necessary and sufficient condition in order that the generator gF of ΣF touches the osculating quadric QΣ of Σ along g, i.e. the quadric generated by the asymptotic tangents of Σ along the points of g, is that

32 3 θθ49+16θ10=0. (3.143)

WILCZYNSKI considered this property as a characteristic of a class of ruled surfaces. To study this class using the calculations of CECH, let us consider the line

rt=+()12ytz, t1y+t2z, which touches Σ in ty1+ t2z and intersects gF in the point ty1+ t2z. The asymptotic tangent ′ of Σ in ty1+ t2z passes also through (ty12+ tz), where one can substitute t1′ and t2′ from (3.48), and the quantities a, b, c are given by (3.98). Thus, the considered point is

′ ()ty12+=tz (−mt1+nt2)y+(−δ nt1+mt2)y+ty1′ +t2z′. (3.144)

In order that r be an asymptotic tangent of Σ, it is necessary and sufficient that the matrix

⎛⎞tt1200 ⎜⎟′′ ⎜⎟n t12n t 22nt1nt2, ⎜⎟ ⎝⎠−+mt12nt −δ nt1+mt2t1 t2 must be of Rank 2. Since the case n = 0 is excluded, then the required condition is reduced to

22 δ nt11−+20mt t2nt2=, or, from (3.98),

22 at11++2bt t2ct2=0. (3.145) 3 The Method of CECH 89

Thus, if the point ty1+ t2z is a fundamental point (cf. § 4.8) with respect to the principal family of curves on Σ , then the line r is an asymptotic tangent of Σ and reciprocally.

Moreover, the points of intersection of the osculating quadric QΣ with gF are to be found on the asymptotic tangents at the fundamental points of the principal family on the corresponding generator g. It follows that the considered points of intersection separate harmonically the two points in which gF intersects the tangent planes of Σ at the harmonic points. Just if Σ belongs to a linear complex ( m = 0 ), the points of intersection of gF with QΣ coincide with the two points in which gF intersects the tangent planes of Σ at the LIE-points. The case of WILCZYNSKI takes place, if the equation (3.145) has double root, i.e. if

m2= δ n2. (3.146)

Thus, one can say that if for all generators g of a ruled surface Σ the osculating quadric QΣ touches the corresponding generator gF of ΣF , then the harmonic curve of Σ is an asymptotic curve. The point of contact of gF and QΣ is to be found on the tangent of the harmonic curve at its point of intersection with g . Actually, the existence of a harmonic curve which is in the same time asymptotic, is a characteristic for a class of ruled surfaces, the G-surfaces considered by MAYER (cf. § 4.9). So, one has the following result:

Theorem 3.11.2: The surfaces of WILCZYNSKI for which the osculating quadric QΣ touches the corresponding generator gF of ΣF , and the special class of the G-surfaces of MAYER (cf. §4.9), characterised by the existence of a harmonic curve which is in the same time asymptotic, are the same. One of these two properties, which is granted, involves the other.

3.12 The conjugate and the bi-conjugate pair of curves on a ruled surface

A. TERRACINI [64] used the method of WILCZYNSKI to study the conjugacy condition for a pair of curves on a ruled surface. He described the surface by the differential equations yl′′ =+y′ lz′ +ly+lz, 1234 (3.147) zm′′ =+12y′ mz′ +m3y+m4z.

From that, the conjugacy condition for the directrix curves cy , cz of the ruled surface Σ is found to be

lm21()l1+m2−−+lm21′′lm21 2(lm41+lm23)=0. (3.148)

For the bi-conjugacy condition, TERRACINI found a special case. Assuming the conditions

lm21==1, lm12+=0, (3.149) are satisfied, then the following two bi-conjugacy conditions for the curves c , c are valid, y z (1) the conjugacy condition (3.148), which from (3.149), takes the form

lm43+=0 (3.150)

3 The Method of CECH 90

(2) the condition

3ll11(′′+−l4 l3+m4)−l1′l4=0. (3.151)

According to the calculations of CECH, we have,

Theorem 3.12.1: The directrix curves c,y cz of a ruled surface Σ described by the equations (Ac) are conjugate if, and only if, the condition aC +cA =0 (3.152) is satisfied (cf. KLAPKA [35]).

This relation is obvious, when one substitutes the coefficients li , mi (i =1,2,3,4) in (3.148) by the corresponding expressions, which one finds by comparing the coefficients of the systems (3.147) and (Ac), assuming that θ′ = 0. Thus, for the bi-conjugacy condition, one has,

Theorem 3.12.2: There exists two necessary and sufficient conditions for the bi-conjugacy condition of the non-asymptotic directrix curves cy , cz of a ruled surface described by the system (Ac):

(1) the conjugacy condition (3.152)

(2) the condition 3ρ′′NN−+ρ 12NB=0 (3.153) where

ρ ==A−C, (3.154) ac and Na=−′′cac+4abc. (3.155)

The previous conjugacy and bi-conjugacy conditions can be generalized for any arbitrary pair of curves cyu+ 1z, cyu+ 2 z on a ruled surface Σ by the following theorem,

Theorem 3.12.3: The curves cyu+ 1z, cyu+ 2 z on a ruled surface Σ, which is described by the system (Ac), are conjugate, when the condition aC∗∗+c∗A∗=0 (3.156)

is satisfied. The non-asymptotic curves cyu+ 1z, cyu+ 2 z on the same surface are bi-conjugate, when

(1) the conjugacy condition (3.156) is valid, and

(2) the condition 31ρρ∗∗′′NN−+∗∗ 2N∗B∗=0 (3.157) is satisfied, where, taking into account the relations (3.16), (3.17) and (3.18), 3 The Method of CECH 91

uf11′ + ()u a∗ =− , uu12−

uu12′′++2 ⎡⎤a+b()uu12+ +cu1u2 2,b∗ =− ⎣⎦ uu12−

uf22′ + ()u c∗ =− , uu12−

Fu()1 A∗ =− , (3.158) uu12−

AB++()u12u+Cu1u2 B∗ =− , uu12+

Fu()2 C∗ =− , uu12− Na∗∗=−′′c∗a∗c∗+4,a∗b∗c∗

and

∗∗ ∗∗AC Fu()12Fu( ) ρρ==∗∗−⇒= =− . ac uf11′′++()u u2f()u2

We notice that, if

fu()1=fu(2)=0, (3.159) then u′ a∗ =− 1 , uu12−

∗ u1′+ u2′ 2bc=−()u12u− , (3.160) u12− u u′ c∗ =− 2 , uu12− where as the expressions for A∗ , B∗ , C∗ remain unchanged. The conjugacy condition takes then the form

uF12′′()u +u2F(u1) =0. (3.161)

From Mayer [49] (cf.§ 4.7), one finds that the harmonic transformation of the pair of directrix curves cy , cz of the ruled surface Σ is the pair cyu+ 1z, cyu+ 2 z, where u1, u2 are the roots of the equations

ac+u2 =0 (3.162) and ac ≠ 0. Thus, it follows

uu+=0, uu = a . (3.163) 12 12 c

3 The Method of CECH 92

Now, if cy , cz are a conjugate pair of curves, then it follows that aC +cA =0, and it follows from (3.16), (3.17), (3.18) and (3.162) the theorem,

Theorem 3.12.4: A pair of distinct non-asymptotic curves on a ruled surface Σ consists of conjugate curves if, and only if, its harmonic transformation pair separates the flecnode pair of curves on Σ harmonically.

If the flecnode curves coincide together, then it is necessary and sufficient that one of the transformed pair coincides with those. This theorem is obvious, when one notices that the harmonisant of the flecnode form (3.18), and of the form on the left hand side of (3.162) is the left hand side of (3.152).

Let cy , cz are a conjugate pair of curves, and let its harmonic transformation is the conjugate pair cyu+ 1z, cyu+ 2z then besides the condition (3.152), also the equations (3.163), (3.156) are valid, where (3.156), from the previous equations and (3.158), takes the form BN = 0, (3.164) where N is defined in (3.155). But N = 0, assuming the ruled surface is referred to its asymptotic lines, leads to k = 0 which means from (3.67) that the ruled surface Σ belongs to a linear complex. Thus, we have

Theorem 3.12.5: Let two distinct conjugate non-asymptotic pairs of curves are given on a ruled surface Σ, where one of this pair is the harmonic transformation of the other, then it follows that: (1) either the ruled surface Σ belongs to a linear complex, or (2) both the considered pairs of curves form with the flecnode pair a three pairs of curves on the surface, which mutually separate each other harmonically. (It is known that in this case just two pairs of the three are real).

Moreover, if the two distinct branches of the flecnode curve on a ruled surface are bi- conjugate, then the surface belongs to a linear complex.

Since, when cy and cz are the flecnode curves, then it must be AC= = 0, B ≠ 0, ρ = 0, therefore the bi-conjugacy condition (3.153) is satisfied if N = 0, i.e. k = 0.

4 Flecnode curves and LIE-curves in affine space

Using the affine geometric formulae of a space curve c in the affine space, studied O. MAYER [48] a ruled surface Σ which has c as an asymptotic line and deduced the equation of its flecnodes. MAYER used the deduced formulae to study some special cases like the case if c is a TZITZEICA’s curve, the case if Σ has distinct planar flecnodes, and the case if the two flecnodes lie in parallel planes. MAYER defined also the harmonic transformation and applied it [49] on a R-family of curves on Σ, which is a set of curves that intersects the generators of Σ in projective rows points. Particularly, the R-family which contains the flecnodes, the LIE-curves and the involute curves; he called it the principal family of Σ.

4.1 The fundamental equations – The flecnode curves

Let X = xs(), Yy= ()s, Z = zs() (4.1) be the equations of a space curve c with respect to its affine arc length as a parameter; one can choose the parameter s in such a way that ()xx′,,′′ x′′′ =1. One will then have the fundamental identities xkiv ++x′′ τ x′ =0, etc. (4.2) where k and τ are the affine curvature and the affine torsion∗) respectively. Let us call the tetrahedron relative to a point M (s) of c which defined by the vectors whose components are ()x′′,,yz′, ()x′′,,yz′′ ′′ , (x′′′,,yz′′′ ′′′ ) and with origin at M as the affine principal tetrahedron. If we designate by u, v, w the coordinates of a point PX= (,,YZ) in the Cartesian system defined by the precedent unit vectors, then one will have the formulae

X =+xx′u+x′′v +x′′′w, etc. in which uX=−()x,,x′′ x′′′ , v =−()X xx,,′′′ x′ , wX=−( x,,x′ x′′ ).

Hence, the necessary and sufficient conditions in order that the point Pu= ( ,,v w) be fixed are

u′ =τ w−1, v′ =kw −u, w′ = −v. (4.3)

Now let Σ be a ruled surface admitting the curve c as an asymptotic line. One can represent it by equations of the form X =+xr(ωx′+x′′), etc. where ω(s) is a projective parameter which fixes the positions of each generator of Σ in the corresponding osculating plane of the curve c.

∗) cf. BLASCHKE [5] p.72 4 Flecnode curves and LIE-curves in affine space 94

The osculating quadric QΣ along the generator g through the point M (s) of c, contains the generator g1 through the point M1(s+ ds), that infinitely small of order three closure. The equations of g1 are

Xx=+r()ωωx′ +x′′ +⎣⎦⎡⎤x′ +r(′ x′ +ωx′′ +x′′′ )ds

1 ′′ ′′ ′ ′ ′′ ′′′ 2 ++2 {}xr⎣⎦⎡⎤()ωτ−x+(2ω−k)x+ωx ds+…, etc. or, in the principal tetrahedron of M ,

1 2 ur=+ωω()1d+r′′s+2 r()ω′−τds+…, v 1 ′ 2 =+rrωωd1s+2 ⎣⎡+r()2−k⎦⎤ds+…, (4.4) 1 2 wr=+dds2 rω s+…

As QΣ contains the generator g and the tangent of c in M , its equation can be finally written in the form (cf. [48] p.29)

2 2vv⎣⎡uk−+ω ()Ω−w⎦⎤+Hw−2w=0, (4.5) where

Ω=−ωω′ + 2 +k, Hh=Ω′′−ω Ω− =−ωω′+3ω′−ω3 −kω+τ, (4.6) hk=−′ τ , one can verify easily that H is the average affine curvature of the ruled surface. From this equation, it can be proved that the osculating quadric QΣ is bound in an invariant affinity to the affine principal tetrahedron of the curve c, if the following conditions are satisfied ω = const., kω −=τ const. (4.7) i.e. the affine curvature and the affine torsion must verify a linear relation with constant coefficients. Reciprocally, if such a relation αk ++βτ γ =0 is satisfied, the curve c is an asymptotic on a ruled surface for which the generator and the osculating quadric are bound in an invariant affinity to the principal tetrahedron of c.

From the equation of the osculating quadric, one can find the equation of the flecnode lines using the following remark: The characteristic of the osculating quadric of a ruled surface is composed of the generator of contact counted twice and the corresponding flecnode tangents. In our case, the characteristic of QΣ is represented by equation (4.5) and its derivative with respect to s :

2 2 22()uk−+ωωvv(Ω−)()u−w−⎣⎡H′ −2k(Ω−k)⎦⎤w=0. (4.8)

By eliminating (u −ωv ) of these two equations, one obtains

1 2 ()2 Hw +Ωvv−2Hw −H′ −2Ωv+2=0. (4.9) 4 Flecnode curves and LIE-curves in affine space 95

This represents obviously two planes which do not contain the generator g, but which must pass through the flecnode tangents; in the intersecting with g, that is to say, by setting in the last equation ur= ω, v = r, w = 0 one obtains the required equation of the flecnode lines:

Hr′ 2 +Ω2r−2=0. (4.10)

Among the ruled surfaces which admit a given asymptotic c, it is appropriate to consider particularly those which satisfy the conditions Ω = 0 or H′ = 0. (A) The condition Ω=0, or from (4.6) ωω′ =+2 k characterizes the ruled surfaces on which the asymptotic c is in the same time the average (mean) curve of the flecnodes (i.e. the locus of the mediums of the segments intercepted by the flecnodes on the generators). As the previous equation is of the RICCATI-type, it follows that:

Theorem 4.1.1: A given curve c is an asymptotic and a mean (average) curve of the flecnodes for ∞1 ruled surfaces; the generators of four of these surfaces form, in each osculating plane of c, a constant anharmonic ratio along the curve c. We are going to call such a family of surfaces as R-families.

(B) The condition H =Ω′ −ω Ω−h=−a (const.), or ωω′′ −+3ω′ ω3 +kω−τ−a=0 (4.11) defines the ruled surfaces for which one flecnode is situated at infinity.

Theorem 4.1.2: There exists thus ∞3 ruled surfaces having a given asymptotic and a flecnode situated in a given plane.

For ω = const., equation (4.11) becomes kω −=τ a′ (const.); which is the same (4.7). Thus we conclude that:

Theorem 4.1.3: If a curve c satisfies a relation of the form αk ++βτ γ =0, (α, β, γ are constants) (4.12) it will be an asymptotic curve on a ruled surface Σ, which has the following properties: (1) the generators of Σ are bound in an affine invariant way to the principal tetrahedrons of c. (2) and the same is true for its osculating quadrics. (3) it has a flecnode curve which is infinitely distant. Conversely, two of these properties lead to the third and the relation (4.12). 4 Flecnode curves and LIE-curves in affine space 96

In particular, the curves having constant affine torsion are characterized by the property (B) of the surface generated by its affine principal normals (cf. MAEDA [47]). The case τ = 0 was found by CECH, and thus the surface under consideration will have a plane directrix; that results from H = 0. Moreover one can prove the following (cf. [48]),

Theorem 4.1.4: If a flecnode of the ruled surface Σ be found in a plane π , the characteristic point of the tangent plane of Σ along the considered flecnode is the pole of the planeπ relative to the osculating quadric of Σ.

4.2 TZITZEICA’s curves

If the conditions (A) and (B) are simultaneously satisfied by a surface Σ, the two flecnodes of Σ will be at infinity and inversely. Besides, in order that the two conditions be compatible, it is necessary and sufficient that the curve c verify the relation hk=−′ τ =a (const.) (4.13) thus the solutions of (A) verify also (B). Let us denote the curves which satisfy this condition by Ta , then for such curves we have the following two propositions: – The asymptotics of a ruled surface has coincident and infinitely distant flecnode curves

are of the type Ta -curves corresponding to the same value of the constant a. – If, for a given curve, one can find a ruled surface, which admits it as an asymptotic and for which the flecnodes coincide in a given plane, then one can find ∞1 of such surfaces, and those surfaces form a R-family.

Actually, these Ta -curves ( a ≠ 0 ) are found by G. TZITZEICA [68] who defined them by means of the relation

τ d 2 = const.

1 where τ and d designate respectively the Euclidean torsion and the distance of the osculating plane from a fixed point. As for the curves To , which he defined by the metric relation (cf. [69]), τθcos2 = const. where θ is the angle of the binormal with a fixed direction. Those To -curves are characterized by that they belong to a linear complex whose axis coincide with a fixed direction. G. LORIA called both kinds of those curves as T-curves. For the T-curves we have the following characteristic geometric property, which is clearly affine and applicable in all the cases:

The Ta -curves are those for which the affine rectifying planes pass through a fixed point, which is infinitely distant for the To -curves.

Let us consider the asymptotic plane of the surface Σ, ()Xx−+,,ωωx′ x′′ ′ x′ +ωx′′ +x′′′ =0. 4 Flecnode curves and LIE-curves in affine space 97

This plane passes through a fixed point O taken as the origin of coordinates, if ω is a solution of the RICCATI-equation: ()ωω′ −−2 ()xx′x′′ ω(xx′x′′′ )−( xx′′x′′′ ) =0 (4.14)

Thus,

Theorem 4.2.1: A curve c is an asymptotic for ∞1 ruled surfaces admit an asymptotic cone with a given centre O, and these surfaces form a R-family.

Moreover, (cf. [48] p.37)

Theorem 4.2.2: If a T-curve contains the point O, the ruled surfaces which admit an asymptotic cone with centre O and the given T-curve as an asymptotic, have the two flecnodes in infinity.

And conversely,

A ruled surface has infinitely distant flecnodes admits an asymptotic cone, and all its asymptotic lines contain the centre of that cone.

Consider a ruled surface Σ having the asymptotic c, and having one plane branch flecnode (that we suppose at infinity) and the other is conical (this term, introduced by PETERSON∗), signifies that the circumscribed developable along the considered curve is a cone). Σ will have a flecnode at infinity if H = 2a (const.), (4.15) and the other flecnode will be defined by the equation

r = 1 . Ω The tangent plane along the last curve, ()Xx−+, ωωx′′x′, ()′−Ωx′+ωx′′′+x′′=0, passes through the point O, the origin of the coordinates, if one has kx()x′x′′ −−ω (xx′x′′′ )(xx′′x′′′ )=0 (4.16)

We notice that this equation is verified for all the values of ω, if the curve c is a T-curve passing through the origin O.

∗) K. M. PETERSON, „ Sur les relations et les affinites entre les surfaces courbes”, Toulouse Ann. (2) 7, 5-43, 1905. 4 Flecnode curves and LIE-curves in affine space 98

Hence, it follows,

Theorem 4.2.3: A T-curve contains the point O is an asymptotic for ∞3 ruled surfaces having one flecnode at infinity, and the other is conical of centre O; the first condition suffices to determine these surfaces.

4.3 Ruled surfaces with distinct planar flecnode branches

Let Z =∞ and Z = 0 be the planes which contain the flecnode curves. One has, along the second flecnode curve zr++(ωz′′z′)=0, Ω−r 1=0, from which by eliminating r, it follows

−Ω = ωω′ − 2 − k = ωz′+ z′′. (4.17) z For this condition, it is also necessary to add that one, which expresses that, the first flecnode is at ∞ ; H =Ω′ −ω Ω−h=2c (const.). (4.18)

The ruled surfaces, which admit the asymptotic c and for which the flecnodes be found in the planes Z =∞ and Z = 0 , correspond to the solutions of the system (4.17), (4.18). Let us find out under which conditions this system is compatible. From (4.17), one finds

ωω′ =+2 ωzz+ ′′ +k, z from which ()ωω2 ++kz′ z′′ +z′′′ Ω=′ − z and by substituting in (4.18), one finds zk′′′ ++z′ ()h+2cz=0. (4.19) If this condition is satisfied by the curve c, all the integrals of (4.17) verify also (4.18). Therefore:

Theorem 4.3.1: The asymptotics of a ruled surface having the flecnodes in the planes Z = ∞ and Z = 0 satisfy the condition (4.19). And inversely, if a curve fulfils this condition, it is an asymptotic for ∞1 ruled surfaces for which the flecnode branches be found in the previous planes, and these surfaces form a R-family.

The condition (4.19) can be geometrically interpreted as follows:

In order that a curve c to be an asymptotic curve on a ruled surface which has two flecnode curves lying in two given planes π1 and π 2 , it is necessary and sufficient that the rectifying points of c relative to the fundamental planes π1 , π 2 be conjugate to two fixed planes ε1 and

ε 2 , which themselves are conjugate to π1 and π 2. 4 Flecnode curves and LIE-curves in affine space 99

Let us suppose that the flecnode e1 of the ruled surface Σ be found in the plane π1 ; g be a generator of Σ , c be the asymptotic through a point M of g, q1 the rectifying line of c relative to the fundamental plane π1, N1 the corresponding rectifying point, ϕ1 the tangent plane of Σ in the flecnode point F2 , and n1 its characteristic line in the displacement of F2 on the flecnode e2. As the plane ϕ1 contains the line q1 , the point N1 is the intersection of the lines q1 and n1. It is known that, when M moves on g, the rectifying line q1 remains tangent to a conic c1 which touches respectively the generator g and the flecnode tangent f2 in the points F1 and P1 where those intersect the plane π1. One obtains then the rectifying point N1 corresponding to the point M of g by intersecting the line n1 with the tangent different from g that one can draw from M to the conic c1. ∗ Inversely, each point N1 of n1 corresponds to a couple of points (M , M ) of g, which describes an involution I that contains the couple of the flecnode points (F12,F ). Moreover, the involution J contains the couple of the LIE-points (L12,L ), that is to say, the couple of points for which the polar planes relative to the osculating linear complex K of Σ are tangents to Σ. To proof that, we have first the following remark: The tangent of a one branch of the flecnode curve, and the characteristic line of the tangent plane along the other branch are reciprocally polar with respect to K. For, one observes that, the polar planes relative to K of two consecutive flecnode points F1, F1′, are precisely the tangents planes at the other corresponding flecnode points F2 , F2′. One concludes of that remark, that the pole of the plane π1 relative to the complex K is the point P1 common to π1 and n1. On the other hand, it is known that the complex K belongs to a bundle of the osculating linear complexes of the asymptotics, whose base congruence is the osculating linear congruence of the ruled surface Σ, whose directrices are the flecnode tangents. That is, according to the definition of the points (L12,L ), the osculating linear complex of the asymptotics issued in these points. It follows that the corresponding rectifying lines are L1P1 and L21P , and that their rectifying points coincide with P1, that which shows our assertion: the involution J is defined by the couples of points (F1, F2) and (L12,L ). Its double points H1, H 2 are simultaneously conjugate to the previous couples; they are the harmonic points (involute points). Moreover, if we suppose that the second branch of the flecnode curve e2 be also found in a plane π 2 , then we have the following theorem,

Theorem 4.3.2: If the flecnode curves of a ruled surface Σ be found in two given planes π1,

π 2 , then the rectifying points of the asymptotics of Σ relative to the fundamental planes π1 and π 2 are conjugate to two fixed planes ε1, ε 2 which contain the harmonic curves of Σ.

4.4 A ruled surface with one rectilinear flecnode branch

Let us represent the ruled surface Σ by the parameteric equations of the form

xu()vv+ x() yu()v+ y()v X = 12, Y = 12, Z = γ 1− u (4.20) 1+ u 1+ u 1+ u where we take v = const. as the generators, and u = const. as the sections of Σ made by the bundle of parallel planes Z = const. Among those, let us consider the two curves

Xx= 1()v , Yy= 1()v , Z = γ (4.21)

4 Flecnode curves and LIE-curves in affine space 100 and

Xx= 2 ()v , Yy= 2 ()v , Z = −γ (4.21a) which correspond respectively to u = 0 and u = ∞. The differential equation of the asymptotic lines can be easily found to be

2λ du ++Dvv2 2∆+D=0 (4.22) dv 21 where

Dx1= (1′′x1′), Dx2= ()2′x2′′ ,

2,∆=()x12′′xx′+(2′x1′′) λ = ()x12′x′ .

Since the asymptotic lines have stationary tangents at the points of intersection with the flecnode curves, one can find from (4.22) the equation of the flecnode curves in the form

⎛⎞D⎛⎞ ⎛⎞D dd⎜⎟1++2uu⎜⎟∆ 2 d⎜⎟2=0. (4.23) ddvv⎜⎟33⎜⎟ dv⎜⎟3 ⎝⎠λλ22⎝⎠ ⎝⎠λ2

Now, let us take the curves u = 0 and u = ∞ as the flecnode curves in the representation (4.20) of the ruled surface. One will have then

D1 D2 3 = const., 3 = const. (4.24) λ 2 λ 2

Inversely, if the plane curves (4.21) and (4.21a) satisfy the conditions (4.24), they will be the flecnodes of the ruled surface formed by the lines which join the couple of corresponding points at the same values of the parameter v on the two curves. The conditions (4.24) are satisfied if the two curves are straight lines which correspond to each other in any arbitrary way, as one has then DD12= = 0, which is obvious.

Let us consider the case where a single flecnode, for example the flecnode e2 , is a straight line ( D1 ≠ 0, D2 = 0 ). Let

Yy=2 =0, Z =−γ be its equations, and let the affine arc length of the flecnode e1 be chosen as the parameter v , which amounts taking D1 = 1; the conditions (4.24) are then reduced to

λ =−x21′′y=l (const.), of which

dv xl2 =− +const. (4.25) ∫ y1′ 4 Flecnode curves and LIE-curves in affine space 101

We notice that the flecnode e1 can be an arbitrary curve in the plane Z = γ , and that the correspondence between the flecnode points depends on two arbitrary constants. Thus we can say:

Theorem 4.4.1: A given straight line and a given plane curve are the flecnodes of ∞2 ruled surfaces, which can be obtained by two quadratures. (A second quadrature be necessary for knowing the affine arc length.) In general, for two given curves, one can find ∞2 ruled surfaces having one of these curves as a flecnode branch.

Let us suppose that none of the flecnodes (4.21), (4.21a) is rectilinear ( D1 ≠ 0, D2 ≠ 0 ).

One can then determine the parameter v in such a way to make one of the quantities D1, D2 , λ be constant; the conditions (4.24) will be thus equivalent to any two of the following three

λ =(x12′′x)=l (4.26)

Dx11=()′′x1′=m1, Dx22= ( ′′x2′) = m2 (4.27) where one of the constants lm, 1, m2 is given and the other two remain arbitrary. Let the flecnode e1 be given, and let us fix the parameter v on that curve by the condition Dm11= .

One can replace the equation D2= m2 by that one which resulted of the elimination of the derivatives y1′′, y2′′ between the equations (4.27) and the derivative of (4.26):

22 λλxx12′′′−+m12x′ xx12′′′+m2x2′ =0. (4.28)

This is a RICCATI-equation which provides us by x2′ ; from which, the equation (4.26) gives us

y2′ and it still remains to carry out two quadrature to find the functions x2 , y2 , which contain consequently five arbitrary constants. Thus,

Theorem 4.4.2: There exists ∞5 ruled surfaces which have a given plane flecnode curve and of which the other flecnode be found in an arbitrary given plane; the determination of these surfaces requires the integration of a RICCATI-equation and three quadratures.

4.5 A ruled surface whose flecnodes are situated in two parallel planes

For the affine arc length of the flecnode curves, one has from (4.26), (4.27)

v 1 1 ′′′3 3 σ11==∫ ()xx1d,vvm1()−v0 v 0 (4.29) v 1 2 ′′′3 3 σ 22==∫ ()xx2d,vvm2()−v0 v0 from which we notice that the corresponding affine arc lengths of the two flecnodes are proportional. 4 Flecnode curves and LIE-curves in affine space 102

Thus, for interpreting equation (4.26), let us introduce the notation of the tangential image∗) of a plane curve

Xx= ()v , Yy= (v),

⎛dx dy ⎞ it is the extremity of the vector ⎜, ⎟ applied at a fixed point O. Now, if one takes the ⎝⎠dσ dσ tangential images of the flecnodes with respect to the same origin O, one sees that the area S of the triangle formed by the point O and the two correspondent tangential images is constant; in fact

1 ⎛⎞ddxx12ddyy21l S =−⎜⎟=2 =const. (4.30) 2d⎝⎠σσ12d dσ2dσ1 3 2()mm12

Thus,

Theorem 4.5.1: In order that two curves situated in parallel planes be the flecnodes of a ruled surface, it is necessary and sufficient that the correspondence determined between these curves by the generators of the ruled surface satisfies the following conditions: (1) The correspondent affine arc lengths be proportional. (2) The area of the triangle formed by a fixed point O and two correspondent tangential images relative to the origin O be constant.

One can give the previous conditions in a remarkable Euclidean form. Let us designate by

s1, s2 and ρ1, ρ2 the Euclidean arc lengths and the Euclidean radii of curvature of the flecnodes e1, e2 respectively, and let θ be the angle of two correspondent tangents. One has

2222 2222 ddsx11=+(′′y1)v , ddsx22=+( ′′y2) v ,

1 ( xx11′′′) 1 ( xx22′ ′′ ) = 3 , = 3 , ρ1222 ρ2222 ()xy11′′+ ()xy22′′+

1 2222 Sx=+2 11′′yx22′′+ysinθ, and taking into account the equations (4.26), (4.27), we notice that the conditions of the previous theorem may be written as

3 ⎛⎞ds m ρ 1) ⎜⎟1 = 11 and ⎝⎠ds22mρ2

2) 1 = nsin3 θ ( n = const.). ρρ12

∗) BLASCHKE [5] p. 29 4 Flecnode curves and LIE-curves in affine space 103

4.6 The harmonic transformation

Consider, on a ruled surface Σ, two curves c1, c2 which intersect an arbitrary generator g in two points M1, M 2 respectively. Let a1, a2 be the two asymptotics through these points, and let us consider, on the consecutive generator g′, a pair of points N1, N2 which is conjugate to the pairs of intersection of g′ with the pairs of curves (cc1, 2) and (aa1, 2) . The pair

(NN1, 2) is well defined if the curves c1, c2 are not asymptotics. When g′ tends towards g, the points N1, N2 will have the limits M1, M 2 distinct from M1, M 2. By applying the same process on all the generators of Σ, one will obtain a pair of curves c1, c2 which is called the harmonic transformation of the pair (cc12,.)

Let

fu=+2U()vv2uV()+W(v)=0 (4.31) be the equation of the pair of curves (cc12,); it is equivalent to two equations of the form

uu= 1()v , uu= 2 (v) which represent separately the curves c1, c2 ; one has

uu+=−2V , uu = W . 12 U 12 U

It can be proved that the equation representing the harmonic transformation curves c1, c2 of the pair of curves f = 0 takes the form (cf. [48] p.52)

⎛∂f ⎞ fI⋅−()f,2ϕϕ⋅δ(f)−2λ⋅J⎜f⋅⎟=0, (4.32) ⎝⎠∂v where

2 ϕ =+Du22∆u+D1 (4.33) and δ , I, J indicate the discriminant, the harmonic, and the Jacobian of the quadratic forms between brackets respectively. We notice that, if one takes the curves v = const. and u = const. as the generators and the asymptotics of the ruled surface respectively, then equation (4.32) takes the form

⎛⎞δ f Jf⎜⎟,. ⎝⎠δv

The harmonic transformation is symmetric; if it is twice applied on a pair of curves, this pair is obtained again. Moreover, we are going to show that,

Theorem 4.6.1: The harmonic transformation of the flecnode curves are the LIE-curves.

Let g, g′ be two consecutive generators; (F1, F2), (F1′, F2′) be their flecnode points; H1,

H 2 be the points where the asymptotics a1 , a2 issued of F1, F2 intersect g′; finally N1, N2 be the points in the same time conjugate to the pairs (F1′, F2′) and (HH1,2). It is sufficient to 4 Flecnode curves and LIE-curves in affine space 104

prove that N1, N2 tend towards the LIE-points of g′ when g tends towards g′. Among the linear complexes which contain the osculating linear congruence along g′, there exists a ∗ complex K for which the pairs of asymptotic tangents at the points (F1′, F2′) and ()HH12, are polar. It is obvious that this complex tends towards the osculating linear complex K′ along g′. The complex K∗ defines an involution I ∗ in which correspond the point of contact and the pole relative to K∗ of a variable tangent plane passing through g′; and according to what precedes, this involution contains the pairs (F1′, F2′) and (HH1,2); its double points will be therefore N1 and N2. On the other hand, the involution I′ defined in the same way by the osculating complex K′ has as double points the LIE-points of g′; and since K′ is the limit of K∗ , I′ is the limit of I ∗, which achieve our demonstration.

We notice that, by supposing λ = const., the equation of the LIE-curves is

∂∂ϕϕ∂ ⎛⎞ ⎛⎞∂ϕ∂2ϕ δϕ()−−22ϕδ⎜⎟λJ ⎜,2 ⎟=0. (4.34) ∂∂vv ⎝⎠∂v ⎝⎠∂v ∂v

As an application on the harmonic transformation, we have the following theorem,

Theorem 4.6.2: If Σ is a ruled surface with plane flecnodes and ε1, ε 2 designate the planes which contain its harmonic curves, then the sections made in Σ by a pair of planes ()α,α′ conjugate to (ε1,ε 2) correspond by a proportionality of the affine arc lengths, and their harmonic transformations are the sections produced by the pair of planes conjugate to both pairs (α,α′) and (ε12,ε ); the osculating quadrics of Σ are conjugate to the planes ε1, ε 2 ; finally the correspondent characteristic points of the tangent planes of Σ along the flecnodes are also conjugate to the planes ε1, ε 2.

4.7 The R-family of curves on a ruled surface

Let us consider a ruled surface Σ described by the equations (3.1) from § 3.1 in homogenous projective coordinates. Assuming normal coordinates, i.e. BA2 − C=±1, and assuming also Σ is referred to its asymptotic lines, i.e. abc= ==0 (cf. § 3.4.2), then the LIE-curves under this conditions form the Jacobian pair of the couples of curves F = 0 and F′ = 0, where the last one represents the harmonic pair on the ruled surface, with F from § 3.2 equ. (3.18).

We will attach to each generator g(v) of the ruled surface a system of coordinates defined by

±=X yY12+zY +y′Y3+z′Y4, (4.35) where

YX1 = ()zy′′z, YX2 =−()yy′z′ ,

YX3 = (yzz′), YX4 =−()yzy′ .

4 Flecnode curves and LIE-curves in affine space 105

Differentiating these formulae with supposing the coordinates X are constants and by replacing the second derivatives of y, z by means of (AC) from § 3.1, we will obtain then

2 Ya11′=−()c−b−j−bY3+c1Y4, Ya′ =− Y− ac−b2 −j+bY, 213()14 (4.36)

YY31′ =− +22bY3+ cY4,

Y42′ =−Y −22aY3− bY4, where we put

aa1 =+′ A, bb1 =+′ B, cc1 = ′ + C.

A system ∞1 of curves on a ruled surface is called a R-family if these curves make a projective correspondence between the divisions that they cut on two arbitrary generators. The R-family can be described by a RICCATI differential equation of the form

uu′ ++αβ2+γu2 =0. (4.37)

A tangent to a curve of the family is ⎡ ′′ 2 ⎤ X =+yuz+r⎣ y+uz−()αβ+2,u+γuu⎦ from which 2 Y1 = 1, Yu2 =−r(2αβ+ u+γu), Y3 = r, Y4 = ur.

The tangents of the curves of the family at the points of a generator form a non degenerate quadric, whose equation can be found by eliminating r and u of the last relations,

22 αβYY33+2Y4++−γY4Y2Y3Y1Y4=0. (4.38)

This equation is reduced to

YY14−Y2Y3=0, (4.38a) if the surface is referred to the considered R-family. This quadric contains two asymptotic tangents; they are the tangents of the curves passing through the points fu()= 0 where f (u) is defined by equ.(3.17) § 3.2. The curves described by these points will be called the fundamental curves of the family. In general, they do not belong to it, but when they do, then they are asymptotics; and reciprocally.

The characteristic of the quadric of tangents is defined by the equation (4.38a) together with that one deduced by differentiation and taking into account (4.36),

22 aY13++22b13YY4 c1Y4−aY1Y3−2b(Y1Y4+Y2Y3) −2cY2Y4=0. (4.39) We notice that this characteristic is composed of the generator g and of a space cubic, which intersects g in the fundamental points and which can therefore be reducible or undetermined.

4 Flecnode curves and LIE-curves in affine space 106

The characteristic cubic decomposes: (1) in an asymptotic tangent and a conic which intersects this tangent as well as the generator g, when

2 22 ()ac11−+24bb ca1−()ac −b (a1c1−b1) =0. (4.40) In this case the R-family is called quadratic.

(2) in two asymptotic tangents and a line g1 which form with g a space quadrilateral, when abc 1==11. (4.41) abc In this case the R-family is called axial. (3) Finally, the characteristic cubic is undetermined, that is in the singular case when ab, , c, A, B, C are nulls.

It can be proved that, the condition for a curve u = const. of the family to be a plane curve is that

222 24()ff11′′−−f f fJ (f ,f1)+4(ac −b −j) f −f1=0 (4.42) where

2 f11()ua=+2b1u+c1u, and Jf(, f1 ) is the JACOBIAN of the quadratic forms f ()u , f1(u). Moreover, the condition in order that a curve u = const. be a conic is

222 24()ff22′′−−f f fJ (f , f2)+4(ac −b −−j) f f2=0 (4.43) where

2 f22()ua=+2b2u+c2u

aa2 =−′ A, bb2 =−′ B, cc2 = ′ − C.

Let us suppose that the line u = 0 be a flecnode curve (A = 0), then from (4.42) and (4.43), the conditions in order that this flecnode be plane, or conic, or coincide with the other flecnode, are respectively 23aa′′ −+a′ 224a()ab′ −ba′ +4aB +4(ac −b −j)a2=0, (4.44) 23aa′′ −+a′ 24a()ab′ −ba′ −4aB +4()ac −b2−j a2=0,B =0.

Two of these conditions lead to the third. Hence,

Theorem 4.7.1: The conditions in order that a flecnode of a ruled surface be plane and double, or conic and double or finally simultaneously plane and conic, are equivalent. 4 Flecnode curves and LIE-curves in affine space 107

4.8 The principal family of curves on a ruled surface

If a R-family contains two pairs of distinct curves which are harmonically transformed one in the other, then it possesses an involution of curves (i.e. all the couples of curves of the family u = const. cut on each generator an involution) invariant by the harmonic transformation. We are going to call it R1 -family. This family is characterized by the existence of a linear homogenous relation with constant coefficients between the quantities a, b, c of the form pa ++qb rc =0 (4.45) calculated in a system of coordinates referred to the generator and to the R1 -family, where ab, , c from § 3.1 equ. (3.12). There exists in this family two curves conjugate to the fundamental curves and reciprocally. These curves have the equation

rq−+upu2 =0, they are the double curves of the invariant involution mentioned before; moreover, they are transformed harmonically in the fundamental curves of the R1 -family. Finally, a R1 -family is well determined by the double curves of its invariant involution. It can be proved that the locus of the axes of the couples of curves which form the invariant involution is a quadric which contains the generator g. This quadric coincides with the quadric of the tangents when

ac11+=0 (4.46) or AC+=0 which shows that, on each generator, the fundamental points correspond to each other in the invariant involution.

Now, let us suppose that a R1 -family contains, in addition to the considered involution, two other couples of curves which correspond to each other in the harmonic transformation. One will have then two distinct relations of the form (4.45), and the quantities a, b, c will be proportional to the constants, that is to say the family will contain two asymptotics (which will be obviously its fundamental curves). In this case, any couple of the considered family is harmonically transformed in a couple of the same family. We say then that one has an invariant family by the harmonic transformation, and designate it by R11 -family. Thus,

Theorem 4.8.1: The necessary and sufficient condition in order that an R-family be invariant by a harmonic transformation is that it contains two asymptotics.

1 One can form with the curves of an R11 -family ∞ invariant involutions by a harmonic transformations, which have in common the couple of asymptotics of the family.

Let us suppose that a couple of the invariant involution is composed of plane curves. If these curves are u = 0, and u =∞, then we find that the considered family is axial, but if a single curve of an axial family is plane (but not fundamental), then all the curves of the family are similarly plane. Hence,

4 Flecnode curves and LIE-curves in affine space 108

Theorem 4.8.2: If the flecnode curves correspond to each other in the invariant involution of a R1 -family, and that a couple of this involution is composed of plane curves, then the considered family is obtained by cutting the surface by a bundle of planes.

As we explained before, the harmonic transformation of the flecnode curves are the LIE- curves. In the normal asymptotic coordinates, this can be immediately verified, since the harmonic transformation of the flecnode couple Fu( ) = 0 has then the equation JF(,0F′)= which represents precisely the LIE-curves (cf. § 4.7). In general the equation of the LIE-curves in any system of coordinates referred to a R-family is

1 Φ≡()uFδ ( )f()u−I()f,FF(u)+2 J(F,F′) =0. (4.47)

One can also deduce the equation of the harmonic curves, JF(, Φ=)0, or ⎣⎦⎡⎤Fu′()+−2J()f,F δ (F) F(u) I(F,F′) =0. (4.48)

If the flecnodes are distinct and non asymptotic, these curves and the LIE-curves determine a

R1 -family, that we are going to call it the principal family of the surface. Moreover, we will call the invariant involution generated by these two couples of curves as the principal involution; it has the harmonic curves as its double curves. The quadric of axes and the quadric of the tangents, relative to the same generator, coincide for the principal family, or else this family is axial. Finally, we have the following theorem (cf. [49] p.19):

Theorem 4.8.3: In order that the sections of a ruled surface by a bundle of planes form a R1 - family, it is necessary and sufficient that the osculating quadrics of the surface be conjugate to two fixed planes π1, π 2 of the bundle. On each generator, the flecnode points are conjugate to the fixed planes π1, π 2. Finally, if two corresponding curves of the principal involution are plane curves, the principal family is cut out by a bundle of planes. (A particular case of this last proposition was found by CARPENTER, cf. § 2.7).

4.9 The G-surfaces

We are going to call a ruled surface whose principal family is invariant for a harmonic transformation as a G-surface. Its principal family contains then two asymptotics (which can also coincide); but, as the principal family of a ruled surface is in all cases a R1 -family, then a G -surface is sufficiently characterized by the condition that a single asymptotic belongs to its principal family.

The two considered asymptotics form a couple of the principal involution, of which it follows that they are conjugate with the harmonic curves (that is to say the double curves of the involution). Conversely, if a ruled surface possesses two asymptotics which verify the previous condition, then it is a G -surface. Thus one can say, 4 Flecnode curves and LIE-curves in affine space 109

Theorem 4.9.1: In order that a ruled surface be of the class G, it is necessary and sufficient that it possesses two conjugate asymptotics with the harmonic curves. Moreover, if a ruled surface possesses two distinct asymptotics which divide in a constant anharmonic ratio a couple of the principal involution (for example the flecnodes or the LIE-curves), then the ruled surface is of class G.

To characterize a G -surface analytically, let us assume that the surface is referred to its principal family. According to CECH, one can normalize the coordinates in such a way that ca= ε , AC==0, B =1, (ε = ±1) or b < 0, a= c, AC==1, (4.49) according to the flecnodes are real or imaginary (cf. § 3.8); the projective invariants h and k will have respectively the following expressions:

ha= 4,ε 2 k=−16εa2b, or h= 4,b2 k=16ab2.

It will be necessary to express that the family u = const. contains the two asymptotics

ba= λ, or kh23= µ , where λ and µ are constants; that is the required condition for the G -surfaces. We notice from equations (3.112) and (3.114) in § 3.9 that the invariants j1, j2 of the flecnode surfaces of a G -surface are equal and reciprocally. Among all the G -surfaces, remarkable are those for which the principal asymptotics coincide. They correspond to

ab±=0, or kh23= 4, and are characterized by the property that the harmonic curve is asymptotic.

An another remarkable particular case is that the surfaces which belong to a linear complex. It is known that, for such surfaces the LIE-curves are asymptotics, and they are characterized by k = 0, (cf. § 3.6.1). Moreover, a surface of a linear complex is transformed in itself by the null polarity associated with the complex; it follows that one of its flecnodes is transformed, by the same polarity, in the circumscribed developable along the other flecnode. Hence, the osculating plane of a flecnode contains the characteristic point of the tangent plane of the surface along the other flecnode. And reciprocally, if a ruled surface have this property, then it belongs to a linear complex. In fact, the osculating plane of the flecnode u = 0 and the characteristic point of the tangent plane along the flecnode u = ∞ are

20aY2−a′Y4=, ()4cc2 ,′ − 4,bc0,2c, and their condition of incidence is ac′′−−a c 40abc =, or simply k = 0. 4 Flecnode curves and LIE-curves in affine space 110

Now, let us consider a ruled surface having a plane flecnode, whereas the other flecnode is a conic (cf. CARPENTER § 2.7). It is clear that, such a surface is a particular case of the G- surfaces; we are going to designate it by G0. Let u = 0 and u = ∞ be the flecnodes, then the conditions that the first will be a conic and that the second will be plane are respectively:

222 24()aa22′′−+a a a(ab2−ba2)+4(ac −b −j)a −a2=0, (4.50) 222 24()cc11′′−+c c c (bc1−cb1)+4()ac −b −j c −c1=0, or, by taking into account the relations (4.49), 23aa′′ −+a′ 224a()ab′ −ba′ +4(εa −b2−j −1)a2=0,

23aa′′ −−a′ 224a()ab′ −ba′ +4()εa −b2−j −1a2=0.

These conditions are equivalent to the following: ab′′−ba =0, (4.51)

23aa′′ −+a′ 224a (εa2−b2−j −1)=0. (4.52)

The first one characterizes the G -surfaces. Moreover, as the two conditions (4.50) , (4.51) involve the third, then one has the following theorem,

Theorem 4.9.2: If a ruled surface possesses two of the following properties: (1) belongs to a class G, (2) have a plane branch of the flecnode curve, (3) have a conic branch of the flecnode curve, then it possesses also the third and it is a G0 -surface.

4.10 The Centro-affine group

The definition of a G0 -surface is invariant with respect to the group Γ of the collineations which leaves fixed the plane of a flecnode and the center O of the circumscribed cone along the other. We will suppose from now on that, the previous plane is at ∞, and that the point O remains at a finite distance (this excludes the G0 -surfaces belonging to a linear complex); Γ will be then the group of affinity that preserves the center O. One can call it centro-affine group. To study the G0 -surfaces in the geometry of this group, it is enough to consider only the sub-group Γ1 of the group Γ.

Some preliminaries of the centro-affine geometry will be necessary. Let, in a Cartesian system with origin O,

Xxii= ()v , (i =1,2,3 ) (4.53) be the equations of a curve which does not lay in a plane passing through the origin O, such that (xx′′x′)≠ 0. (4.54) 4 Flecnode curves and LIE-curves in affine space 111

We will define the centro-affine arc length of the curve (4.53) by the invariant integral

1 sx= ∫ (x′′x′)3 dv and we are going to take the parameter v as the arc length thus defined, that is which supposes (xx′′x′)=1 or (xx′x′′′ )= 0.

One will be able then to put x′′′ =Kx′ +Tx (4.55) where Tx= (′′x′′x′′), Kx=−()x′′x′′′ are the invariants of the group Γ1 of orders 3 and 4 respectively. When the parameter v is arbitrary, the expressions of T and K are (cf. [49] p.24)

2 ()x′′xx′′′′ −5 ⎡ ()xx′x′′′ ⎤ T = , K=1 ()xxx′ ′′ 3 ⎢5 −4()xx′′x′′′ −(xxx′ ′′′ )⎥ (4.56) 2 3 3 xx′′x′ ()xx′′x′ ⎣⎢ () ⎦⎥

T is the invariant already considered by TZITZEICA [69], whose Euclidean expression is T = 1 . τd2 We will attach to a point M (x) of the curve (4.53) a principal trihedral of origin M, defined by the unit vectors x′′, xx′, ′′′. Designating by Y the coordinates of a point Px( ) relative to the principal trihedral, then

X =+xx′Y12+x′′Y+x′′′Y3.

Taking into account the relations (4.55), one can write the equations of the curve in the vicinity of the point M (x) as

′′vv11′23′v1′ ′′′′v4 Xx=+x++26x ()Kx++Tx 24⎣⎦⎡⎤Tx+(T+K) x+Kx +... or 1 34 YK1 =+vv6 +(T+K′)v+…, 1124 YK2 =+22vv4+…, 1134 YT3 =+62vv4T′ +…

The first two of the previous equations represent the projection c′ of the given curve c on its osculating plane in M , made parallel to the direction OM. If one brings together the developments analogue to the affine geometry of plane curves, one sees that the invariant K and the centro-affine principal normal of c coincide with the curvature and the affine normal of c′ respectively. 4 Flecnode curves and LIE-curves in affine space 112

Now, let us consider a ruled surface Σ, tangent along the curve c to the cone which projects this curve from O. Let the curve c be represented by (4.53), then the surface Σ will be represented by equations of the form X =+xu(ωx+x′) (4.57) where ω is a function of v. Let us calculate the quantities ab, , c; A, B, C for the surface Σ. One can identify the equations (4.57) with the equations (3.1) from § 3.1 by putting

yxii= , y4 =1; zxii=+ω xi′, z4 = 0 (i =1, 2, 3 ). (4.58)

One will have then

ez2θ ==()y′′zω from which

θ′ = ω′ . 2ω

Let us introduce in the equations (Ac) in § 3.1, the values of yi , y4 , zi , z4 and their derivatives drown from (4.58), we obtain finally that

a = 1 , b =1 ω 2 −ω′ , cH= 1 , 2 4ω () 2ω

3ω′ 1 ′′ ′ 22′ A =− , B =−2 (63ωω ω −6ω ω +4ωT ), (4.59) 4ω 4ω

C=−1 ()3ω′H2ωH 4ω 2 where HT=−ω′′3.ωω +−ωK One has then the following equations for the asymptotics and the flecnodes respectively 22ωωuH′++u22()−ω′u+ω=0, (4.60) and ()23ωωHH′′−−u22()6ωω′′−3ω′−6ω2ω′+4ωTu+3ωω′=0 (4.61)

∗ Let G0 be a class of the G0 -surfaces, for which the plane of a flecnode is at infinity, and the centre of the circumscribed cone along the other is the point O. If the flecnodes are u = 0 (that is to say the curve c) and u =∞, then from (4.61), one must have ω = const., TK−=ω const.

4 Flecnode curves and LIE-curves in affine space 113

Thus,

∗ Theorem 4.10.1: The conic flecnode of a G0 -surface satisfy a linear relation with constant coefficients between its centro-affine invariants T and K,

1 KT=m +n ( m ≠ 0 ). (4.62)

Inversely, if a similar relation is verified by a curve (4.53) referred to its centro-affne arc ∗ length, this is a flecnode line for a G0 -surface, represented by the equations X =+xu(mx+x′). (4.63)

∗ The asymptotic lines of the G0 -surface represented by (4.60) are given by the equation 22um′ +−()22nu+mu+1=0

∗ which one can easily integrate. Therefore, one knows the asymptotics of a G0 -surface, as soon as one knows the conic flecnode and its centro-affine arc length (which is obtained by a quadrature). The two asymptotics represented by the equation

()mn22−+u2mu+1=0 or u = −1 , (4.64) mn± divide obviously the flecnodes in a constant anharmonic ratio. These are therefore the two principal asymptotics of the surface, which is referred to its principal family. It can be proved ∗ that these two principal asymptotics (supposed real) of a G0 -surface are T-curves. Moreover, ∗ if the principal asymptotics of a G0 -surface are real and distinct, then they correspond to each other by a proportionality of their affine arc lengths, and centro-affine arc lengths. One can prove that these last two propositions are also of a projective nature, i.e., the correspondent ∗ projective arc lengths of the two principal asymptotics of a G0 -surface are proportional.

Finally, as an answer on the question, under which conditions the flecnode surfaces of a ∗ ∗ G0 -surface be also G0 -surfaces, we have the following theorem,

∗ ∗ Theorem 4.10.2: If a flecnode surface of a G0 -surface is also a G0 -surface, then it will be the same for the other flecnode surface and for all the successive flecnode surfaces; and all these surfaces belong to a group of unimodular affinities.

4.11 Space curves of constant affine curvature

Let us consider a point x of a space curve, and let us denote by t, p and w respectively the tangent vector, affine principal normal and affine binormal of WINTERNITZ (cf. SCHIROKOW [58] p.76). If we denote the affine curvature and the affine torsion of the curve by k and τ respectively, then we have the following equations (cf. [58] p.77) (,tp,w)=1, (4.65)

x′′==tt,,pp′=kt+w,w′=τt+3kp (4.66) 4 Flecnode curves and LIE-curves in affine space 114 where accents denote differentiation with respect to the affine arc length of the curve.

If we put p×=wt∗ , wt×=p∗ , tp× = w∗ (4.67) we have (tp∗∗,,w∗)=1 (4.68)

p∗∗×w=t, wt∗∗×=p, tp∗∗× = w (4.69) with the multiplication table

tp∗∗w∗ t 10 0 (4.70) p 01 0 w 00 1 and thus

tk∗∗′ =− p−τ w∗, p∗∗′ =−t−3kw∗, (4.71) wp∗∗′ =− . Now, if we put x×=tt, x×=pp, x× w= w, (4.72) then we get tp′′==,,pkt+w+w∗w′=τ t+3kp−p∗. (4.73) Now, putting ε =+at at , π =+ap ap, ω = aw + aw (4.74) where a and a denote constant vectors, thus we get

ε′ = π , πε′ =+kaω+w∗ , ωτ′ =+ε3,kaπ−p∗ (4.75)

∗∗ ωλ′′ =+00εµπ+ν0ω+at()+6,kw ∗ ∗ ωλ′′′ =+11εµπ+ν1ω+ak()−10 p+9k′w where

2 λτ0 =+′ 3,k λ1 =+ττ′′′94kk +k , 2 µτ0 =+3k′, µ1 =+32k′′ τ′ +12k, (4.76)

ν 0 = 3,k ν1 =+6.k′ τ 4 Flecnode curves and LIE-curves in affine space 115

Let us call the ruled surface described by the affine binormal of WINTERNITZ of a space curve the affine binormal surface of WINTERNITZ of the curve.

The affine binoraml surface of WINTERNITZ of a space curve is a developable surface, if , and only if, the affine curvature of the curve is equal to zero. And in the case where the affine curvature is equal to zero, the affine binormal surface of WINTERNITZ is a tangent surface of the curve x − w τ (cf. [58] p.86). In particular, if k = 0, τ = const. ≠ 0, the considered ruled surface is a cone, and if k ==τ 0, it is a parabolic cylinder. We are going hereafter to exclude the case in which the affine curvature is equal to zero.

Putting ax=+(λw)×a, (λ: a scalar) (4.77) in (4.74), we get

ελ=− ap∗ , πλ= at∗ , ω = 0. (4.78)

Substituting (4.78) in the last three equations of (4.75), and then equating the members on the right to zero, one obtains al∗ = 0, am∗ = 0, an∗ = 0 (4.79) where lk∗∗=−31λτt()+λp∗, ∗∗∗∗ mt=+()16µλ00−λλp+kw, (4.80) ∗∗ ∗∗ nt=−µλ11()10k+λλ p+9k′w.

Equations (4.79) are the necessary and sufficient conditions that the straight line with a, a as PLÜCKER’s line coordinates should meet four consecutive affine binormals of WINTERNITZ, where a is given by (4.77).

Eliminating a from (4.79), we get ()lm∗∗,,n∗= 0, i.e. 31kλτ+ λ0

12+=µλ00λλ k 0,

µλ1110kk+ λλ 9 ′ which may be rewritten in the form

2 αλ01++αλ α2=0, (4.81)

4 Flecnode curves and LIE-curves in affine space 116 where we have put

22 23 α0 =−+66kkττ′′ k′′ 9k′ τ−9kk′τ′ +3k′τ+27kk′ −4kττ′, 23 ατ1 =−66kk′′k′−49kτ′+k′+36k, (4.82)

α2 = 3.k′ By virtue of the first two equations of (4.79), we may take

al=×∗m∗, so that we have

at=+γ12γp+γ3w, (4.83) where

γτ1 =−61k ()λ+ , 2 γλ2 =−18k , (4.84) 232 γτ3 =−()33kk′′τ+τ−9kλ+()3k′+2τλ+1.

Thus the flecnodes on the affine binormal surface of WINTERNITZ of the curve x are x + λw, where λ being given by (4.81). And the directions a of the flecnode tangents of the considered surface are given by (4.83).

By virtue of (4.81) and (4.82), we see that the given curve is a flecnode curve on the affine binormal surface of WINTERNITZ, if, and only if, its affine curvature is constant and not equal to zero. Thus, we have the following theorem of MAEDA [39], [40]:

Theorem 4.11.1: In order that a space curve should be a flecnode curve on its affine binormal surface of WINTERNITZ, it is necessary and sufficient that the affine curvature of the curve be constant and not equal to zero.

5 The Method of BOL

5.1 The fundamental equations

Let us choose two points on a generator g of a ruled surface Σ and denote them by

q1()v , q2 (v) (5.1) For a torsal generator of this ruled surface, we have

(,qq12,qq12′′, )= 0. (5.2)

We are going to exclude this case of our study, so that by an appropriate normalization of the vectors qi ()v , we can achieve that

(,qq12,qq1′′,2)= κ, κ = const. ≠ 0 (5.3) and that the relation

q1()v+ uq2()v, (5.4) for every fixed value of the parameter u will always describe an asymptotic line on the ruled surface.

As q1(v) and q2 (v) describe asymptotic lines on Σ, then we have

(,qq12,qq1′′,1′)= 0, (qq1,,,222q′ q′′ )= 0 (5.5) that gives

(q1,,,qqq2′′21′) + (q12,,qq12′,q′′)=0. (5.6) By differentiating (5.3), we have

(,qq12,qq12′′,′ ) + (qq12, ,qq12′ ,′′ )=0. (5.7) From (5.6) and (5.7), we find that

(q1,,,qqq2′′21′)=0, (q12,q,q12′,q′′) =0. (5.8)

As the vectors qq12,,q1′,q′2 are linearly independent, so one can put q1′′ as a linear combination of these vectors. Let

qq1′′ =+ϕ11 1 ϕϕ12q2 +13q1′ +ϕ14q′2. (5.9)

If we set that in (5.5) and (5.8), then we get ϕ13 = ϕ14 = 0, which means that q1′′ is a linear combination of just q1 and q2. The same can be found obviously for q′2′.

Moreover, putting qs11′ = , q′2= s2, and if we assume that u = const. are asymptotic lines and (5.3) is valid, then the fundamental equations describing the ruled surface referred to its asymptotic lines as directrix curves will take the form q′ = s, sq′ =+ϕ ϕ q, 11 1111122 (5.10) qs′22= , sq′22=+ϕϕ1122q2 with ϕ11, ϕϕ12 , 21, ϕ22 as functions of the parameter v. 5 The Method of BOL 118

Setting the quantities jf, 01, f, f2 as j =+1 ()ϕ ϕ , 2 11 22 (5.11) 1 ff01=−2 ()ϕ 1ϕϕ22,1=12,f2=ϕ21. so equations (5.10) will take finally the form q′ = s, sq′ =+()j ff+q, 11 10112 (5.12) qs′22= , sq′22=+fj12()−f0q.

s1 (v )

s2 (v )

q1 ()v

q2 (v ) Fig. 5.1

We are going to choose the points q1, q2 , s1, s2 as vertices of the moving tetrahedron (cf. Fig. 5.1), so that from (5.3) we have

(qq12, , s1, s2)= κ ≠ 0. (5.13)

The lateral planes of this tetrahedron may be described as Qq=−1 (,q,s), Qq= − 1 (,q,s), 11κ 21 21κ 22 (5.14) 1 1 Sq11= κ (,s1,s2), Sq22= κ (,s1,s2 ) so that one gets the following multiplications table

QQ12S1S2

q1 0001

q2 00−10 (5.15)

s1 01− 00

s2 1000.

Using (5.12) one can find the derivatives of the vectors representing the lateral planes as follows

QS1′ = 1, SQ10′ =−()jf 1−f1Q2=ϕ22Q1−ϕ12Q2, (5.16) QS′22= , SQ′22=−fj1+()+ f0Q2=−ϕϕ21Q1+ 11Q2.

For an arbitrarily point p we have

pq=+p01 pp1q2+2s1+p3s2 (5.17) where (,,p01pp2,p3) are the local coordinates of p. 5 The Method of BOL 119

The equation of an arbitrarily plane

AQ=+aaa01 1Q2+2S1+a3S2 (5.18) in the local coordinates takes the form

ap30−−+a2p1 a1p2 a0p3=0 (5.19)

To find the effect of parameter transformation, let us set

v= f (v∗ ) (5.20) with

v ∗ φ′ d = φ()v , A()v = 1 . (5.21) dv 2 φ

Hence,

ddqqi−−11i ==φφsi (5.22) ddvv∗∗ so that

ddqq12 −2 (,qq12, , )= φ (qq12, ,qq1′,′2). (5.23) ddvv∗∗

Generally, the condition (5.13) remains not fixed by the parameter transformation. In order that this condition remains fixed, we have to couple the parameter transformation with a ∗ renormalization of the vectors qi (v) by putting qi= ρqi. We are going to call this coupled transformation the ∗−transformation. Applying this transformation, we get

∗ ∗−dqi 1 si==φ (ρ′qi+ρq′i) (5.24) dv ∗ and hence

∗∗ ∗∗ddqq12 42− (,q1q2, , )= ρφ (qq12, ,qq12′,′ ). (5.25) ddvv∗∗

1 Setting ρ = φ 2 , we get

1 −1 ∗ 2 ∗ 2 qi= φ qi, ssi=φ (i+Aqi). (5.26)

From (5.24), we have

∗ − 3 dsi 2 ∗ =+φ []sq′′iiAA+si−A()si+Aqi dv (5.27) − 3 2 ⎡⎤′′2 =+φ ⎣⎦sqii()AA− 5 The Method of BOL 120 and hence,

ds∗ 1 −∗22⎡⎤′ ∗ ∗ =+φϕ()11 AA−qq1 +ϕ12 2 dv ⎣⎦ (5.28) ds∗ 2 −∗22⎡⎤′ ∗ ∗ =+φϕ21qq1 ()ϕ22 +AA−2 dv ⎣⎦ Using (5.11), one can find the transformation formula for j in the form jj∗−=+φ 2(A′ −A2) (5.29) while the fi are seminvariants of weight 2 with

∗−2 fi= φ fi. (5.30)

5.2 The osculating quadric – the flecnode congruence

For the asymptotic lines parameter, i.e. u = const., it is valid that

xq=+12uq ⇒ xq′′ =+1′′ uq′′2 (5.31) and from (5.10)

xq′′ =+()ϕϕ11 u21 1 ++(ϕ12 uϕ22 )q2. (5.32)

The condition for an inflexion point is x′′ ≡ 0m(odx,x′ ) (5.33) but x′′ is a linear combination of x and x′ if, and only if,

ϕ12 +=uuϕϕ22 (11 +uϕ21). (5.34)

It follows that a point q1+ uq2 is a flecnode point on the generator if, and only if,

22 Φ=()uu,v −ϕϕ12 +(11 −ϕ22 )+ϕ21u=f2u+2f0u−f1 =0 (5.35) which means that the flecnode points coincide if, and only if, the discriminant

2 ∆= f0+ff12 (5.36) vanishes, where it can be proved that ∆ is a seminvariant of weight 4.

The parameter for which ∆=±1, is called by WILCZYNSKI the projective arc length of the ruled surface. Clearly, it is defined only for the ruled surfaces whose flecnode points never coincide, and it may be represented by

s=∆∫ 4 dv. (5.37)

According to these calculations, we can conclude that if f0 , f1 and f2 vanished, then it follows from (5.12) that, for every fixed value of the parameter u, the relation 5 The Method of BOL 121

′′ ()qq12+=uj(q1+uq2) (5.38) is valid, which means that the curves u = const. on the surface are straight lines, from which the surface is a quadric. Moreover,

Theorem 5.2.1: If the functions ff01()vv, (),f2(v) and j(v) are given, then the ruled surface is uniquely determined, up to a projective transformation, as the solutions of system (5.12) are with these initial values uniquely determined.

The equation of the osculating quadric along a generator v= v0 in the local coordinate system takes the form

pp03−=p1p2 0. (5.39)

Parametrically, it may be represented as

zu(,w)=+ss12u +wq1+uwq2. (5.40)

The polar plane of a point

aaa01qq++12 2s1+a3s2, (5.41) with respect to the osculating quadric, is represented by the equation

aa01QQ++12 a2S1+a3S2, (5.42) since this polar plane has from (5.39) the equation

ap30−−+a2p1 a1p2 a0p3=0 (5.43) and from (5.19), this represents the plane (5.42).

As we explained before (cf. § 2.4), the osculating quadric contains two sets of lines, the concordant set of lines, which consists of the lines

(sq11+w, s2+wq2) (5.44) and its transversal set of lines

(qq12+u , s1+us2). (5.45)

The whole of the transversal lines of all osculating quadrics constitute a congruence which has the surface as its only focal surface, whereas the whole of the concordant lines build the flecnode congruence. The flecnode congruence is the locus of all the lines (ss1 , 2) with different values of the parameter v.

As an arbitrarily developable of the flecnode congruence contains a line of every osculating quadric, so we can choose the coordinate system such that the developable is to be formed by the lines()ss1, 2. 5 The Method of BOL 122

Now let

ps=+12us (5.46) is the associated focal point, with u depends naturally on v, i.e., by different values of the parameter v, p describes the associated focal curve. Hence

ps≡ 0 (mod 1, s2) ⇒ ss′′1 +≡u 210(mods,s2) (5.47) and from (5.10)

()ϕϕ11 ++21uuqq1 (ϕ12 +ϕ22 ) 2 ≡0(mods1,s2 ). (5.48)

Thus, we get the following conditions

ϕ11 +ϕ21u =0, ϕ12 +ϕ22u =0. (5.49)

Multiplying the first condition with u and subtracting the result from the second one, so we find the following quadratic equation in u

22 −ϕϕ12 −()22 −+=ϕ11 uuϕ21 f2u+2f0u−f1 =0. (5.50)

The appropriate values of the parameter u determine however from (5.50) the flecnode points on the associated generator of the ruled surface. So we can conclude that if s1+ us2 is a focal point of the congruence, then the point q1+ uq2 of the same transversal line is a flecnode point of the ruled surface. Generally, the flecnode tangents build the two flecnode surfaces of our ruled surface. These flecnode surfaces are the focal surfaces of the flecnode congruence, as all the focal points of the congruence are located on these surfaces.

The tangent plane of the flecnode surface at the point (5.46) can be described by the points

()ss12,,q1+ uq2 (5.51) or from (5.16), through

SS1 + u 2. (5.52)

We have to notice that this tangent plane coincides together with the tangent plane of the osculating quadric in the same point, which is also obvious geometrically.

Now we are going to choose the coordinate system so that the point (5.46) describes an asymptotic line on the focal surface. This is the case if, and only if,

′′ ()SS12++u(s1us2)=0. (5.53)

Because of (5.12), (5.15) and (5.16), it follows that uj′⋅=0. (5.54)

Hence, if the coordinate system is to be chosen such that j = 0 (5.55) 5 The Method of BOL 123

then the lines (ss1, 2) will touch every flecnode surface along an asymptotic line. That means the asymptotic lines on both flecnode surfaces correspond to each other. So we can conclude that the flecnode congruence is a W-congruence, as we already explained before. We are going to call the coordinate system, for which j = 0 is valid, a CARTAN’s coordinate system, and its associated parameter a CARTAN’s parameter.

5.3 The ruled surfaces in line coordinates – The osculating congruence

From the differential equations (5.10), it follows that the differential formulae of the six edges of the tetrahedron (cf. Fig.5.1) are

′ (qq12, )=−()qs1, 2 ()q2, s1 ′ ()qs11, =⋅ϕ12(qq1, 2) ′ ()qs22,,=−ϕ21⋅(q1q2) (5.56) ′ ()qs12,,=+(s12s)ϕ22⋅(q1,q2) ′ ()qs21,,=−(s1s2)−ϕ11⋅(q1,q2) ′ (s1,,s2 )=⋅ϕϕ11 ()q1 s2 −12 ⋅(qs2 ,2 )−⋅ϕ21 (q1,s1 )−ϕ22 ⋅()qs2 ,1 .

These formulae may be simplified if we use, instead of (qs1, 2) and (qs21,), the two linear complexes

vq=−()12,,s ()q2s1 (5.57) wq=+()12,,s (q2s1) so that, we have

1 ()qs12, =+()vw 2 (5.58) 1 ()qs21,.=−2 ()v−w

The differential formulae (5.56) will hence take the form

′ ()qq12, = v ′ ()qs11,,=⋅f1(q1q2) ′ ()qs22,,=−f2⋅(q1q2) (5.59)

vs′ =+2,()12s2j ⋅(q1,q2)

wq′ =−2,f01⋅()q2 ′ ()ss12,=+jf v 0.w+f1⋅()q2,s2−f2⋅(q1,s1).

5 The Method of BOL 124

For these six vectors we have the following multiplication table

()qq12,,(q1s1)(q2,s2)v w ( s1,s2)

()qq12,00000κ

()qs11,00−κ 000

()qs22,0−κ 0000 (5.60) v 000−2κ 00 w 0000−2κ 0

()ss12,0κ 0000.

The derivatives of a generator

pq= (1, q2) (5.61) may now be found as follows pv′ =

ps′′ =+2,()12s2j (q1,q2)=2,(s12s)(modp) 1 2 pq′′′ =−ff21(),,s1+ 0w+ f1()q2s2(modp,p′) (5.62) 1 iv ′ 2 pq=−ff21′ (),,s1+ 0′w+ f1()q2s2(modp,p′,p′′′)

1 v 2 pq=−ff21′′(),,s1+ 0′′w+ f1′′()q2s2(modp,p′,p′′,p′′′).

In Π5 the points p, p′, p′′ constitute a plane, which also from (5.61) and (5.62), may be described by the points

()qq12,, v, (ss12,). (5.63)

2 This plane intersects M 4 (cf. chapter 7) in a conic section, whose points represent the concordant set of lines of the osculating quadric, as an arbitrarily line (5.44) of this set may be described in the form

2 ()ss12,++wwv (q1,q2). (5.64) The transversal set of lines may also , from (5.45), be described by

2 ()qs11,,++uuw ()q2s2. (5.65)

The four points p, p′, p′′, p′′′ determine in Π5 a 3-dimenstion space, which correspond in Π3 to a linear congruence, the osculating congruence of the ruled surface. The directrix lines of this congruence are represented in Π5 by the points which are polar with p, p′, p′′, p′′′ with 2 respect to M 4 . They are hence the lines (5.65) which suffice the condition

⎡⎤2 ′′′ ⎣⎦()qs11,,++uuw ()q2s2 ⋅p=0. (5.66)

According to (5.60) and (5.62), it follows the condition

2 Φ=()uf20u+2fu−f1=0. (5.67)

But the values of u, which suffice this condition, correspond from (5.35) to the flecnode points. 5 The Method of BOL 125

Hence we have

Theorem 5.3.1: The flecnode tangents associated to our generator are the directrix lines of the osculating congruence.

Now, let u be a root of the flecnode relation (5.67) which correspond to a directrix line, then it must be ′ ()qq12+≡uu0m(odqq12+, s1+us2) (5.68) ′ ()ss12+≡uu0m(odq1+q2, ss12+u) and u′ suffices the relation

2 fu20′′+−22fu f1+u′(fu2+f0)=0. (5.69)

From (5.68) we find that

uu′qq21≡+0m()od q2, s1+us2, (5.70) ()fu02++fqq1(f1−uf0)2+u′s2≡0m(odq1+uq2, s1+us2).

Because of (5.67), these conditions are then exactly satisfied if, and only if, u′ = 0. (5.71) Hence one can say that,

Theorem 5.3.2: A directrix line of the osculating congruence is stationary if, and only if, it touches the associated flecnode curve.

The condition (5.71) is identically satisfied in v if, and only if, the associated flecnode curve is an asymptotic line. In this case the considered directrix line will be fixed on the whole ruled surface, and the flecnode curve must coincide with it. All the generators of the ruled surface cut this fixed line which means that they belong all to a special linear complex. So we conclude that, if the flecnode curve is an asymptotic line on the surface, then it must be a straight line. This happens if, and only if, the ruled surface belongs to a special linear complex, and in that case, this straight line will be the directrix of the complex. We notice that the roots of (5.67) are

uf=−11±f2 +ff=−f±∆. (5.72) 1,2 ( 0 0 1 2 ) ( 0 ) ff22

So it will be one root (f0+ uf2) if, and only if, ∆ = 0, i.e., if and only if both roots coincide together. If this is not the case now, then (5.71) is valid , because of (5.69), if and only if

2 fu20′′+−20fu f1′=. (5.73) Hence, if both the flecnode points are different, then (5.71) is valid for every common root of the relations

2 Φ=()uf,2v 20u+fu−f1=0, (5.74) 2 Φ=v ()uf,2v 20′′u+fu−f1′=0. 5 The Method of BOL 126

The osculating congruence is stationary if, and only if, piv belongs to the linear space in 5 Π which is spanned of p, p′, p′′, p′′′, hence from (5.59) and (5.62), if f0′, f1′, f2′ are respectively proportional to f0 , f1, f2 , i.e. if

fii′= ρ f . (5.75) So we conclude that,

Theorem 5.3.3: The osculating congruence is stationary if, and only if, both formulae (5.74) are proportional.

5.4 A projective classification of ruled surfaces – The characteristic curve

If we exclude the isolated positions, at which a flecnode curve touches an asymptotic line on the ruled surface, from considerations, then according to the previous results one can give the following projective classification of the ruled surfaces:

(I) Quadrics: The flecnode curve is indeterminate. (II) Both branches of the flecnode curve are rectilinear: That is the case if, and only if, the ruled surface belongs to a linear congruence. (III) Only one branch of the flecnode curve is rectilinear, but not the other: In this case belongs the ruled surface to a special linear complex, but not to a linear congruence, and the rectilinear branch will be the directrix of that complex. (IV) Non of both branches of the flecnode curve is a straight line.

In both cases (II) and (IV) the branches of the flecnode curve will be: (A) real and distinct. (B) coincide in a real curve (a straight line). (C) conjugate complex.

As it is known that the curved asymptotic lines of a ruled surface map its generators characteristic curve projectively and successively, so one can interpret xf00==()vv, x1f2( ), them as elements of a binary domain. According xf22= ()v to O. HESSE [25], this domain can be represented by the tangents of a conic section k in an u2 (v ) auxiliary plane π. If now on a ruled surface a P(v ) curve-pair is given, then this pair will intersect every generator in two points, and to the asymptotic lines through these points belong two tangent of the conic k, which can be uniquely v k u1 () well-defined by their intersection point P. Now, 2 if we repeat that for every generator of the ruled xx01+ x2= 0 2 surface, then the point P will describe in the xu01= , x==u, x2−1 plane π a curve, which is called by BOL [7] the hyperbolic image of a curve-pair on the ruled Fig. 5.2 surface (cf. Fig. 5.2). 5 The Method of BOL 127

Specially, if we choose the flecnode pair of the ruled surface, then its hyperbolic image, which is well-defined by the ruled surface alone, is invariant up to a mapping of the hyperbolic geometry with respect to the absolute conic k, and it is called by BOL [7] the characteristic curve of the ruled surface. Analytically, the previous classification is characterized by the behaviour of the quadratic relation

2 Φ=(,ufv) 20u+2fu−f1. (5.76)

Let us consider in an auxiliary plane the curve whose projective coordinates x0 , x1 and x2 are

x00= f()v , xf11= ()v , xf22= ()v . (5.77)

We are going to call this curve the characteristic curve of the ruled surface. In the same plane let us consider a conic section k which is described parametrically as

2 x0 = u, x1 = u , x2 = −1 (5.78)

2 so its equation takes the form xx01+=x20. The tangent of the conic k in the point (5.78) pass through the point (1, 2u, 0), so its line coordinates are

2 X 0 = 2,u X1 =−1, X 2 = u . (5.79)

Hence the equation of the conic k in line coordinates takes the form

2 X0+4X1X2=0. (5.80)

Now, for the parameter values of those tangents of the conic k, which pass through the point (5.77), we have

2 fu20+−20fu f1=. (5.81)

Clearly these values represent the flecnode points of the generator which correspond to the parameter v. We notice that both values of the parameter u coincide together, if the point (5.77) lie on the conic k. Hence one can conclude the following: (1) A ruled surface of class (II) is characterized by that its characteristic curve is a fixed point. (2) A ruled surface of class (III) is characterized by that its characteristic curve is tangent to the absolute conic k. (3) For a surface of class (IV-A) lie the characteristic curve in the exterior domain of k, whereas by the class (IV-C) lie the characteristic curve in the interior domain of k. (4) For a surface of class (IV-B), the characteristic curve is a part of the arc of k.

5.5 The osculating linear complex

If the osculating congruence is not stationary, then the five points

p, p′, p′′, p′′′, piv (5.82) 5 The Method of BOL 128 span a hyperplane in Π5. This hyperplane represents the osculating linear complex K of the 2 ruled surface Σ along a generator g. The pole u of this hyperplane with respect to M 4 may also be considered as an image of the osculating linear complex. As u is polar with p, p′, 2 and p′′, with respect to M 4 , then we can represent it in the form

uw=+n0n1(q2, s2)−n2(q1, s1). (5.83)

From (5.63), this vector (5.83) is then polar to p′′′ and piv if, and only if

20nf+nf+nf =, 00 21 12 (5.84) 20nf00′′++n2f1 n1f2′=.

From (5.74), the triples (,f01 ff, 2) and (,f0′ ff1′′, 2) are linearly independent, if the osculating congruence is not stationary. Hence the ratios between n0 , n1 and n2 are through (5.84) definitely specified. If we consider n0 , n1 and n2 as the coordinates

X 00= 2,n X12= n , X 21= n (5.85) of a straight line in the auxiliary plane, then from (5.84) we find that this straight line passes through the points f = (,ff01 , f2) and f ′ = (,ff01′′ , f2′) which determine the tangent of the characteristic curve (5.77). Hence, we conclude that the osculating linear complex is represented by the tangent of the characteristic curve. It degenerates if, and only if, uu⋅=0, 2 i.e. from (5.83) and (5.60), iff nn12+n0=0, therefore from (5.85), it must be

2 XX12+=40X0 , (5.86) hence, according to (5.80), the straight line representing the osculating linear complex touches the absolute conic k. So, we can say that the osculating linear complex of a ruled surface along one of its generators degenerates if, and only if, the tangent of the characteristic curve touches also the absolute conic k.

If the ruled surface has two non-rectilinear coincident flecnode curves, then its characteristic curve will coincide with the absolute conic k. So one concludes that all the osculating linear complexes of a ruled surface, whose flecnode curves are coincident and non-rectilinear, are special. Conversely, if all the osculating linear complexes of a ruled surface, which does not belong to a linear complex, are special, then the flecnode curves of this surface coincide together. Moreover, The osculating linear complex is stationary if, and only if, the characteristic curve have a point of inflexion. For, from (5.62), u ⋅ pv = 0 if, and only if

nf00′′++n2f1′′ n1f2′′ =0 (5.87) but this condition is compatible with (5.84) if, and only if

ff01f2

()ff, ′′, f′==f0′f1′f2′0 (5.88)

f01′′f′′f2′′ hence iff the characteristic curve (5.77) has an inflexion point. 5 The Method of BOL 129

Moreover we have,

Theorem 5.5.1: The ruled surface belongs to a linear complex if, and only if, its characteristic curve is a straight line. This complex is special if, and only if, the characteristic curve is a straight line which touches the absolute conic k.

Let us find which straight lines

2 mu=+(,qq12 s1+us2)=(q1,s1)+uw+u(q2,s2) (5.89) of the transversal set of lines belong to the osculating linear complex. It must be that u ⋅=m 0, which leads to the condition

2 nu20+−2nu n1=0. (5.90)

But this condition implies that the straight line, whose line coordinates are (,X 01 XX, 2), passes through that points of the absolute conic (5.78), which correspond to the parameter u . Hence, we conclude that,

Theorem 5.5.2: Generally there exists two lines of the transversal system of lines of the osculating quadric, which belong to the osculating linear complex of the ruled surface. These lines are described by the intersection points of the tangent of the characteristic curve with the absolute conic k. The intersection points of these lines with the generator are the LIE- points of the generator.

A ruled surface which belongs to a non-special linear complex u is characterized by the property that the LIE-points on it describe two special asymptotic lines on the surface, as we already explained before. These asymptotic lines are in the same time complex curves, whose tangents belong to the linear complex u. Moreover, if the image point u()v of the osculating linear complex of a ruled surface belongs to a linear subspace of Π5 , then the ruled surface belongs to a linear complex and u()v is a fixed point (cf. §7.1).

The tangent of the asymptotic line through a point qq12+ u is described by

2 (,qq12++uu s1s2)=(q1,s1)+uw+u(q2,s2). (5.91)

The conic section, which describes in Π5 the transversal set of lines of the osculating quadric, lie hence in the plane, which spanned by the points

(,qs11 ), w, (,qs22). (5.92)

The planes E()v are the osculating planes of the space curve in Π5 described by u()v and which represent the osculating complexes. If the osculating linear complex is fixed, then all the planes pass through the image point of this fixed complex. But from (5.59) we notice that

(,qs11)′ = f1p, w′ =−2 f0 p, (,qs22)′ = − f2p. (5.93)

Hence, every linear combination of the points (5.92) with constant coefficients describes in Π5 a space curve, whose tangent passes through the image point p of the generator.

5 The Method of BOL 130

5.6 The Plane hyperbolic geometry

To explain the relation between the theory of ruled surfaces and the plane hyperbolic geometry, let us consider a non-degenerate conic section k in the projective plane, and its equation is represented in the form

2 xx, =∑ gxik i xk =0. (5.94) ik,0=

Considering k as the absolute conic, then the coordinates of any point, which does not lie on k, can be normalized such that

xx, ==ε 0 ±1. (5.95)

The point x describes a curve x(v) in the plane, and by differentiating (5.95), it follows that

xx, ′ = 0 (5.96) i.e. x′ is polar of x with respect to the absolute conic. The arc length of the curve can be described by

d,sh = xx′′dv (5.97) hence

xx′′ v. sh = ∫ ,d (5.98)

Assuming that the curve x(v) and the absolute conic k have no common tangent, then it is always valid that xx′′,≠ 0. So, we can find a vector t proportion with x′such that

tt, ==ε1 ±1. (5.99)

Thus, we have x′ = αt, α ≠ 0. (5.100)

From (5.95), (5.96), (5.99) and (5.100) it follows that x and t are two vertices of a polar triangle of the absolute conic k. The third vertex can be described by n, where n is normalized such that

nn, ==ε 2 ±1. (5.101)

So we can build the following multiplication table

xtn x ε 00 0 (5.102) t 00ε1

n 00ε 2 . 5 The Method of BOL 131

We have to notice that in this hyperbolic case two vertices of the polar triangle are always in the exterior region of the absolute conic (see Fig. 5.3), hence for ε i it is always valid that two of them are positive, i.e.

ε 01 εε 2=−1. (5.103) The differential formulae of the three vectors x, t and n can now be found in the form

x′ = αt, tx′ = −+ε 2αε0γn, nt′ = −γ , γ ≠ 0. (5.104) Applying the ∗−transformation, then the vectors x, t and n remain invariant, and α ∗= φα−1 , γ ∗= φγ−1 . (5.105) The invariant s = ∫αdv (5.106) which is uniquely determined up to an additive constant and the sign, is called the hyperbolic arc length of the curve, and the absolute invariant γ ρ = (5.107) α is the hyperbolic curvature of the curve. If one takes the hyperbolic arc length as a parameter, then it will be α = 1, γ = ρ. (5.108)

When x(v) describes a curve, then n (v ) describes its polar curve with respect the absolute conic k. The line (xn, ) is conjugate to the tangent (xt, ) with respect to k, it is thus the hyperbolic normal of the curve (and of its polar curve). The point m =+nρx (5.109) is called the hyperbolic centre of curvature of the curve at the considered point. From (5.104), then m′= ρ′x (5.110) which shows that the curve m(v ) is touched by the normal (xn, ) of the curve; it is its hyperbolic evolute. If ρ′ = 0, then the point m is fixed along the curve, the normals (xn, ) pass thus through a fixed point. Such curves are called the circles of the hyperbolic geometry. They are characterized by that its curvature ρ is constant. A point of the curve, at which ρ is constant, is called a hyperbolic vertex.

Now let the points of the curve x(v) be given as the intersection of two tangents of the absolute conic k as we explained in § 5.4. Let the two points of contact on k be described by the two vectors p1(v) and p2 (v) (cf. Fig. 5.3) . Assuming that these points are real and distinct, then the intersection point x(v) must lie in the exterior region of the absolute conic, i.e.

xx, =ε 0 =1 (5.111) hence from (5.103) it follows that

ε1=−ε 2. (5.112) 5 The Method of BOL 132

p1 (v )

t v x v () ( )

p2 ()v

n ()v

Fig. 5.3

As p1(v) and p2 (v) are different points on k, then they can not be polar to each other, i.e.

pp12,≠ 0, so we can normalize these vectors such that

pp, = 1 (5.113) 12 2 and for the renormalization, it is possible that

p11= σ p1, p2= σ 2p2 (5.114) with

σ12σ = 1. (5.115)

Hence, the multiplication table of the three vectors x, p1 and p2 takes the form

xp12p x 10 0 (5.116) 1 p1 00 2 1 p1 002 .

As x lies on the tangents of the arc p1(v) and p2 (v) of the absolute conic, so we can set

pp11′ =+a1d1x, pp′22=a2+d2x (5.117) and as x′ is polar to x, then we can put

xp′ =−2b21+2b1p2. (5.118)

By differentiating and using (5.116), we get the following relations

aa12+=0, bd11+=0, −+bd22=0. (5.119)

5 The Method of BOL 133

Putting 2ba01==−a2, hence the differential formulae of the three vectors x, p1 and p2 take the form

xp′ =−22bb21+ 1p2,

pp10′ =−2bb11x, (5.120)

pp′20=−2bb2+ 2x.

If we choose α ≥ 0, then ε1 and α are specified by the relation

2 αε1==xx′′, −4bb12. (5.121)

Moreover we find that

t=−2(α −1 bbp+p), 21 12 (5.122) −1 np=+2(α bb21 1p2).

From (5.104), as

n′ =−γ t, ⇒ tn, ′ =−εγ1 (5.123) moreover, from (5.122),

−1 np′=−22α ⎣⎡bb02( 1b1p2) +b2′p1+b1′p2⎦⎤ (5.124) hence we deduce that

′ 1 ⎛b1 ⎞ γ =+2lb0 2 ⎜og⎟. (5.125) ⎝⎠b2

The renormalization (5.114) can be established such that p1′ is proportional to x, from which is then b0 = 0, and hence the formulae (5.120) take the form

xp′ =−22bb21+ 1p2,

px11′ =−b , (5.126)

px′22= b .

By the renormalization, then

−1 pˆ1= kp1, pˆ 2= k p2 where k = const. ≠ 0 (5.127) and ˆ ˆ −1 bk1= b1, bk22= b. (5.128) Putting

1 b1′ 1 b2′ A1 = , A2 = , (5.129) 2 b1 2 b2 then (5.121) and (5.125) take the form

2 1 ⎛⎞b1 ′ −=αε14,bb12 γ ==⎜⎟log A21−A , (5.130) 2 ⎝⎠b2 5 The Method of BOL 134 hence, the arc length and the curvature of the curve will be then described by

s2bbdv, ±1 A A . =± ∫ 12 ρ =−()21 (5.131) 2 bb12

Thus, if one knows b1 (v ) and b2 (v ) by the normalization for which b0 = 0, then the curve will be uniquely determined up to a hyperbolic transformation. Moreover, from (5.126) we notice that the curve x(v) has a common tangent with the absolute conic if b1 = 0 or b2 = 0. If we take the hyperbolic arc length as a parameter, then from (5.130),

4(bs12)b(s)=±1, ργ()ss= ( ) =−A1A2 (5.132) and by differentiating,

AA12+=0 (5.133) thus, it is valid that

ρ ==2A1−2A2 (5.134) and by a suitable choice of the integration constant, one finds

ρ s ds − ρ s ds 1 ∫ () 1 ∫ () be1 = 2 , be2 =±2 (5.135) For the condition of a hyperbolic vertex point, i.e. ρ = const., it will be then, from (5.131)

22 A11′−=AA2′−A2. (5.136)

If we choose the vectors x, p1 and p2 as base vectors of a projective coordinate system, then for an arbitrarily point p it is valid that

px=+pp02p1+p1p2. (5.137)

It follows from (5.116) that the equation of the absolute conic takes the form

2 pp , =+pp01p2=0 (5.138) and parametrically it takes the form

2 p0 = u, p1 = u, p2 = −1. (5.139)

Hence, every point on the absolute conic can be, for every value of the parameter v, described by

2 qx(,uuv)=−p12+up. (5.140)

From (5.120) it follows that

2 qq′=−2(bu20b)+(x−2up1)(u′−b2u −2b0u+b1) (5.141) i.e. q′ is proportional to q if, and only if,

2 ub′ =+20u 2bu−b1=β (u). (5.142)

Thus, we conclude that (5.140) describes a fixed point on the absolute conic if, and only if (5.142) is valid. 5 The Method of BOL 135

5.7 DV-set of curves on a ruled surface

By DV-set of curves on a ruled surface one understands a one-parameter system of curves, which map the generators of the ruled surface protectively and successively (It is called by MAYER the R-family of curves, cf. § 4.7). Through every point of the surface passes exactly one curve of the set, and every curve of the set meets each generator just one time. If we describe the ruled surface by (5.4), these curves can be described in the form

uf= (v ). (5.143)

A DV-set of curves is characterized by the following properties: (1) A DV-set is uniquely determined if three of its curves are given.

(2) The curves (5.143) of a DV-set are described by the solutions of a RICCATI differential equation of the form

ua′ =+()vvu2 b()u+c(v) (5.144) and conversely equation (5.144) determines a DV-set by every choice of the functions a(v), b(v) and c()v , assuming that they are all continuously differentiable. (3) If a DV-set is given, then the directrix curves and its normalization are specified such that the curves of the DV-set are described by u = const. (4) A DV-set is characterized by that the tangents of the ∞1 set of curves along each generator belong to a quadric surface, which is called the tangent quadric of the DV-set along the generator.

If we make a normalization such that u = const. describes the curves of a DV-set, then the parameter u will be not just a projective parameter on every generator, but can be also considered as a projective parameter of the curves of the set, exactly as we have made before by the asymptotic lines in § 5.9.

If p(u) is the parametric representation of a conic section k in an auxiliary plane, assuming that u is a projective parameter, so we can map every curve of the set on a point of k, which corresponds to the same value of the parameter u.

If beside the DV-set, a curve-pair is given on a ruled surface, then this pair meets every generator in two points. We can define this pair by an image curve in an auxilarily plane, as we explained in § 5.4. Let α1(v) and α 2 (v) are the parameter’s values of the two intersection points on the generator, so we have to find the two points

pp1(v) = (α1(v)), pp2()v= (α2(v)) (5.145) on the conic section, then draw the tangents through them, and the intersection point of the tangents is a point of the image curve. We are speaking then about the hyperbolic image of a curve-pair on a ruled surface with respect to a DV-set of curves.

Now let us build the accompanying coordinate system, i.e. the vertices of the triangle of reference , then the vertex n(v ) will describe also a curve in the auxiliary plane. We are going to consider this curve as an image of a curve-pair with respect to our DV-set. This pair is called the median pair of the first one with respect to the DV-set. Their points of intersection 5 The Method of BOL 136

with a generator are the median points on it. The associated parameter values u1 and u2 on the absolute conic correspond to the points of contact of the absolute’s conic tangents through n(v) (cf. Fig. 5.3), i.e. the points of intersection of the side (,xt) of the triangle with the absolute conic, as this side is the polar of n(v) with respect to the absolute conic.

As x and n are conjugate with respect to the absolute conic k, then the points of contact of the conic’s tangents from x and n are harmonically separated. Thus, we conclude that the median points on a generator separate the intersection points of a given curve-pair with that generator harmonically.

Similarly, to the curve t(v) corresponds a curve-pair on the ruled surface. This pair is called the harmonic pair with another given pair with respect to a DV-set. This pair separates both the given pair and its median harmonically. Their points of intersection with each generator correspond to the points of intersection of the side (,xn) with the absolute conic. It is known that only one side of a polar triangle of a non-degenerate conic meets this conic in non-real points.

The hyperbolic image of the flecnode pair on a ruled surface with respect to the asymptotic lines as a DV-set is the characteristic curve of the surface, as we explained before. Its median curves are the LIE-curves with respect to the asymptotic lines. The flecnode curves, the LIE-curves and the harmonic curves belong to a one parameter set of curves, which maps the generators of the ruled surface protectively and successively, which is the principal family of curves on the ruled surface ( cf. § 4.8). If on a ruled surface two DV-sets of curves are given, then there are generally on each generator two positions at which the curves of both sets touch each other. These positions are the contact positions of both sets. As the asymptotic lines build a DV-set on the ruled surface, then there are generally on each generator two positions at which the curves of a given DV-set on the surface are touched by the asymptotic lines. They are called the osculating positions and they are characterized by that the osculating plane of a curve of the set at such positions coincides with the tangent plane of the surface. These positions may be real and distinct, real and coincident, or imaginary. If an arbitrary DV-set on a ruled surface is given, then we can choose the representation

zq(,uuvv)=+12() q(v) (5.146) such that u = const. describes the curves of this set. By a normalization of q1 and q2 we can achieve that

(qq12, , q1′′, q2)= κ, κ = const. ≠ 0. (5.147)

Putting qs11′ = , q′2= s2 and choosing q1, q2 , s1, s2 as vertices of the moving tetrahedron, then the differential formulae of these vectors take the form (see Fig. 5.4) q′ = s, sq′ =+α αβq+22s+βs, 11 1111122 111 122 (5.148) qs′22= , sq′22=+αα1122q2+22β21s1+β22s2.

5 The Method of BOL 137

s 1 s2

q1

q 2

Fig. 5.4

Because of (5.147), it is valid that

β11 +β22 =0. (5.149)

Applying the ∗−transformation, one finds that

1 1 ∗ 2 ∗ − 2 qk= φ qk, ssk=φ (k+Aqk) (5.150) where φ(v ) and A(v) as explained before in (5.21). Moreover we find also that

αφ∗−=−22(2α AAβ+′ −A), α ∗−=−φα2 (2Aβ), 11 11 11 12 12 12 (5.151) ∗−22∗−2 αφ22 =−(2α22 Aβ22 +AA′ −), α21 =−φα(221 Aβ21)

∗−1 βik = φβik . (5.152)

For abbreviation, let us put

α(,uuv)=+αα2 ( −α)u−α, 21 11 22 12 (5.153) 2 β (,uuv)=+ββ21 (11 −β22 )u−β12 , and

2(a v)=+α11 α22. (5.154)

If the quadratic equations of α and β, and the invariant a are given, then the ruled surface and its DV-set are uniquely determined up to a projective mapping. 5 The Method of BOL 138

Applying the ∗−transformation, then

β ∗= φ−1β, α ∗−=φα2 (2−Aβ) (5.155)

aa∗−=+φ 2(A′ −A2). (5.156)

For a curve

uF= (v) (5.157) on the ruled surface, we have from (5.146) and (5.148) that

z′′=+ss12uu+q2,

z′′ =+()αα11 uu21 qq1 ++(α12 α22 +u′′)2 +2(β11 +uβ21)s1 +(β12 +uβ22 +u′)s2 (5.158) 2 ′ =+2 ⎣⎦⎡⎤ββ12 uu( 22 −β11) −β21 +usq2 (mod 1,q2 , z).

Hence the asymptotic lines on the surface are described by

u′ = β (,uv) (5.159) and therefore the equation

β (,u v)= 0 (5.160) describes the osculating positions of the DV-set. They are then indeterminate if, and only if,

β vanishes identically, i.e. β12 =β21 =0, and from (5.149), then iff

βik = 0. (5.161)

The given DV-set composed then of the asymptotic lines on the ruled surface if, and only if,

(5.161) is valid for every value of the parameter v. We have to notice that if βik = 0, the formulae (5.148) take then the same form as (5.10), and it will be a= j, and α will be identical with the form Φ in (5.35). From (5.149) and (5.151), it is valid along an asymptotic line on the surface that

β ′ =+ββv 2(β11 +uβ21). (5.162)

For the asymptotic lines, (5.158) can be therefore represented in the form zs′ =+us+βq, 12 2 (5.163) zq′′ =−()βαv 2 (mod z,z′).

The flecnode points on a generator are therefore described by the equation

α −βv =0. (5.164)

Our ruled surface is then a quadric surface if, and only if,

α(,uv)= βv (u,v) (5.165) is valid identically in u and v. 5 The Method of BOL 139

If one applies the parameter transformation (5.20), the line

∗∗ ()ss12()AA,vv , ( ,) will describe the congruence which is formed by the concordant lines of all tangent quadrics ; one calls it the concordant congruence of the DV-set. It can be easily proved that, if the two equations

β (,u v )= 0, α (,u v)= 0 have a common root for every value of the parameter v , then one of the two focal surfaces of the concordant congruence is a ruled surface, which is generated by the asymptotic tangents of the curves of the DV-set. Both focal surfaces of that congruence are ruled surfaces if, and only if, β (,u v)= 0 and α(,u v)= 0 are proportional. In this case, the DV-set is called a stratified set. If a curve of a stratified DV-set is an asymptotic line, then it is a straight line, and accordingly it is a flecnode curve on the ruled surface. Moreover, If a curve-pair is given on a ruled surface, whose median curves are defined, and divide the flecnode curves harmonically, then there exists a well-defined stratified DV-set, whose osculating curves are that given curve-pair.

Let a two DV-sets on the ruled surface are given, and let the cross ratio formed by the curves of these two DV-sets with the asymptotic lines be designated by D, hence, if this cross ratio is constant over the whole surface, then the two DV-sets are called isogonal. If D = −1, then they are conjugate.

Theorem 5.7.1: A ruled surface is well-defined up to a projective transformation by giving the hyperbolic image of the flecnode curves referred to the set of the asymptotic lines, and by giving the value Dt0 ()of the cross ratio, which the stratified isogonal set makes with the principal set (cf. § 4.8). With Dt0 ()=1, then the stratified principal sets are themselves identified. For these given quantities, there exists always a ruled surface, assuming that the flecnode curves are distinct (cf. BARNER [3]).

Analogue to (5.14) we describe the side planes of the moving tetrahedron by the plane vectors Qq=−1 (,q,s), Qq= − 1 (,q,s), 11k 21 21k 22 (5.166) 1 1 Sq11= k (,s1,s2), Sq22= k (,s1,s2).

Hence the following multiplication table is valid

qq12s1s2

Q1 000 1

Q2 00−10 (5.167)

S1 01− 00

S2 100 0.

Each asymptotic line on the surface corresponds to a fixed point on the absolute conic if, and only if, equation (5.159) is identical with (5.142), i.e., if, and only if, we put 5 The Method of BOL 140

b1= β12, b01=−β 1β22, b22= β 1. (5.168)

Moreover, setting

2,a01=−α 1α22 a2= α21, a1= α12 (5.169) then the differential formulae (5.148) will take the form q′ = s, sq′ =+()aa +aq+2bs+2bs, 11 101120112 (5.170) qs′22= , sq′22=+aa1()−a0q2+2b2s1−2b0s2.

5.8 A curve-pair on a ruled surface

Let a curve-pair is given on a ruled surface, and let us choose it as the pair of directrex curves of the surface, then the differential formulae (5.148) are valid to represent the surface. But this representation is not uniquely specified, as still the renormalization

qˆ1 = ρ11q, qqˆ 22= ρ 2 (5.171) is possible. In order that the condition (5.149) remains fixed, it must be

ρ12ρ ==k const. (5.172) since

sˆ11=+ρ s1ρ1′q1, sˆ 22=+ρ sq2ρ2′ 2. (5.173)

By differentiating, then

sqˆ′′=+ρ (2ααq+βs+2βs)+2ρs+ρ′′q 1 1 11 1 12 2 11 1 12 2 1 1 1 1 (5.174) ˆˆ =+ααˆˆ11qqˆˆ1 12 2 +22β11sˆ1 +β12 sˆ2 and substituting from (5.171) and (5.173) in (5.174) and comparing the coefficients, we find that

ˆ ρ1′ ˆ ρ1 ββ11 =+11 , ββ12 = 12 . (5.175) ρ1 ρ2 ˆ Thus, one can establish the normalization in such a way that β11 = 0, and because of (5.149), ˆ then also β22.

Assuming that this renormalization is already applied, then

β11 =β22 =0. (5.176)

From (5.168), then

b0 = 0. (5.177)

Hence, the differential formulae take the form q′ = s, sq′ =+()aa +aq+2bs, 11 1011212 (5.178) qs′22= , sq′22=+aa1()−a0q2+2b2s1. 5 The Method of BOL 141

From (5.159), the asymptotic lines on the ruled surface are then described by

2 ub′ =−21ub. (5.179)

On the osculating tangent through the point q1+ uq2 lies therefore the point

2 ss12++ub()1−b2uq2=s1−b1q2+u()s2−b2q1 (mod q1+uq2). (5.180) Putting

ms11=−b1q2, ms22= − b2q1 (5.181) then the line

(,qq12++uu m1m2) (5.182) is the osculating tangent through q1+ uq2 and the line (,mm12) lies on the osculating quadric of the ruled surface. From (5.178), if follows that

mq10′′=+()aa 1+()a1-b1q2+b1s2, (5.183) m-′′22=+()ab2q1()a−a0q2+b2s1.

Moreover, it can be proved that

f0= a0, f11= a− b1′, f22= a− b2′, ja= + b1b2 (5.184) where f01, f, f2 and j are defined in (5.11).

Hence the differential formulae of the moving tetrahedron of the curve-pair, whose vertices are the vectors qq12, , m1 and m2 take the form qm′ =+b q, mq′ =+()jf +fq+bm, 1112 1011212 (5.185) qm′22=+b2q1, mq′22=+fj1(.−f0)q2+b2m1

It is clear that b1 = 0 means that the curve q1(v) touches an asymptotic line on the surface, also b2 = 0 means that q2 (v) touches an asymptotic line. Applying the ∗−transformation, we find from (5.26), (5.29), (5.150) and (5.156) that

11− qq∗==φφ22,(m∗ m+Afq),∗−=φ21f, b∗−=φb, kk k kk k kk k (5.186) j∗−=+φφ22()jAA′′−,a∗−=2(a+AA−2).

−1 By the renormalization (5.171) with ρi = const. and if we put ρρ21 = k, then u= kuˆ (5.187) and ˆ ˆ ˆ −1 ˆ ˆ −1 f0= f0, f1= kf1, f22= kf, bk1= b1, bk2= b2 (5.188) while j remains unchanged. From (5.147) and (5.185) it follows that

(qq12, , m1, m2)==k const. (5.189) 5 The Method of BOL 142

Analogue to (5.166) we introduce the plane vectors Qq=−1 (,q,m), Mq= 1 (,m,m), 11k 21 11k 12 (5.190) 1 1 Qq21=−k (,q2,m2), Mq22= k (,m1,m2) so the following multiplication table, which correspond to (5.167), is valid

qq12mm1 2

Q1 00 0 1

Q2 00−1 0 (5.191)

M1 01− 0 0

M2 10 0 0.

Using (5.181) we find that Q1 and Q2 are the same vectors in (5.166), and that

MS11=−b1Q2, MS22= − b2Q1. (5.192)

For the plane vectors we find from (5.190) and (5.185) the following differential formulae QQ′ =+b M, MQ′ =−()jf −fQ+bM, 1 12 1 1011212 (5.193) QQ′22=+b 1M2, MQ′22=−fj1+()−f0Q2+b2M1.

We notice that by the transition from point coordinates to plane coordinates it is valid the following substitution rule

⎛⎞qmkkbjk fkak ⎜⎟. (5.194) ⎝⎠QMkkbjk −−fkak2 fk For the median curves of our pair, it is valid from (5.179) that

2 bb12−=u 0. (5.195)

So the median curves, if they are real, will be described by

b1 qq12± (5.196) b2 and if they are conjugate complex, will be described by

b1 q1± i q2. (5.197) b2

As the harmonic curves separate both our pair and the median pair harmonically, then they suffice the equation

2 bb12+u=0 (5.198) i.e. the harmonic curves are also described by the equations (5.196) or (5.197). Using (3.195), then for the median curves it is valid that

uA′ =−(2A1)u. (5.199) 5 The Method of BOL 143

The median curves touch then the asymptotic lines if, and only if,

A12= A, (5.200) as from (5.179), for the asymptotic lines through the median points it is valid that u′ = 0. The median curves of our pair are then asymptotic lines if, and only if, (5.200) is valid for all values of the parameter v.

Equation (5.199) describes the principal set on the surface (cf. § 4.8) , which explained before, as this equation suffices both the curve-pair and the median pair. Setting

d1= 2b1, db22= 2 (5.201) hence the principal set can be described by the relation

−1 dd21 ==u const. (5.202)

The osculating positions of the set are characterized by that (5.179) and (5.199) give the same value of u′, i.e. for the osculating positions it is valid that

2 −+bA11()−A2u+b2u=0. (5.203)

The osculating curves of the principal set are identical with the median pair of the harmonic curves of our pair. For, the curves of the principal set, which pass through the points of the generator sufficing (5.203), separate the harmonic curves harmonically and touch the asymptotic lines in these points.

Finally, applying the ∗−transformation, then for A1 and A2 it is valid that

∗−1 Akk=−φ ()AA (5.204) φ′ where A = 1 . 2 φ

5.9 Ruled surfaces through surface strips

It is known that a surface strip can be defined as a space curve, with a given plane at each of its points, and this plane contains the curve tangent at that point, i.e. this plane is tangent of the curve. The strip-curve is described by x(t) with t as an arbitrarily parameter, and the strip plane at t is described by X(t) . We are going to assume always that the strip plane does not coincide with the osculating plane of the strip-curve at any position t.

To find the vertices of the tetrahedron accompanying our strip, we are going to use the basic formulae of a surface strip explained by BOL ([6], § 59, § 60, § 66).

BOL chooses the points x, t= x′, yx=−′′ ax (5.205) as vertices of the tetrahedron, and the planes 5 The Method of BOL 144

X, T= X′, YX=−′′ aX (5.206) as its side planes (cf. Fig. 5.5). For the fourth vertex, he chooses the striction point (central point) of the developable of the strip z =χ(XX, ′′, X′)=χ(X, T, Y) 5.207) and the fourth plane is the osculating plane of the strip-curve Zx=χ( , x′′, x′)=χ(x, t, y) (5.208) where (xt, , y, z)= χ, (XTY, , , Z) = χ, χχ = −1 (5.209) and χ, χ are constants different from zero. For these vectors the following multiplication table is valid xtyz X 00 10 T 0−100 (5.210) Y 10 00 y Z 0001− . Z ≡ osculating plane We have to notice that the edge (xt, ) of the T Y tetrahedron is tangent of the strip curve, while the x tangent edge (xz, ) or (XT, ), which is the intersection of two successive strip t X planes, is called the conjugate tangent to the conjugate tangent (xt, ) of the strip. tangent z ≡ central point

Fig. 5.5

The differential formulae for these vectors take the form xt′ = , XT′ = , ty′ =+ax, TY′ =+aX, (5.211) yz′ =+gat+hx, YZ′ =+ga T−hX, z′ = gx ZX′ = g .

For these vectors, the following transformation formulae are valid

xx∗ = φ , XX∗ = φ , tt∗ =+2,Ax TT∗ =+2,AX (5.212) yy∗−=+φ 12(2AAt+2x), YY∗−=+φ 12(2AAT+2X), z∗ = z, ZZ∗ = . and 5 The Method of BOL 145

gg∗= φ −2 , gg∗= φ −2 , hh∗= φ −3 , aa∗−=+φ 22(2A′ −2A). (5.213)

We notice that g, g and h are semi-invariants, i.e. its vanishing has a geometric significance, (1) g = 0 means, from (5.211), that the striction point z is fixed, and if g vanishes identically, then all the strip planes pass through a fixed point. (2) g = 0 means that the osculating plane Z is fixed, and if g vanishes identically, then the strip curve is a plane curve. (3) h = 0 this is a sextacktic position or coincidence position of the strip, and if h vanishes identically, then the strip is called a coincidence strip. (4) If g and g vanish together, then the strip curve is a plane curve, and the developable of the strip is a cone. (5) If g and h vanish together, then the strip curve is a conic section. (6) Finally, if gg== h=0, then the strip touches a quadratic cone along a conic section.

If we now draw in the strip plane a straight line through each point of the strip curve, then we get a ruled surface, which contains the strip curve and be touched by the strip planes. Assuming that this line coincides neither with (xt, ) nor with (xz, ), then it may be described by gx=+(,t) α(x,z) (5.214) where α ≠ 0 is an arbitrarily function of the parameter t. A curve on our surface, which meets every generator in just one point, may be described by qt()tt=+α z+σ ()x. (5.215)

1 Applying a parameter transformation with choosing a parameter such that A= 2 σ ()t, then from (5.211) we find that qt=+∗α z∗ (5.216) hence assuming that the parameter t is established in such a way that (5.216) is valid, then our curve may be described by the relation qt=+α z. (5.217) Differentiating this relation, we get qz′′=+α y+(ag+α)x (5.218)

qz′′ =+()αα′′ ga+(2+g)t+[(a+gα)′ +α′ g+h]x (5.219) ≡+[]αα′′ ga−(2 +gα) z+[(a+gα)′ +α′ g+h]xq(mod ).

Our curve is then an asymptotic line on the ruled surface if q′′ lie in the tangent plane (,qx , q′), i.e. if α′′ +−gaαα(2 +g ) =0 or if 5 The Method of BOL 146

2.ag=+α′′ α −1 −gα (5.220) α

To find the flecnode point, it must be qq′′′≡ 0 (mod ,q), i.e. if ()ag++α ′′h+α g=0, or from (5.220) if

′ α′′ ++ggαα−1 +22h+α′g=0. (5.221) ( α )

Applying the parameter transformation on the conditions (5.220) and (5.221), one finds that, as α is an absolute invariant, i.e. α ∗ = α, then

∗−1 ′ αφt∗ = α, ∗−2 ′′ ′ αφtt∗∗ =−(2αAα), (5.222)

∗ ⎛⎞α ∗∗ ′′ tt =−φ −3 αα′′ 22AA′ ′ −α′ −4Aα′′ −2Aα′ . ⎜⎟∗ {( ) ( ) ( ) } ⎝⎠α ααα αα

From (5.213) and (5.220) we get the condition

24aA+−′ 4A21=α′′ −2Aα′ +gα − −gα α α or 44AA′ =−22Aα′′+α ′+gαα−1−g −2a (5.223) α α this condition means that the point

qt∗∗=+ααz∗=t+z+2Ax (5.224) describes an asymptotic line on our ruled surface. For the parameter t∗ , the condition (5.221) is valid if

′ α′′−−22AA′ααα′′′−4A′′+8A2 α ( αα) ( α) αα (5.225) ++(ggαα−−11)′′−4A(gαα+g) +2h+2αg=0.

Substituting from (5.223) in (5.225), we get a quadratic equation in A, whose roots are the required flecnode points.

The strip curve is the flecnode curve on our ruled surface, if this quadratic equation α′ 2 turned to be linear one, i.e. if the coefficient 6 α of A vanishes. Hence we conclude that the strip curve is a flecnode curve on our ruled surface if, and only if α is constant. In this case the strip is called a flecnode strip. Moreover, it is valid that

5 The Method of BOL 147

Theorem 5.9.1: Every strip is a flecnode strip on ∞1 ruled surfaces. The lines of every two surfaces of these ruled surfaces build with the tangent of the strip-curve and the conjugate strip tangent along the strip a constant cross ratio.

For α = const., equation (5.223) takes the form

211 − AA′ =+4 (gα −gα−2α), (5.226) whereas equation (5.225) takes the form

gg′′αα−−1+−4(Agα1+gα)+2h=0. (5.227)

Thus, we conclude that the second flecnode point coincides with the strip point if ggα−1 +α=0. (5.228)

Both flecnode curves on our ruled surface coincide together if the condition (5.228) identically satisfied. This is possible, with α = const., if the ratio between g and g is constant. There is then a ruled surface through our strip, on which the strip curve is the only flecnode curve, if g g =const. ≠0. (5.229)

Such strip, for which (5.229) is valid, is called a bi-flecnode strip. Moreover, a surface strip, for which (5.228) is identically satisfied, is called a pangeodetic strip; thus, one concludes that,

Theorem 5.9.2: A flecnode strip of a ruled surface is pangeodetic if, and only if, the strip- curve is a double counted flecnode curve (cf. Theorem 3.3.1).

Assuming, as a special case, that the curve q1(v) of a given pair on a ruled surface is a flecnode curve. As the flecnode points are described by the relation

2 fu20+−2fu f1=0, (5.230) therefore it is characterized by the condition f1 = 0. Making the normalization such that b0 = 0, then from (5.185) and (5.193), the differential formulae take the form qm′ =+b q, mq′ =+()jf +bm, 1112 10112 (5.231) qm′22=+b2q1, mq′22=+fj1()−f0q2+b2m1. and QQ′ =+b M, MQ′ =−()jf +bM, 1 12 1 10112 (5.232) QQ′22=+b 1M2, MQ′22=−fj1+()+ f0Q2+b2M1.

Moreover we assume that q1(v) does not touch any asymptotic lines on the surface, i.e. we assume that

b1 ≠ 0. (5.233) 5 The Method of BOL 148

The flecnode tangents (,qm11) through the points of q1(v) build the flecnode surface associated with q1()v . This surface have a common surface strip with our ruled surface along the flecnode curve q1()v ; i.e. a flecnode strip.

Let us choose the parameter v such that b1 is constant, and setting

qq11= , mq11= b 2,

1 f0 (5.234) qm21= , mm22=+q1. b1 b1 hence the differential formulae of these vectors take the form

qm11′ =+b1q2, mq11′ =−()bb2f01+b1m2,

j ⎛⎞f0′ j (5.235) qm22′ =+q1, mq′22=+⎜⎟fb1+()1b2+f0q2+m1. b1 ⎝⎠bb11

These formulae have the same form as (5.231) with

j f0′ b1= b1, b2 = , jb= 12b, f0= − f0, ff22= + (5.236) b1 b1 i.e., q1(v) is also a flecnode curve on the flecnode surface.

As f1 = 0, then from (5.36) the invariant ∆ takes the form

2 ∆= f0 . (5.237)

The osculating plane of q1(v) contains, from (5.231), the points

q1, qm11′ =+b1q2, qm11′′ = 2(b 2modq1) (5.238) this plane will be then, from (5.190), described by

MQ11− b 2. (5.239)

From the substitution rule (5.194), then the striction point of the common flecnode strip of both ruled surfaces is described by

mq11− b 2. (5.240) Each striction point of a flecnode strip is then polar with the corresponding osculating plane with respect to the osculating quadric of a ruled surface Σ generated by (,qq12); alike naturally with respect to the osculating quadric of (,qq12). Using the formulae (5.211), (5.212), that leads for our flecnode strip to set

d21xq= , d tm=+m, 211 (5.241) db21ym=()2+m2,

d21z =−mm1.

5 The Method of BOL 149

These vectors related to the tetrahedron accompanying the ruled surface (,qq12) are described by

d21xq= , dbtm=+q, 2112 (5.242) db21ym=+2,2f0q1

db21z =−mq12, where d2 is explained in (5.201). It is proper by our parameter choice to introduce a constant factor, so that both sides in (5.241) maintain the same weight. According to the approach (5.241), it follows from (5.231) that the differential formulae of the vectors (5.242) take the form xt′ = , ty′ =+ax, (5.243) yz′ =+gat+hx, z′ = gx with

aj=+b1b2 (5.244)

gj=− + f0+b1b2, gj =+f0−b1b2 (5.245)

hf=+0′ 2b1f2. (5.246)

As the formulae (5.243) have the same form as (5.211), so it represent a strip, where the strip curve x()v= q1(v) is our flecnode curve, the strip plane X contains the points x, t, z, hence the points q1, m1, m1, and it coincides with the common tangent plane of both ruled surfaces.

The quantities g, g and h are then the invariants of the flecnode strip. We have to notice that these formulae are valid for a parameter for which b1 is constant. From (5.186),

∗−1 bb11= φ , (5.247) by differentiating , then

∗ db1 −2 =−φ (2b11′ Ab). (5.248) dv ∗

∗ Now, if v is the parameter for which b1 is constant, then the left side vanishes, and hence for this parameter it is valid that

A = A1 (5.249) where A1 is explained by (5.129).

Hence the vector q2 (v) is proportional to (m11+ A q1), then setting 1 q21()v =(m+A1q1) (5.250) b1 5 The Method of BOL 150

1 so q2 (v) will have the weight 2 as q2 ()v , and by a parameter transformation for which b1 is constant, then (5.250) turned to the form explained in (5.234). The relations

q1= q1, b1= b1, f0=−f0 (5.251) which, from (5.234) and (5.236) are valid by choosing a special parameter, remain also valid by choosing an arbitrarily parameter v.

From mq11+b2===q1′′q1m1+b1q2, it follows that

mq11=−A1+b1q2. (5.252)

Taking into account that the rest of the formulae (5.236) as well as (5.244), (5.245), (5.246) are valid by a parameter transformation, and that for the strip invariants the transformation formulae (5.213) are valid, so we find further that

2 2 j =−bb12 A1′ +A1, j =−bb12 A1′ +A1 (5.253)

aj=+j, gj=−j+f0 , gj = −+jf0 (5.254)

hf=+01′ 2bf2−4A1f0. (5.255)

Because of (5.212), the formulae (5.242) take the form

d11xq= , dA(2tx+=)m+Aq+bq, 1111112 (5.256) 2 dA11(2yt++2A1x)=2b1(m2+A1q2)+f0q1,

dA11z =+mq11−b1q2. or it may be rewritten in the form

d11xq= , dAtm=−q+bq, 111112 (5.257) dA11ym=−221+ b1m2+ f0q1,

dA11z =+mq11−b1q2.

Finally, from the last formula in (5.234), we find that

f0 mq21+=A2m2+A1q2+q1 b1 hence 1 2 mm22=+AA1q2−⎣⎦⎡⎤1m1+()A1−f0q1. (5.258) b1

The lines (mm1, 2) belong to the flecnode congruence of the surface Σ, since these lines lie on the respective osculating quadric of the ruled surface. The curve m1 (v ) of the focal surface (,qm11) is a focal curve of the congruence if, and only if, it is touched by the line

(mm1,2), i.e. from (5.231), iff

jf+0 =0. (5.259) 5 The Method of BOL 151

Specially, q2 (v) is a focal curve of the flecnode congruence of Σ, when (5.245) is valid for a parameter, for which b1 is constant, since, for a parameter of this kind, m1 describes the curve

q2 (v). Similarly, the curve q2 (v) is also a focal curve of the flecnode congruence of the ruled surface (,qq12), when jf+0 =0 for a parameter, for which b1 is constant, thus from (5.236), when

fb01−b2=0. (5.260)

As the left-hand side is a semi-invariant, the condition (5.260) is valid for every choice of the parameter.

Similarly, the lines (mm1, 2) build a torse, when

jf−0 =0. (5.261)

For, in this case, the torse condition (mm12,,mm12′′,) = 0 , from (5.231), is satisfied.

Analogue to that, the lines of the flecnode congruence of (,qq12) through the points q2 (v) build a torse of the second set of this congruence, when jf− 0 = 0 for a parameter, for which

b1 is constant, thus from (5.236), when

fb01+=b20. (5.262)

This condition is also invariant for a parameter transformation and is therefore valid by an arbitrary choice of v.

Because of (5.236), one can write (5.246) in the form

hf=+(22f)b1. (5.263)

This relation is semi-invariant and is valid therefore for every choice of the parameter. The case f0 = 0 means that (5.230) has a root u = ∞, thus that q2 (v) is the second flecnode point of (,qq12). Similarly, f2 = 0 means that q2 (v) is the second flecnode point on (,qq12). If h = 0, then that is a sextaktic position of the strip. Hence,

Theorem 5.9.3: The second flecnode points of both ruled surfaces for a certain value v 0 of the parameter v correspond to each other if, and only if, the flecnode strip has a sextactic position. The second flecnode curves of both ruled surfaces correspond to each other if, and only if, the flecnode strip is a coincidence strip.

5.10 Ruled surfaces with two distinct flecnode curves

Assuming a ruled surface Σ with a distinct pair of flecnode curves is given, and choosing this pair as the directrices of the surface, then in this case we have to put f2 = 0 in (5.231) and (5.232), as the roots of (5.230) must be 0 and ∞. Hence the differential formulae (5.231) and (5.232) will take the form qm′ =+b q, m′ =+()jfq+bm, 1112 10112 (5.264) qm′22=+b2q1, m)′20=−( jfq2+b2m1

5 The Method of BOL 152 and QQ′ =+b M, M′ =−()jfQ+bM, 1 12 1 10112 (5.265) QQ′22=+b 1M2, MQ′20=+()jf 2+b2M1.

By a parameter transformation, then from (5.186) and (5.204)

11− qq∗∗==φφ22,(m m+Afq),∗=φ−2 f, kk k kk 00 (5.266) ∗−12∗− 2∗−1 bbkk==φφ,(j j+A′ −A),Ak=φ(Ak−A) with

1 bk′ Ak = , k =1,2, bk ≠ 0. (5.267) 2 bk

Let us define A0 by

1 f0′ A0 = (5.268) 4 f0 which is always possible, as f0 ≠ 0. From (5.266) we have

∗ df0 −3 =φ (4f00′−Af ). (5.269) dv ∗

If b1 = 0, then the flecnode curve q1(v) touches an asymptotic line on the surface, moreover if b1 vanishes identically, then the flecnode curve degenerates into a straight line. Similarly b2 = 0 means that q2 (v) touches an asymptotic line. As f0 ≠ 0, then the flecnode curves are distinct.

The transversal lines (,qm11) and (,qm22) through the flecnode points are the focal lines of the osculating congruence of our ruled surface. They are intersected by the lines

(,qq12), (,qm12), (,qm21), (,mm12). (5.270)

As the osculating congruence is spanned by the vectors pq= (,12q),p′, p′′, p′′′, whereas the osculating complex is spanned by p, p′, p′′, p′′′, and piv , then the later must also contain the derivatives of the vectors (5.270), then

(,qm12)′ =+bj1 (q2,m2) (m1,m2)+(−f0)(,q1 q2)+b2(qm1, 1) hence it contains also the vector

b21(,qm1)+ b1 (q2,m2). (5.271) The osculating linear complex is described then by

Kq=−b21(,m1)b1(q2,m2), (5.272)

2 5 since this vector is polar with the vectors (5.270) and (5.271) with respect to M 4 in Π . Differentiating (5.272), we get

Kq′′=−b21(,m1)b1′ (q2,m2) (5.273) hence, when bk ≠ 0 and from (5.267), the osculating linear complex is fixed if

A1= A2. (5.274) 5 The Method of BOL 153

From (5.267), this condition is valid identically in v if, and only if

b 1 = const. (5.275) b2

The complex ruled surfaces, which belong to a non-degenerate linear complex, are then described by (5.274) or (5.275). Moreover the complex ruled surfaces, which belong to a degenerate linear complex, are characterized by that b1 = 0 or b2 = 0. For, the linear complex

(5.272) degenerated if, and only if that b1 = 0 or b2 = 0.

The line (,qq12+u m1+um2)of the osculating quadric through the point q1+ uq2 belongs to the osculating linear complex if, and only if

2 bu2−=b10. (5.276)

This equation describes the LIE-points on the generator. They generate the LIE-curves on the surface, which coincide with the median pair of the flecnode curves. Moreover, the harmonic curves on the surface are described by

2 bu2+=b10. (5.277)

For, from (5.195), we find that (5.276) describes the median curves of the directrix curves, i.e., the flecnode curves. The principal set on the surface is described by

−1 dd21u= k, k = const. (5.278) where d1 and d2 are defined in (5.201).

The connecting line between the points pq=+pppq+m+pm 01 12 2 1 3 2 qq=+qq01 1q2+q2m1+q3m2 belongs to the osculating linear complex if, and only if, the vector (pq, ) is a linear combination of the vectors (5.270) and (5.271). This leads to the condition

bp10()q2−−p2q0 b2(p1q3−p3q1)=0. (5.279)

If the osculating linear complex is not special, then equation (5.279) describes the associated null system. From (5.191), the plane which correspond to the point p in this null system is described by

bp21QQ1++b1p0 2 bp23M1+b1p2M2. (5.280)

Particularly, to the points q1, q2 , m1 and m2 correspond respectively the null planes Q2 ,

Q1, M2 and M1; the line (,qm11) or pp13= = 0 have the image (,QM22), hence (,qm22). Both flecnode surfaces are then permuted by the null system of the osculating linear complex. If the ruled surface belongs to a non-degenerate linear complex, then the associated null system permutes the flecnode curves, and hence both flecnode surfaces. 5 The Method of BOL 154

To study the relation between the flecnode surfaces of Σ, let the quantities which represent the flecnode curve q1(v) will be given the index 1, and those for q2 (v) the index 2.

We are going to denote the invariants of the flecnode surface through q1(v) by overbarred symbols, whereas the invariants of the other flecnode surface by the symbol ∧.

The following relations are then valid, by an arbitrarily parameter and from (5.257), for both strips:

d11xq= 1, d22xq= 2, d tm=+a, d tm=+a, 11 1 1 22 2 2 (5.281) df11yb=+2,1 1q1 df22yb=−2,2 0q2

d11z =−ma1 1 d22z =−ma2 2 where aq=−bAq, aq= bA− q, 112 11 22122 (5.282) bm11=−bA2 1m1, bm22=−bA12m2.

Moreover we find that ˆ aj1 =+j, aj2 =+j, ˆ gj1=− + j+ f0, gj2=− + j−f0, (5.283) ˆ gj10=−j+f, gj20=−j−f 2 ˆ 2 jb=−12b A1′ +A1, jb=−12b A2′ +A2. (5.284)

Describing A0 by (5.268), then from (5.255) and as ff12= = 0, we have

hf10=4(A0−A1), hf20= 4(A2− A0). (5.285)

Then from (5.285) it follows that

hh12+=4(f0A2−A1). (5.286)

But, from (5.274), (A2− A1) vanishes exactly if the osculating linear complex of the ruled surface is fixed. Thus, it is valid that,

Theorem 5.10.1: If both flecnode strips of a ruled surface, in the points of the same generator, have a sextactic position, then the osculating linear complex of the surface along this generator is fixed. If the osculating linear complex along a generator is fixed and one of the flecnode strips have a sextactic position, then this is valid also for the other one. If both flecnode strips of a ruled surface are sextactic, then this surface belongs to a linear complex.

Using a coordinate system in which b1 is constant, then the second flecnode point of the surface (,qm1 1)is described by b qq+=u m+1 q (5.287) 121u 1 provided that u satisfying the transformed equation of (5.230), in which we set f1 = 0, then

fu2+2f0=0. (5.288) 5 The Method of BOL 155

From (5.236) and as f2 = 0 then 22fbfb u =− 01= 0= 1 . (5.289) f2 f00′ 2A

The second flecnode point will be then described by

m1+ 2A0q1 (5.290) in a coordinate system in which b1 is constant. In an arbitrarily coordinate system, it will be described by

m10+−(2AA1)q1 (5.291) for, the vector (5.291) is, because of (5.266), a semi-invariant and coincides with (5.290) if b1 is constant, i.e. if A1 = 0. Moreover we have,

−1 Theorem 5.10.2: If the coordinate system be chosen such that f0b1 is constant, then the point m1 will describe the second flecnode point of the surface (,qm11).

For, from (5.267) and (5.268), this coordinate system is characterized by

20AA01−=. (5.292)

Analogue to (5.291), the second flecnode point of the surface (,qm22)is described by

m20+−(2AA2)q2. (5.293)

Both flecnode points lie then on the same transversal line of the osculating quadric if, and only if A1= A2, i.e. iff the osculating linear complex of the ruled surface is fixed.

We have to notice that there are no ruled surface, whose flecnode curves are distinct and in the same time both are conic sections. For, it would be gg12= ==h1h2=0. From (5.285) ˆ it follows that A1= A2, then also from (5.284) j = j, and from (5.283) f0 = 0 which is a contradiction (cf. Theorem 2.7.1).

The line Lm= (1, m2) of the moving tetrahedron generates a ruled surface called the accompanying surface. By differentiating L, we get

Lq′ =+()jf01(,m2)−(j−f0)(q2,m1)

Lq′′ ≡+2(fb02 1,m1)2fb01(q2,m2)+(j′ +f0′)(q1,m2) 22 (5.294) −−( jf′′02)(qm, 1) +2( j−f0)(q1,q2) (mod L)

2 Lq′′′ ≡+()24fb02′ fb02′ (1,m1)+()j′′ +f0′′+4fb01b2−2f0j−2f0 ()q1,m2 2 ++()2 fb01′′4 fb01 ()q22,4mq+−()jf′′ + 0′′+ f0b1b2−2f0j+2f0()2,m1

22′ +−3()jf01()qq, 2 ()mod L, L′ .

The point m1 describes a flecnode curve on the accompanying ruled surface if, and only if, through it passes a straight line whose image in Π5 is polar with L, L′, L′′ and L′′′. This leads to the following condition 5 The Method of BOL 156

223 2 ()j −=f00Cf′′ b1 (5.295) where C is constant.

If one takes as a basis a fixed coordinate system, then (5.295) leads to a first order differential equation. The required coordinate system depends then besides C also on another parameter. If m2 describes also a flecnode curve on the accompanying surface, then from

(5.295), b1 and b2 will have a constant ratio, i.e. the ruled surface belongs then to a linear complex. The accompanying surface is called by WILCZYNSKI the derivative ruled surface (cf. § 2.4).

From (5.264) it follows by repeated differentiation that

qq11′ =+b 2m1,

qq10′′ =++()jf b1b21+2b1A1q2+2b1m2,

2 qq10′′′ =+(4jf′ ′ +A1b1b2+2A2b1b2)1+b1(4A1+2A1′ +3j−f0+b1b2)q2 ++( jf+3bb)mm+6Ab , 0121 112 (5.296) iv 2 2 2 q10=+⎣⎡ jf′′ ′′+()j+f0+b1b2(6A1′ +10A1+2A2′ +4A2

+12A12Aj++62f0+b1b2)]q1 ⎡⎤′′′2 ′ ++⎣⎦jj3 +2A11+4A(2A1+3A1+3 j−2 f0) +4b1b2(2A1+A2) b1q2 2 ++(2 jf2 0′′+16b1b2A1+8b12bA2)mm1+(8A1+16A1+4 j+4b12b)b2 2.

Hence the local coordinates of a point on the flecnode curve are jf++bb j′′++f 42bbA+bbA q =+1 012vv23+ 0 121 122+… 10 23! 24AA′ ++2 3j−f+bb qb=+vvAb23+b11 012v+… 11 1 1 1 1 3! (5.297) jf++38bb j′′++f bbA+4bbA q =+vv01234+ 0121122v+… 12 3! 12 24AA′ ++2 j+bb qb=+vv23Ab+b11 12v4+… 13 1 1 1 1 6

Analogue, for a point on the second flecnode curve we have

24AA′ ++2 3j+f+bb qb=+vvAb23+b22 012v+… 20 2 2 2 2 3! j−+f bb j′′−+f 24bb A +bb A q =+1 012vv23+ 0 121 122+… 21 23! (5.298) 24AA′ ++2 j+bb qb=+vv23Ab+b22 12v4+… 22 2 2 2 2 6 j−+f 34bb j′′−+f bb A +8bb A q =+vv01234+ 0121122v+… 23 3! 12 5 The Method of BOL 157

Now for an arbitrarily point on the generator corresponds to the parameter v, it is valid that

pii=+qu12qi.

Hence, the osculating quadric will be described by

3 v 2 pp −=pp ⎡⎤34fbvv+fu+f′−3 bb A−A u −3fuv+… . (5.299) 03 12 6 ⎣⎦01 0 (0 2 12()1 2)0

5.11 Ruled surfaces with coincident flecnode curves

Assuming that q1(v) is the double counted flecnode curve on the surface, then in this case

f0 = 0 and the differential formulae (5.231) and (5.232) will take the form qm′ =+b q, mq′ =+jbm, 1112 1112 (5.300) qm′22=+b2q1, mq′22=+fj1q2+b2m1 and QQ′ =+b M, MQ′ =+jbM, 1 12 1 1112 (5.301) QQ′22=+b 1M2, MQ′22=−fj1+ Q2+b2M1.

The second directrix curve q2 (v) can be arbitrarily chosen. We are going also to assume that the flecnode curve q1(v) does not touch any asymptotic line on the surface, i.e. b1 ≠ 0. The flecnode surface of our surface will have also a coincident flecnode curves.

If the parameter v is chosen such that b1 is constant, then from (5.234) we have

qq11= , m1= b1q2, 1 (5.302) qm21= , mm22= b1 and from (5.236) j b1= b1, b2 = , jb= 12b, f2= f2. (5.303) b1

The accompanying vectors of the common flecnode strip will be, from (5.241), described by d xq==q, dbym= 2, 111 112 (5.304) db11tq=+ ()2q2, db11z =−()qq22 and for the invariants of the strip, we have from (5.244), (5.245) and (5.246) that

aj=+j, gj=−j, gj = − j. (5.305)

hb= 212f. (5.306)

From (5.305) g = g, then it follows that

Theorem 5.11.1: If the double counted flecnode curve of a ruled surface is a plane curve, then it is a schade contour with respect to a central illumination and conversely. 5 The Method of BOL 158

Finally, using (5.300) and analogue to (5.297) and (5.298), if we choose the parameter and the coordinate system such that h, b1 and f2 are constants, then an arbitrarily point on the flecnode curve can be described by jb++b j′ h+2bbA q =+1 12vv23+ 12 2 +… 10 23! 34jb++b j′ h+4bbA qb=+vb 12vv34+b 12 2 +… 11 1 1 3! 1 24 (5.307) jb++32b j′ h+8bbA q =+vv12 34+ 12 2v+… 12 3! 24 jb++b 51j′ h+0b2bA qb=+vv24b 12 +b 1 2 2v5+… 13 1 1 661 0 and

22Ab ++f A′b 4A22b +bb +3jb qb=+vv22 2 23+22 2 2 12 2v+… 20 2 23! ′ 1 bb + j 4bb12++j h q =+1 12 vv23+ 2 +… 21 23! (5.308) 68bA++f bA224bA′ +2bb +4jb qb=+vv2322 2 +22 22 12 2v4+… 22 2 3! 12 31bb ++j 6bb A 2j′ +h q =+vv12 34+ 12 2 v+… 23 3! 24 and the osculating quadric

3 v 22 pp−pp =⎡⎤33hbvv++……u h ++u 2 f +… (5.309) 03 12 6 ⎣⎦()521 ()()2

5.12 The fundamental mapping –The circular ruled surfaces

The osculating plane of the flecnode curve q1 (v ) contains the points q1′ and q1′′ from (5.296); thus it intersects the line (mm1, 2) in the point

hm11=−A1b1m2. (5.310) Similarly, the point

hm22=−b1+A2m2 (5.311) represents the point of intersection of the same line with the osculating plane of the second flecnode curve q2 (v ). It is obvious that, the two asymptotic tangents through the points aq=−Abq, 11112 (5.312) aq22=−bA1+ 2q2, of the generator (qq12,), pass also through the points h1 and h2. This construction can be used to explain a mapping of the ruled surface in itself, in which each generator will be mapped projectively on itself (cf. Fig. 5.6). 5 The Method of BOL 159

m1 m2

h1 h2

a a1 q1()v q2 ()v 2 Fig. 5.6

Applying the ∗−transformation , thus remains the two osculating planes unchanged, but ∗ ∗ the line (mm1, 2) will be changed to (mm1, 2) . Similarly, the points h1, h2 , a1 and a2 will be ∗ ∗ also changed. We can thus establish v in such a way that a1 coincides with an arbitrary ∗ point P, different from q1, on our generator. The uniquely determined position of a2 can thus be considered as an image of P by our mapping. Such a mapping is called the fundamental mapping of a ruled surface. Therefore, to find the image of an arbitrary point

P, different from q1, on our generator, one draws the tangent of the asymptotic line through

P and intersects it with the osculating plane of q1 (v ). Through this point of intersection, one draws the concordant line of the osculating regulus and intersects that with the osculating plane of the other flecnode curve. If this point of intersection lies on the asymptotic tangent through a point P of our generator, then P represents the image of P by the fundamental mapping.

From (3.266), one finds that the effect of using the parameter v ∗ is that

−1 ∗ 2 aq11=−φ ⎣⎡⎤()AA1−b1q2⎦, (5.313) −1 ∗ 2 aq22=−φ ⎣⎦⎡⎤bA1+()2−Aq2.

Both points have then on the generator the coordinates

b1 AA2 − u1 =− , u2 =− (5.314) A1 − A b2 thus between them exists the relation

−1 bu22−=b1u1 A1−A2 (5.315) or

bu21u2+−()A2 A1 u1−b1=0. (5.316) The relations (5.315) or (5.316) describe therefore the fundamental mapping analytically. It represents an involution if, and only if,

A12= A (5.317) thus, from (5.267), iff b 1 = const. (5.318) b2 5 The Method of BOL 160

Which means that the ruled surface belongs to a non-degenerate linear complex. In the same time, and from (5.131), this condition means also that ρ = 0, i.e. the characteristic curve is a straight line.

To find the condition for which two asymptotic lines on the ruled surface correspond to each other by the fundamental mapping, let us first consider these asymptotics, which from (5.179) are represented by

2 2 ub12′ =u1−b1, ub22′ =u2−b1. (5.319) From (5.315), it follows by differentiation that

−−1222 22A2b2u2−+A1bu11 A2′′−A1+b2(b2u2−b1) +bu11 (b2u1−b1) =0, and thus, taking into account (5.315), one finds

22 AA11′−=A2′−A2. (5.320) Comparing with (5.136), then (5.320) means that ρ = const., i.e. ρ′ = 0. Thus, the characteristic line is a circle, and the surface is a circular ruled surface. Therefore, one can say,

Theorem 5.12.1: If the fundamental mapping maps an asymptotic line of the ruled surface to another asymptotic of the same surface, then this is valid for each asymptotic line and the surface is a circular ruled surface.

The double points of the fundamental mapping satisfy the equation

2 bu22+−()A A1u−b1=0 (5.321) which one finds by setting uu12==u in (5.316). Let D be the double ratio, which these double points make with the flecnode points u = 0, u = ∞, then one finds that

2++D 1 =±4ρ 2. (5.322) D This relation gives a geometrical meaning of the curvature ρ. Thus,

Theorem 5.12.2: The double ratio of the double points of the fundamental mapping with the flecnode points have a constant value on each generator if, and only if, the surface is a circular ruled surface.

The double points generate a pair of curves on the ruled surface called the double curves of the fundamental mapping, for which we have,

Theorem 5.12.3: The double curves of the fundamental mapping are asymptotic lines on the surface if, and only if, the surface is a circular ruled surface.

5 The Method of BOL 161

From the definition of the median curves and the harmonic curves, one concludes that

Theorem 5.12.4: The double curves of the fundamental mapping are the median curves of the harmonic curves, i.e. they are the locus of the points at which the curves of the principal set touch asymptotic lines (cf. § 3.15). Moreover, the double curves of the fundamental mapping are identical with the LIE-curves if, and only if, the ruled surface belongs to a linear complex.

The fundamental mapping can also be extended to the flecnode surfaces, in which the points a1 and m1 correspond to each other. Thus it develops a mapping of the flecnode surface on the origin ruled surface, in which the corresponding generators are mapped on each other. From (5.266), (5.313) one finds that the points

−1 ∗ 2 mm11=+φ ⎣⎦⎡⎤()vvAq1(), (5.323) −1 ∗ 2 aq11=−φ ⎣⎦⎡⎤()AA()vv1()−b1()vq2()v, for a fixed value of the parameter v correspond to each other by this mapping for the same value of the projective parameter A on both generators. Moreover, from (5.257) we notice ∗ ∗ that, the connecting line between the two points a1 and m1 , which correspond to each other by the fundamental mapping, passes through the central point z of the flecnode strip along the flecnode curve q1 (v ). The projective mapping defined by (5.323) for a fixed v can thus be considered as a central projection of both generators one on the other with this central point z as a centre of projection (cf. Fig. 5.7).

m m 2 h 1 1

a1 q1 q2 z

Fig. 5.7

For b1 = const., we notice that a1≡ q2, and from (5.302) one finds that the line connecting the two points q2 and q2 passes through the central point (5.240) of the flecnode strip. If one chooses the directrix curve q2 (v ) on the given ruled surface, then from that the curve q2 (v ) is uniquely determined ; one finds each point q2 by connecting q2 with the corresponding central point, and then intersects this line with the generator of the flecnode surface. q2 (v ) can be chosen as an arbitrary curve, which has no common points with the flecnode curve q1 ()v , and which does not touch any generator of the surface. Similarly, one can achieve that

q2 , under appropriate restrictions, describes an arbitrary curve on the flecnode surface. 5 The Method of BOL 162

5.13 The flecnode sequel of surfaces

Using the previous calculations of BOL, one can deduce easily the following properties of a flecnode sequel of a ruled surface, which we are going to summerize without proof (cf. § 2.15): (1) A real periodic flecnode sequel consists always of an even number of members. (2) A flecnode sequel terminates if the surface belongs to a special linear complex, and only in this case. (3) If a ruled surface of the sequel belongs to a non degenerate linear complex, then the null system of this complex transforms the sequel into itself, where the flecnode surfaces of the complex surface at the same level correspond to themselves. (4) If two consecutive surfaces of the flecnode sequel belong to a non degenerate linear complex, then all surfaces of the sequel will have the same property. The surfaces with even number will be projectively equivalent, and alike those with odd number.

(5) All complexes Ki of the ruled surfaces of a flecnode sequel, which consists of complex surfaces, will be described in Π5 by the points of the same straight line g. Consequently, these complexes will have a common linear congruence C , which described by g. Every

line of C remains fixed by the null systems associated with all complexes Ki . It remains also fixed by the projectivities, which transform the surfaces and the flecnode strips of the sequel into one another. (6) All surfaces of a flecnode sequel of a ruled surface are complex surfaces if, and only if, −1 −1 there exists a coordinate system, in which all the quanitities f01b , f02b and j are constants. (7) There exists periodic sequels consist of complex ruled surfaces, whose period is an arbitrary even number ≥ 6. (8) A periodic flecnode sequel consists always of at least six surfaces. (9) Each flecnode sequel of period six consists of complex ruled surfaces. (10) There exists no flecnode sequel of odd period, which consists of complex ruled surfaces. 6 The flecnodal and LIE-curves on a ruled surface of fourth degree

6.1 Ruled surfaces of fourth degree and STURM-type III

A rational ruled surface of fourth degree will be cut from each plane in space in a 4th order rational curve. This curve have in general three different double points. From that, then the order of the double curve d and the class of the double developable ∆ are the same and equal to three. If this structure is irreducible, then this surface is called, according to STURM, a ruled surface of fourth degree third art ΣIII ; such a surface belongs to a non singular linear complex.

Each plane of the torse ∆ contains always two generators and a conic section of ΣIII ; any two of such conics will be cut from the generators of the surface in a projective row of points.

Conversely, if cy and cz are two conics, whose planes ρ and σ respectively do not coincide, and between the points of cy and cz there exists a projectivity β, then this projectivity generates a rational ruled surface of fourth degree (one says cy ~ cz in β ). The connecting lines of all the corresponding points in ρ and σ fulfil an axial congruence (3,1) of the torse of 3rd class ∆, which also touches ρ and σ. All the other planes of this torse cut ρ and σ in lines, which envelope two osculating conic sections c1 and c2 of ∆. Let the ruled surface belongs to a linear complex , then one can consider it as the intersection of this linear complex with the axial congruence, or dually with the congruence of the bisectors of a space th cubic c3. We have to notice that in this case the LIE-curves of ΣIII are asymptotic lines of 6 order (cf. KRAMES [37] p.255, VOSS [70]).

It is known that two axes of a 3rd class torse are to be cut by the planes of this torse in a projective row of points. Therefore, it follows that, the conic sections, which lie on a ruled surface of 4th degree 3rd art, cut on all the generators of this surface a projective row of 1 2 points. As ΣIII contains ∞ conic sections, then it can be generated by ∞ methods through two projective conic sections, or through all the lines of an axial congruence of a 3rd class torse, which intersects a conic section, whose plane belongs to this torse. Let γ be the null system associated with the linear complex , then one can replace each point of ΣIII by its null plane or each tangent plane through its null point with respect to γ , so ΣIII goes into itself. The double torse ∆ corresponds thereby the irreducible double curve of third order d.

So one can say, the surface ΣIII is self dual. Because of that, ΣIII can also be generated from all the bisectors of a 3rd order space curve, or from the lines of intersection between the corresponding tangent planes of two 2nd class cones, between them (in general) a projectivity relation (cf. [37] p.246). Hence, one can say, each non singular linear complex have with the bisector congruence (1,3) of a space cubic, or with the axial congruence (3,1) of a 3rd class torse a ruled surface of 4th order STURM type III in common. Each of such a congruence contains conversely ∞5 of such surfaces.

Each osculating plane of the space cubic c3 is a bi-tangent plane of the surface, and it contains always a conic section and two generators of the surface. Let us take this plane as ρ. Now, the two generators in ρ intersect in the pole of ρ with respect to K; it is in the same time the only double point of the surface in ρ, and doesn’t lie on cy . Thus one has the following theorem, 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 164

Theorem 6.1.1: In the plane of a conic c2 , the double point of the surface which doesn’t lie on c2 , is the pole of this plane with respect to K. When one makes from it the tangents of c2 , then the points of contact are the LIE-points which lie on c2.

KLAPKA [32] depended on the following theorem to construct the flecnode points on ΣIII ,

Theorem 6.1.2: Let p be an arbitrary generator of a ruled surface ΣIII , which lie in the axial congruence C of a space cubic c3 , and let p′be an infinitesimally close generator of p, then through these two lines a regulus R, which lie in C and touches the surface, is determined. Through each line of the regulus passes a pair of osculating plane of c3 which intersects p in a pair of points of an involution J, to which belongs the flecnode pair of p.

To give a proof of this theorem, let us consider the flecnode quadratic form f (t) from §

3.4.1 equ. (3.46), where when ft()= 0, then the point ty1+ t2z is a flecnode point on (yz). We are going to assume that the determinate ω =≠(yzy′′z )0, is not constant. This assumption is against the assumption of CECH, since CECH’s assumption

is not valid for algebraic surfaces (cf. § 3.1). The tangent of the curve cty12+tz, i.e. the line ′ t=+(,t12ytz (t12yt+z)), is an asymptotic tangent of the surface in the point ty1+ t2z, when ty⋅=()z′′ 0, i.e. when

22 2()yzy′z′ (t21t′ −=t1t2′ )(yy′y′′z)t1 +{( yzy′z′′ ) −( yzy′′z′ )}t1t2+( zz′yz′′ )t2 or simply when

2(ωϕ(tt21′′−=t1t2)t). (6.1)

Then, it will be

22 22ωϕty=+()z(t) ω{(yy′′)t1+⎣⎦⎡⎤(zy)+(yz′)t1t2+(zz′)t2} (6.2)

The asymptotic tangent t is a flecnode tangent, when ty⋅=()z′′′ 0, (6.3) i.e. when t1: t2 is a root of the flecnode form

′ ′′ ′ 1 ′ ′′′ 2 f ()t =+2ω{⎣⎦⎡⎤(yy y z ) 3 ()yy y z t1 ′ ′′ ′ ′ ′ ′′ 11′ ′′′ ′ ′′′ ++⎣⎡()zy y z (yz y z )+33()zy yz +(yz y z)⎦⎤t12t (6.4) ′ ′ ′′ 1 ′ ′′′ 2 ′ ++⎣⎦⎡⎤()zz y z 3 ()zz yz t2 } −ωϕ(t) or simply, f ()tt=2ωψω()−′ϕ(t). (6.5) 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 165

Now, let us explain the meaning of the two equations ϕ(t) = 0 and ψ (t) = 0.

For tt12: = const., the curves cty1+t2z form a system of parametric lines. If t1: t2 is a root of the equation ϕ(t) = 0, then the tangent of the parametric line in ty1+ t2z coincides with the asymptotic tangent of the surface. On each generator there exists, when the discriminate of ϕ ≠ 0, two points which have this property. KLAPKA called them the parametric points. If the points y and z are functions of a parameter t, then the parametric points are not changed if one substitutes the parameter t by a new one, or if one multiplies y and z by the same function ρ(t) ≠ 0. By such transformations, the flecnode form f (t) will be multiplied by a non vanishing factor. Similarly for the determinant ω, which vanishes only for the parameter value corresponding to a torsal generator, whereas ω′ changes but remains always as a linear combination of ω and ω′. Thus, from (6.5) the form ψ (t) f ()tt+ω′ϕ() ψ ()t = (6.6) 2ω remains by such transformations as a linear combination of the expressions f (t) and ϕ()t , thus the null points of the form ψ (t) build on the generator (yz) an involution of the second degree, to which belongs the pairs of the flecnode points and the parametric points. ,

When cy and cz are two projective plane curves in the planes ρ and σ respectively, where the projectivity relation is determined by the parameter t, then the parameter curves cty1+t2z (tt12: = const.) are plane curves and these planes build a torse of third class. The intersection of these planes with the torse build an axial congruence C of a space cubic c3 , in which lie the surface with the curves cy and cz . It can be proved that the null points of the form ψ (t) are the points through which pass two parametric curves in the osculating planes of c3 , which intersect in a line r of the regulus of contact R, i.e., since the parametric curves are plane curves, then the null points of the form ψ ()t are the intersection points of the generator (yz) with both the osculating planes of the cubic c3 , which pass through the line r.

From the previous theorem, the construction of the flecnode points can be simply made as follows:

As we explained before the projective row of points on cy and cz build a collineation between the two planes ρ and σ. Let l be the intersecting line of the two planes, where there are no points on this line that self-corresponding in this collineation. Let the projectivity β between cy and cz be given by three pairs of points, then also the intersection points of the tangents of cy and cz will correspond to each other in a point correspondence, and the collineation will be then totally complete. The corresponding pair of lines in this collineation, which intersect each other on l, will envelope two conic sections c1 and c2 , which correspond to each other also (pointwise and tangentwise) in the same collineation. Now to find the self-corresponding points U and V of the involution J, let us construct the tangent ty of cy . This tangent intersects c1 in the points of contact of the tangents m and n of c1, see Fig. 6.1. When m~ m and n~ n in β, then the planes (nn) and (mm) intersect the generator (yz) respectively in the required points U and V , which separate the flecnode points harmonically.

6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 166

t ρ y n cy

Y

U c1 m

V l

m

Z

cz n σ

Fig. 6.1

The construction of the LIE-points on a non-torsal generator (yz) of the surface ΣIII is given from the following theorem,

Theorem 6.1.3: The tangent planes of the surface in the parametric points T1 and T2 on the generator (yz) intersect the surface in two conic sections c1 and c2 respectively. c1 intersects (yz) not only in T1 but also in a double point Q2 of the surface, similarly c2 intersects (yz) in T2 and also in Q1. The LIE-points L1 and L2 are the self-corresponding points of the involution determined by the pairs T1, Q1 and T2 , Q2.

Actually, this theorem is equivalent to the following theorem: the LIE-points L1, L2 , the pair

T1, T2 , and the pair Q1, Q2 belong to the same involution. If this is valid, then the following cross ratio relation is fulfilled (cf. Fig. 6.2)

()LL12T1Q1 = (LL12T2Q2). (6.7)

c 2 T 2 ()yz L 2 Q 2 Q1

L1

T 1

c1

Fig. 6.2 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 167

Let us consider an arbitrary plane through a generator p, and let F and P be two different points in the plane and don’t lie on p, then the point L1 and its corresponding point

L2 , for which (6.7) is valid, lie on a conic section m, Fig. 6.3, passing through the points

FP,,,RS,L1, where R and S are the points of intersection of the lines F∨ T1 and FQ∨ 1 with the lines P∨ T2 and P∨ Q2 respectively.

The position of the points FP, , R, S don’t depend on L1, when L1 changes, the conic section

m builds a set of conics with the basis FP, , R, S and the pair L1, L2 builds DESARGUES’s involution∗). The degenerated conic sections FR∨ , P∨ S and FS∨ , P∨ R cut on p the pairs T1, Q2 and Q1, T2 respectively, such that the LIE-points belong to the involution that determined by these pairs. They are the common points of the involution, which is determined by the pairs TT12, QQ12 and TQ12, TQ21, i.e. the LIE-points are the self-corresponding points of the involution, which means that the previous two theorems are equivalent as we assumed. , p

Q1

Q 2

L 2 S

T1 L 1 T2

R P

m F

Fig. 6.3

Let the equations of the projective conic sections cy , cz be in the form

y1 = 0, zt1 =+(tα ), yt2 =+()ta, z = 0, 2 (6.8) ybt2 , 2 3 = zt3 =+(γ ), 2 2 yt4 =+()c. zt4 = β where c ≠ 0, γ ≠ 0, a ≠ α ≠ 0, and the projectivity is determined by the parameter t. Let p be the generator (yz) which corresponds to the parameter value t = 0. The point ty1+ t2z is a LIE-point on p, when the tangent

qt=+()12ytz, t1y′′+t2z, i.e. the line

22 qy=+()y′′t1{( yz)(+y′z)}t1t2+( zz′)t2, (6.9)

∗) DESARGUES’s theorem says, the pairs of points in which a fixed line is met by the conics of a pencil are pairs of an involution ([60], p. 161). 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 168

iv is a linear combination of the lines (yz), ()yz ′, ()yz ′′, ()yz ′′′ and ()yz , i.e. when t12: t annulled the form

ωω12 13 ω14 ω23 ω24 ω34

pp12 13 ... .

pp12′′13 ... . = 0, (6.10) ...... ()iv ()iv pp12 .... 34 where for simplicity, we put yz −=y z p =yz ik ki ik ()ik ω =+ty22y′′tt yz +zy′+tzz′. ik 11()ik 2{()ik ( )ik } 2( )ik Thus, since

yy′′=yy =yy′′===yy yy′0, yy′=−ac2 , ()12 ()13 ()14 ()23 ( )34 ( )24 zz′′′==zz zz =zz′=zz′=0, zz′=−αγ 2 , ()12 ()14 ()23 ()24 ()34 ()13 yz′′+=zy 0, yz′+zy′=0, yz′+zy′=0, ()12 ()12 ()13 ()13 ()24 ()24 yz′′+=zy −c22αγ,,yz′+zy′=−a yz′+zy′=2γc γ−c , ()14 ()14 ()23 ()23 ()34 ()34 ( )

22 pp12 ==13 p14 ==pp23 24 =0, p34 =−γ c,

2 pp12′′==13 p24′=0, p1′4 =−αc, 2 (6.1) pa23′′==γγ, p34 −2 c()γ+c,

pp13′′ ==24′′ 0, p1′′2 =−aαα, p14′′ =−c()2 +c, 22 pp23′′ =+γα()2 γ, 3′′4 =−()c+γ+4cγ,

p1′′′21=−()ap+αα, ′′′3=− b, p1′′′4=−(α+2c),

pa23′′′ =+2γα, p24′′′ =β, p34′′′ =−2()γ+c,

()iv ()iv ()iv pp12 ==14 −1, p13 =−b, ()iv ()iv ()iv pp23 ==1, 24 ββ, p34 =b−1, then one finds the form

2224 22 2222 24 22 ()ac−−αγ{abc()αγt11−2abcαβγtt2+αβγ(a−c)t2}. (6.12)

The point ty1+ t2z is a parametric point when

tt12= 0 (6.13) and is a double point when

22 22 22 2 2 ct()αγ−−1αβγt2=0, and abct12−γ ()a−=c t 0, 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 169 i.e. when

2 4 22 2 2⎡⎤2 2 2 2 2 4 22 abc(αγ−−)t1cγ⎣⎦()a−c (α−γ)+αβab t1t2+αβγ()a−c t2=0. (6.14) One notices that the quadratic forms (6.12), (6.13), (6.14) are not linear independent, which gives the required proof. ,

6.2 An application on a general normal surface of a quadric along a plane section

The normal surface Σ of a second degree surface Φ along a plane cut c is identical with the normal surface of the second degree cone Γ1 circumscribing Φ along c (cf. [55], p.290). We can thus concentrate on the study of the normal surface of the second degree cone. A line through the vertex S of Γ1 normal to its tangent plane fulfils a normal cone Γ2 of Γ1. The generators of these both cones are in a 1-1 correspondence to each other. If one moves the generators of Γ2 along the corresponding generators of Γ1, till they intersect c, then one gets the generators of Σ. Thus, Γ2 can be considered as the director cone of Σ. The projectivity determined by the generators of both cones between the points of c and the improper curve th c∞ of Γ2 generates the required normal surface, which is in general a ruled surface of 4 degree and STURM-type III.

y l

p1′

p′ Q1 Z L2′ t1 t

Y1 Q′ 1 ′ F1 Q′ x 2 Z L1′ T′ 1 T′ X X 2 Q F′ R 2 Y 2 r

t Q2 2 Q p2′ S′ q

Fig. 6.4

The normal surface has a double curve of third order; its conic sections lie in planes, which envelope a developable surface of third class. Now, let the plane cut of the quadric cone is an ellipse c in the horizontal plane Π1. Let x be the major axis of the ellipse, y be 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 170 the minor axis, and let the cone vertex S be given by its horizontal projection S′ and its vertical projection S′′ (in Fig. 6.4 one sees just S′ ). For the general case, S′ must lie neither on x′ nor on y′. A plane through a general generator of the surface is normal to the ellipse c and the contour of the surface in the horizontal projection is its evolute. If one makes two tangents of c from S′, then the normals of the ellipse c at the points of contact Q and R are the two lines q and r respectively, which are two generators of the surface in the horizontal plane. The planes through the conic sections of the normal surface, to which belongs also the plane at infinity, intersect each of them in the tangents of a parabola (cf. [55], p. 295). Thus, let the envelope of the lines q, r, x, y be a parabola l in the horizontal plane.

Let us first construct the parametric points on a generator p through an arbitrary point Q on c, and whose horizontal projection is p′. From Q one can construct two tangents t1 and t2 of l. Now, if X ≡∩qx, Z ≡∩qy, Yq11≡ ∩t, Yq22≡ ∩t, and X ≡ px′∩ , Z ≡py′∩, then the horizontal projection of the required points T1′, T2′ fulfil the following equations QY QT′ ZYZT′ 1= 1 (also 11= ), XY1 XT1′ XY1 XT1′

QY QT′ ZYZT′ 2= 2 (also 22= ), XY2 XT2′ XY2 XT2′ since, it is known that three arbitrary plane conic sections of the normal surface cut on each generator two segments, whose ratio is constant (cf. [55], p.298). Thus the construction of the parametric points follows as in Fig. 6.4

– The construction of the double points Q1 and Q2 of the surface on the generator p :

t1 and t2 intersect c besides in Q also in the points Q1 and Q2 respectively. Let the

normals of c in these points be p1′ and p2′ respectively. Then, the points of intersection

of these lines with p′ are the horizontal projections Q1′ and Q2′ respectively of the required double points.

– The construction of the LIE-points:

The self-corresponding points of the involution defined by the pairs T1′, Q1′ and T2′, Q2′

are the horizontal projections of the LIE-points L1′ and L2′. (In Fig. 6.3 this involution is

elliptic, thus the points L1′ and L2′ are schematic, similarly the flecnode points F1′ and

F2′)

– The construction of the flecnode points F1′ and F2′: The flecnode points can be constructed when one intersects the parabola l with the tangent t of c in the point Q. If these points of intersection are real, then the tangents of l in these points are the traces of the planes ρ and σ of the conic sections respectively.

In the vertical plane Π2 , let the parabola k be the envelope of the lines x, z, u, v, where the last two lines are the generator of the surface in the vertical plane. Thus, one can easily construct the traces of the planes ρ and σ in the vertical plane as tangents of the parabola k. The generator p intersects both planes in the points U and V which divide

the flecnode points harmonically. The horizontal projections F1′ and F2′ of the flecnode points are thus the self-corresponding points of the involution determined by the pairs

L1′ , L2′ and U′, V ′. (Obviously, it is enough to find one point U′ or V ′, since

()UVT′′12T′′=−1). 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 171

If the intersection between t and l is not real, then one can make the construction using the middle of the above mentioned involution: Let us construct the tangent t1 of the parabola l, which is parallel to t. From the point of intersection t1 ∩ x, let us construct another tangent

n1 of the parabola k. As t1 and n1 are the horizontal and vertical traces of the planes respectively; these traces intersect p in the middle of the involution.

From the previous construction, one notices that the position of the LIE-points and the parametric points in the horizontal plane do not depend on the coordinates of the vertex S. It follows that all the surfaces of the normals, which one gets by the variation of S along a vertical line, have a common horizontal projection of the LIE-curves.

6.3 STURM-type IX and IV

These two types of ruled surfaces of fourth degree are dual to each other. The surfaces type IX resp. IV can be considered as the locus of the common lines of a special linear complex with directrix d and the axial congruence resp. the bisector congruence of a space cubic.

Surfaces of type IX has also d as a triple directrix. A surface ΣIX can also be constructed through a torse of 3rd class and a pencil of planes projective to it, when both have no common rd plane together. Moreover, ΣIX can be constructed by a cubic involution on a torse of 3 class or also by two conic sections and a straight line which intersects both of them. Each plane of the bitangent developable of ΣIX contains two generators and a conic section which passes through the point of intersection on d (cf. [37], p.256). If ρ is the plane of a conic section cy , then the other planes intersect ρ in lines which envelope another conic section l. Theorem 6.1.2 is valid for these surfaces, and it is sufficient to construct the flecnodes on an arbitrary generator p of the surface:

Let Y be a point of this generator on cy , then one can construct the tangent ty of cy in this point. This tangent intersects l in two points M , N at which the tangents of l represent the traces of two planes, which intersect p in two points U, V. Then, the harmonic conjugate of the intersection point of p and d with respect to U and V is the required flecnode point on p.

Dually, for a surface of type IV, one can construct a pair of tangent planes λ, µ whose points of contact separate the flecnode on d and the required flecnode harmonically.

We have to notice that since ΣIX or ΣIV belongs just to a special linear complex, then its directrix line is its only LIE-curve (cf. § 1.6).

6.4 STURM-type V

This type of surfaces is self-dual. The double curve consists of a conic k and a straight line cy which is a double directrix, and it must meet k. From each point of cy pass two generators which meet the residual conic cz of a section by any bi-tangent plane, so that an involution is generated on cz . The bi-tangent plane through a pair of generators from each point of cy envelopes a second degree cone of vertex V , where cz is a conic on this cone. The surface

ΣV can also be constructed by a (2,1)-correspondence between a conic section and a straight line in a general position, or by a (2,1)-correspondence between the tangent planes of a cone nd of 2 class and a general pencil of planes. Moreover, ΣV can be considered as the locus of a 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 172 straight line, which always touches a cone of 2nd class and intersects a fixed tangent of this cone as well as a conic section whose plane touches the cone.

One can then set

y1 = 0, zt1 =+()t1,

y2 = 0, zt=+l, 2 (6.15) 2 yt3 =−p, zt3 = ,

yt4 = . z4 = 0.

Let the edges of the tetrahedron of reference xx12= = 0 resp. xx24= = 0 be two arbitrary generators p resp. q of the surface; these lie in the plane of the conic section cz and intersect in a point S ()0,0,1,0 on cy . The values of the parameter t, which correspond to these positions, are t = 0 resp. t =∞. Moreover, it is valid that ω ==()yzy′′z −()t 22+p (t +2,lt +l) (6.16) ϕ =+⎡⎤22lt 2 t l −p −lp t −ltt t . ⎣⎦{}()() 122

One can find ψ from ϕ by differentiation taking t1 and t2 as constants. The points z(0) resp. z(∞) are obviously the parametric points on p resp. q ; let us denote them by P resp. Q. The line P∨ Q passes through V and is therefore a generator of the quadratic cone. The parametric equations of the flecnode curve – i.e. its branch which is different from cy -take the form

423222 x1 =−{t()l p−l −46plt−lpt−4lpt+lp( pl−pl−)}t(t+1), 423222 x2 =−{}t ()l p −l −46plt −l pt −4l pt +lp()p −lp −l ()t +l , 43 2 xl3 =+{}tlt−2lpt−lp(t−p) (6.17) +−{}t 42()l p −l −4 plt3−6l 2pt 2−4l 2pt +−lp()p lp −l t,

43 xl4 =+{}tlt−2.lpt−lpt

Let us denote the required flecnodes on p resp. q by F resp. G, and let R be the pole of

P∨ Q w.r.t cz , then one can describe the first three coordinates of the mentioned points as follows

x12xx3 Q 100 S 001 Vpl0 (6.18) R 111 Gp10−−l2 1

Ql1 10− 1 Up0 l 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 173

where Q1 is a singular point lying in the intersection of q and k. One can construct point

U as follows: the tangent of cz in P – i.e. the asymptotic tangent of the surface – intersects q in a point 1; the line S∨ R intersects P∨ Q in a point 2. If one projects the vertex V from the point of intersection of 1∨ 2 with p on q , then one finds U. The condition of U is obvious of the following theorem,

Theorem 6.4.1: The points U and G form a pair in an involution, which is determined by the self-corresponding point Q and the pair S, Q1.

Let x1 and x3 resp. y1 and y3 be the coordinates of the points H resp. K which together with Q separate the pair U, G resp. S , Q1 harmonically, then it is valid that xl− x p 13 =−1, 2 xl13−−x ()l p−l resp.

y1 3 =−1, yy1 −−()1 l i.e.

x y 1 = l −1, resp. 1 = l −1, (6.19) x3 2 y3 2 so that HK≡ .

From the known points Q, S, Q1 and U on q, one can construct the flecnode just with a ruler. An analogue construction is valid for the flecnode on p and for each arbitrary generator of the surface.

6.4.1 An application on the arched surface of an oblique passage

This surface has three directrices: two arbitrary congruent circles k1 and k2 lie in two parallel (vertical) planes; and a line n normal to the planes of the circles passing through the mid- point of the line connecting the centers of the two circles∗).

An arbitrary generator of this surface can be considered as the connecting line of two points with the following Cartesian coordinates: x =−ar+ cosω, x = ar+ cosψ , yb= , yb=− , (6.20) zr= sinω zr= sinψ where, if

k = r+ a, r− a

∗) For the geometrical characteristics of this surface, s. WIENER [77] , p.475. 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 174 then the following relation between ω and ψ is valid

ψ ω tan 2= k tan 2. (6.21)

The parametric equations of the curves k1 ≡ cy resp. k2 ≡ cz can be represented as follows

2 22 yr1 =−a−()a+rt, za1 =+r+(a−r) kt, yb=+1,t2 zb=− 1+k22t, 2 () 2 () (6.22)

yr3 = 2,t zk3 = 2,rt 2 22 yt4 =+1. zk4 =+1.t

ω where t = tan 2 . Hence, one can find the parametric equations of the flecnode curve l in the form

2 xa1 =−21k()ϕϕ(1+6+ϕ), ⎡⎤2 22 xb2 =−()ϕϕ11()+k()1++2ϕ(k−1), ⎣⎦ (6.23) 2 xk3 =−21r()k()1+6ϕϕ+t,

2 3 xk4 =−()11()+ϕ , where ϕ = kt 2. Thus it follows that: The horizontal projection l′ of the flecnode curve is a rational cubic.

The contour of the surface in the horizontal plane consists of the two edges t1′, t2′ and a curved line, which can be simply proved that it is a hyperbola (cf. WIENER [77], p.479) whose asymptotes are y′ and a. We are going now to show that the cubic l′ has a cuspidal point at infinity in the direction of the asymptote a. A cuspidal tangent u∞ , which is the line at infinity and an inflexion point at the origin O′.

To prove that, let us consider the equation of the surface, w.r.t the Cartesian coordinate system in Fig. 6.5, in the form

2 ⎡⎤22 2⎡22 222⎤ Fa≡+⎣⎦xyb()x+z =b⎣rx+(r−a) z⎦ (6.24)

(cf. LORIA [44], p.239). For eliminating z, one has

∂F = 0, ∂z and excluding the torsal generators in the horizontal plane, then the equation of the hyperbola takes the form

2 44ab2x2−−+a()r22a xy b(r22−a )=0, (6.25) from which, the equation of the asymptote a is 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 175

abx −−(r 22a )y =0, thus its direction = ab . ra22−

1 k1′′ k2′′

2 O′′ T′′ 1

l′′

T2′ k2′

N F t1′ l′

O′ t′ 2 f P

k1′ M T′ 1 e Q a m s b

i

y′ ≡ v

Fig. 6.5 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 176

From (6.23) one finds that the point D∞ , whose parameter value ϕ = −1, of the cubic l′ is a ∞ double point, and that the equation x4 = 0 has a triple root. The Cartesian coordinates of D are then

22 x12:xx: 4=+2(ra):b(k−1):0=(r−a):ab:0, (6.26) thus the point at infinity lies in the direction

x2 ab = 22, x1 ra− which verifies the first part of our statement.

The point on l′, which corresponds to the parameter value ϕ =1, is obviously the origin

O′. If λ11xx+λ2 2=0 is the equation of the tangent of l′ in this point, then it must be valid that

2222 λϕ12()ra+−(1k)(1+6+ϕ)−λb⎡⎤(1+k)(ϕ+1)+2ϕ(k−1) =0, { ⎣⎦}ϕ =1 thus

2 λλ12:2=−bk⎣⎦⎡⎤3(1+k):8ak, hence, the direction of the tangent i is

2 λ bk⎡⎤23−+(1k) −=1 −⎣⎦. (6.27) λ2 8ak

The equation of the intersection of i and l′ - with x1 and x2 from (6.23) – is therefore

λ11xx+=λ2 2 0, that is

()1+k 2 (ϕ −1)3 , (6.28) which verifies the second part.

Moreover, we are going to show that l′ touches the two torsal generators of the surface in the horizontal plane, and we are going to find the points of contact. First we show that one of both torsal generators corresponds to the parameter value ϕ = 0, b and its direction is − a .

6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 177

Let us denote the point on l′ for which ϕ = 0 by T′, then from (6.23), its coordinates are

xa1 =−2,k 2 x2 =−bk()1,+ (6.29) x3 = 0, 2 xk4 =−1.

For ϕ = 0, one has also from (6.23), dx 1 =−10ak, dϕ

dx 2 2 =−bk⎡⎤12+ 2 + 1−k, (6.30) dϕ ⎣⎦()() dx 4 =−31 k 2 dϕ () then the point at infinity on the tangent t1′ of l′ in T1′ is described by

dx − 3x, dϕ i.e. the point whose coordinates are −4,ak 4bk, 0, 0 (6.31)

′ b thus the direction of the tangent t1 is − a .

From (6.29) one can easily construct T1′: Since t1′ is already known, then it is enough to find the x -coordinate of T1′, which is from (6.29)

22 x1 −2ak r −a x == 2 = . (6.32) x4 1− k 2r

Thus, it follows that, if the projections of the two circles k1 and k2 intersect in a point 1, then one can measure from the origin O′′ in the negative direction of x -axis a distance 2r to get a point 2. From point 1 one constructs a perpendicular to 1∨ 2 which intersects the x -axis in the projection T1′′ of the required point (cf. Fig. 6.5).

For the second torsal generator t2′, it is symmetric with t1′ w.r.t the origin O′ and its point of contact T2′ is similarly symmetric with T1′.

If the projections of the two circles k1 and k2 don’t intersect, one can nevertheless construct point 2, then the required point T1′′ can be determined as corresponding to 2 in the involution on x defined by the set of circles determined by k1′′ and k2′′.

One can also find T1′ as the mid-point of the distance on t1′, which lies between the asymptotes a and y′, since T1′ is the point of contact of t1′ with the contour hyperbola. 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 178

One can now construct the cubic l′, since the cuspidal point D∞ with its tangent u∞ , the inflexion point O′ and the tangent t1′ with its point of contact T1′ are known. This construction depends on the following : The intersection between two pencils in a [1,2]- correspondence is a curve of third order, which has the vertex of the 1-correspondence-pencil as a simple point and has the vertex of the 2-correspondence-pencil as a double point (cf. WEYR [76] p.124 ).

∞ Thus let T1′ be the vertex of the 1-correspondence-pencil, D be the vertex of the 2-corresp.- pencil, and let the direction conic c (cf. WEYR [76] p. 35) be determined w.r.t the inflexion point O′, then according to WEYR ([76] p. 124): If the point, corresponding to which the direction conic is determined, is an inflexion point, then the direction conic touches a line which separates with the inflexion tangent the two lines connecting the inflexion with the two vertexes of the two pencils harmonically, where the point of contact is the intersection point of this line with the common line of the two pencils. Hence, the direction conic c touches the cuspidal tangent u∞ in the cuspidal point D∞ , and another line v, which passes through O′ and together with the inflexion tangent i separates ∞ the pair of lines connecting O′ with T1′ and D harmonically. Moreover, the point of contact ∞ of v with c lies on the line T1′∨ D. The conic c touches also the lines t1′ and T1′∨ O.′ Hence, the conic c is determined by the last mentioned pair of tangents, the tangent u∞ with its point of contact D∞ , and the condition that the tangent in its intersection point with ∞ ∞ T1′∨ D ( which is different from D ) passes through O′.

If one applies BRIANCHON’s theorem, then the tangent v of the direction conic, which is a parabola whose axis parallel to a, can be constructed as follows: Let e be a line passing through O′ parallel to t1′, and f be a line passing through T1′ parallel to a. Then one projects the intersection point e∩ f in the direction T1′∨ O′ on t1′, thus the line connecting this last projected point with O′ is the required tangent v ≡ y′, cf. Fig. 6.5.

In order to construct the inflexion tangent i, it is enough to find the line which with v separates the pair T1′∨ O′ and a harmonically. Thus one can determine the direction parabola c; the condition that we mentioned at the beginning can be substituted by the tangent v with its point of contact V , so that other points of the cubic l′ can be constructed using the construction of the direction conic.

The construction could be more simple if one has the two points of intersection of a line s passing through O′ perpendicular to a, with the cubic l′. In this case the curve l′ will be determined by the projection of a 1-corresp. pencil with vertex point O′, a 2-corresp. pencil ∞ with vertex point D and a direction conic w determined w.r.t T1′ (cf. WEYR [76], p. 148).

The direction conic w is a parabola whose axis is parallel to a, it touches T1′∨ O′ and also the inflexion tangent i in its intersection point with f . From the point s∩ f let us make two tangents of the parabola w and intersect these two tangents with T1′∨ O′ in two points. By projecting these two points from D∞ on s, we get M and N, which are the required intersection points of s with l′. It will be O′M = O′N.

6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 179

Let us determine the direction parabola h w.r.t M. Its axis is parallel to a and touches s and the inflexion tangent i in a point on a line b passing through M parallel to a. Since s is a vertex tangent of h, one can easily construct the focal point F of this parabola. So, one can easily construct other points on the cubic l′ as follows: Let us make an arbitrary line m parallel to a. Let Pm≡ ∩ s, and let us construct n perpendicular to P∨ F through P, so n intersects b in a point Q, whose projection from O′ on m is a point on the cubic l′.

The construction of a tangent tZ of the flecnode curve l′ in an arbitrary point Z in the horizontal projection depends on the basis that this curve is a W-curve . It follows that the division ratio of the point of contact of the tangent w.r.t its intersection points with a and i is always the same. This ratio is obvious if one has the tangent t1′ with its point of contact T1′.

Thus, one can construct tZ′ as follows: Let the line T1′∨ Z′ intersects a in a point A, and t1′ intersects i in I. Then the line A∨ I intersects the line through Z′ parallel to t1′ in a point V , and if the line through V parallel to a intersects i in a point W , then the line

Z′′∨≡WtZ .

6.5 STURM-type VI and XI

These two types are dual to each other. A surface of type VI can be considered as the locus of the lines which determine a [2,1]-correspondence between a line cy and a conic section cz in a plane ρ. The double curve of ΣVI consists of the directrix cy and the conic cz . The surface nd. ΣVI has in general a torsal line of 2 order, which is in the same time a double directrix, and two torsal lines of 1st order. The bitangent developable consists of the triple counted pencil of planes whose axis is cy . The line cy intersects the conic in a point Ut( = 0), to which – if we consider it as a point of cz - corresponds a pair of points on cy , one of them is already U and the second is Ct1 (=∞), the cuspidal point of the surface. There exists another two cuspidal points of the surface on cz , C2 and C3 , whose connecting line is m. We notice that cy is a simple directix and therefore a generator of the surface (cf. Fig. 6.6).

Let V be the intersection of tU and m, where tU is the tangent of cz in U; Q be the intersection point (different from U ) of the polar of V w.r.t cz with cz , and if one chooses the tetrahedron of reference so that

U ()1, 0, 0, 0 , C1 ()0,1,0,0 , V (0,0,1,0), Q(0,0,0,1) then one can take the parametric equations of the directrix curves of the surface VI as follows

22 yz11==1, (1+t) , yt==, z0, 22 (6.33) 2 yz33==0, 2t(t+1), 2 yz44==0, t.

Thus, the flecnode form (cf. § 3.4 equ.(3.46)) will be

42 3 f ()tt=−16 {(t1)t21−t}t2. (6.34) 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 180

If t ≠ 0, then the point

x =−(1t23)y+z, i.e. the point whose coordinates are

22 x1 =−⎣⎦⎡⎤(1tt)+4, xt=−(123), 2 (6.35) 2 xt3 =+2(1 ),

x4 =1 is the flecnodal. The line connecting x with a point A, where At(), t2 −1, 0, 0 , (6.36) intersects ρ in the point B()4,tt 0, 2(2 +1), t. (6.37)

The locus of the lines A∨ B is a surface µ of fourth degree with two directrices cy and cb , since the parameter t defines a [2,2]-correspondence between the points A on cy and the points B on cb . The flecnode line cx is a part of the intersection between the two surfaces µ and ΣVI . Instead of the point B, cb ≡ m; if z1,,… z4 are the coordinates of an arbitrary point, then from (6.37)

zz14−40=, is the equation of cb , and for the intersection points of cz and cb , one has (1t 22−)=0. (6.38)

These points are the cuspidal points C2 and C3 , since the equation ω ==(yzy′′z )0, for the parameter t corresponding to the torsal lines, cy takes the form p1

tt22(1−=)0. Y1 A p For each generator p of ΣVI , there corresponds for each value of the parameter t a point A on cy and Y tu V a point B on m. If one can construct A and B, then one finds the flecnode point on p as the point of U C intersection of A∨ B and p. If Z resp. Y are the X 3 intersection points of p with cz resp. cy , then one can construct B by projecting Z from U on m (cf. C2 Fig. 6.6). B m To construct A, let us consider another generator Q Z cz p1 which passes through Z and has a double point with p. If t is the parameter of p, then 1/t will be Fig. 6.6 6 The flecnode – and LIE-curves pair on a ruled surface of fourth degree 181

the parameter of p1 which intersects cy in Y1. Let the double ratio

()UC,;1 X,I = Dx , (6.39) where X is an arbitrary point on cy and I (1, 1, 0, 0 ), then it will be

DDAY+=1 DY. (6.40)

The point Y is then the summation of the points A and Y1 w.r.t U and C1 as basis points.

The pair A, Y1 belongs to an involution that determined by the pair U, Y and the self- corresponding point C1. One can thus construct A and the flecnode point with just a ruler.

The point A can be constructed as follows: l

l1 Let the three lines l, l1, l2 pass through the points Y, C1, U respectively, where the line l V l2 is different from cy , and the three lines R l, l1, l2 are not concurrent. Let l intersects l1 in V , l1 intersects l2 in R and Y1 ∨V resp. S W C1 ∨W intersects l2 resp. l in W resp. S, then the line S∨ R intersects cy in the cy required point A, cf. Fig. 6.7. U Y1 A Y C1

Fig. 6.7 For the surface XI, one can use a dual construction to find the tangent plane in a flecnode point of an arbitrary generator of the surface, and the construction will be then reduced to the search of the point of contact of this tangent plane.

The other types of ruled surfaces of fourth degree, according to the classification of STURM, belong to linear congruences, therefore its flecnode curves coincide with the directrices of these congruences.

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space

25 3 Using KLEIN’s and PLÜCKER’s point model M 4 ⊂Π of the projective line space G of Π one can treat ruled surfaces, understood as one-parametric line sets, as curves in a 2 hyperquadric M 4 , therewith following V. HLAVATY [26], R. SAUER [58], KANITANI [28,29, 30], F. ANZBÖCK [1] and G. WEISS [72,73]. Let us recall the basic ideas and symbols in the following introductory chapter. Finally we are going to give a treatment of the LIE-curves problem of a ruled surface using the so called PLÜCKER-coordinates only, what seems to be new.

7.1 Explicit method of determining flecnode and LIE curves

Seemingly all other authors treat the problem of finding flecnode and LIE curves via special parametrisations of a ruled surface Σ and geometrically distinguished moving frames and they end up with differential equations for those curves. I want to follow an approach initiated and partly performed by G. WEISS, which allows a direct calculation of those curves on the basis of a PLÜCKER-coordinate representation of Σ .

7.1.1 The KLEIN-model of line space and the differential geometry of curves

The well known basic idea of PLÜCKER and KLEIN was to treat lines of Π3 as fundamental elements and give a model, where these fundamental elements can be seen as points in a five- dimensional projective space. The interchange between the two points of view can be formulated as a mapping Γ of a line p ⊂Π3 to a point P ⊂Π5. According to WEISS [72] p is interpreted as the image and kernel of a singular null-polarity γ . So Γ can be extended to a mapping of all (regular and singular) null-polarities γ of Π3 to the point set of Π5 , which is represented in coordinates based on adjoint projective coordinate systems in Π3 and Π5 as follows:

Definition 7.1.1.1: The bijective mapping Γ=: {γ} →P5 of the set of regular and (not trivially) singular null- polarities of Π3 to the point-set of Π5 is called the (extended) KLEIN-bijection. In the arithmetic Model Π()4 of Π3 , γ is represented by a set of proportional skew symmetric matrices ()qjk and the KLEIN-image γΓ = P is the point 6 (,qqqqqq010203233112 ,, ,,)= :(,,,,,) qqqqqq 123456 of Π(). This homogenous sextuple is called the PLÜCKER-coordinates-sextuple of γ , respectively of p = kerγ in case of γ being singular.

Thus the KLEIN-bijection Γ, restricted to the set of not trivially singular null-polarities maps 25 the line-set G onto the point-set of the so-called KLEIN-quadric M 4 ⊂Π , a hyperquadric 2 of index 2 with two 3-parametric sets of planes as “generators”. The equation of M 4 is described by

xxαα,:2(=++= xxxxxx14 25 36 )0. (7.1) 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 183

The bijection Γ transforms the group PGL( Π3 ) of all projective collineations of Π3 into the + 52 25 group PGL (Π , M 4 ) of those automorphic collineations of M 4 ⊂Π , which do not 2 interchange the two sets of two-dimensional generators of M 4 . Hence, by means of Γ one can handle the differential geometric problems of a ruled surface Σ as problems of a curve 2 5 ΣΓ in the quadric M 4 of Π , thus following ANZBÖCK [1], HLAVATY [26], SAUER [58], WEISS [72] and POTTMANN and WALLNER [53].

25 Let μ be the polarity of the KLEIN-quadric M 4 ⊂Π . The μ -polar hyperplane Qμ of a point

Qq= (),α then is described by

qxαα,:=++++= qxqxqx14 25 36… qx 63 0. (7.2)

In the inner product brackets (7.2) we sometimes will use symbols for points instead of their coordinates, i.e.

XY,:=== xαα , y , ((), X x α Y ()). y α (7.3)

5 2 The μ -polar pair ()QQ, μ represents a harmonic homology of Π keeping M 4 fixed ; such a homology is the Γ-transform of a regular null-polarity γ in Π3. The set of fixed lines of γ is a regular linear complex K, usually called the null-system of γ . The mapping Γ maps this 22 null-system K onto the quadric M 34:.=∩QMμ Regularity of γ (or K ) is expressed by

qqαα,0.≠

S (3) μ

f2Γ (4) f Γ (1) 1 Sμ Sμ

2 (2) V1 X ()u0 Sμ (4) X ()u0 S X ()u0 2 (3) V2 .....quadratic cone 2 S V1 S (2) ΦΓ S (1) X ()u0

X ()u0

2 M 4

Fig. 7.1

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 184

Let the homogenous PLÜCKER-coordinates ( xuα ()) , α =1,… ,6, of the generators x (u ) of ΣΓ and of the points X (uxu )= ( )Γ of ΣΓ be C r -functions ( r ≥ 5) of a parameter u ranging within an open interval I ⊂ . Then in each point X ()u0 of ΣΓ there exists a set of (k ) osculating k -spaces Su(),0 spanned by X ()u0 and the first k derivative points. These osculating k-spaces are invariantly connected to ΣΓ with respect to the renormings of the projective coordinates ()xuα () as well as to the parameter transformations and to the + 52 collineations χ ∈ΠPGL( , M 4 ) ; they are geometric properties of Xu()0 ⊂ΣΓ . Non-trivial k osculating k -spaces Su()0 have dimensions 1 to 4 and can be summerized as follows: (cf. Fig.7.1).

(1) – The tangent S() u=∨ Xu () Xu (): 000 (1) We assume that Rank()XX00 ,= 2 , i.e. x()u0 is a regular generator of Σ . As S touches Σ⊂Γ M 2 we distinguish the cases S (1) is a tangent of M 2 and S (1) is contained in M 2 , 4 4 4 Definition 7.1.2:

A regular generator x()u0 of a smooth ruled surface is called torsal, if the tangent of the 2 curve xu()0 Γ is contained in the KLEIN quadric M 4 , and non-torsal otherwise.

p

Fig: 7.2

S (1)Γ−1 represents a pencil of linear complexes K , which is spanned by the complexes −1 −1 (1) −1 X 0Γ and X 0Γ . For a regular non-torsal generator, S Γ is a parabolic pencil of complexes, and the common lines of all complexes form a parabolic line congruence

N ()u0 , which consists of all tangents of Σ along x().u0 This parabolic congruence has

the generator x()u0 as the only focal line. The regular null-polarities γ1 to that pencil of

complexes, restricted to the generator x(),u0 act identically and these restrictions form a

projective mapping of the pencil of planes through x()u0 onto the point-set of x(),u0 (cf. Fig.7.2) which is called CHASLES’ contact projectivity (or CHASLES-correlation) β along

x():u0

βγ::=→EEx00xx P 0 . (7.4)

22 The parabolic congruence N()u0 has a common quadratic cone NM24⊂ as Γ -image, (3) (1) (1) (3) which spans the μ -polar 3-space QS:= μ of S . The points Y of Q fulfil the conditions YX,,0.00== YX (7.5)

(3) (1) 2 In case of a torsal generator xu()0 ∈Σ the 3-space Q contains S and intersects M 4 in two planes of different type, one representing the Γ-image of the field of lines in the

torsal plane x(),u0 the other representing the bundle of lines through the cuspidal point

of x().u0 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 185

()2 – The osculating plane S ()uXuXuXu=∨∨ () () (): 0000 We assume that Rank (XXX000 , , )= 3. For a regular non-torsal generator the intersection (2) 22 21− Su()041∩= M : V is a conic section osculating the curve x()u Γ in X 0 , and V1 Γ 2 3 represents the concordant regulus of a ring shaped quadric Λ ()u0 in Π , which has

second order contact with the surface Σ in all points of x()u0 and is called the LIE- (2) (2) 2 quadric of Σ along x().u0 The μ-conjugate subspace Q to Su()0 intersects M 4 in 2 2 another conic section V1 representing the transversal regulus of the LIE-quadric Λ ()u0

mentioned above. These transversal generators are the asymptotic tangents along x()u0 and fulfil the conditions

21− VugGXGXGX10Γ==∈===:,,,0.RG(){ | 000} (7.6)

()3 – The osculating 3-space S ()(u= XXXXu ∨∨∨ )(): 00 (3) −1 We assume that Rank(,,,)4.XXXX0000= Then Su()0 Γ represents a three dimensional space of linear complexes. The “support” of this osculating space consists of 2, 1 or 0 lines within the osculating transversal regulus. These lines have the intersection (1) (3) 2 points of QS:= μ with M 4 as Γ -images and are called the flecnode tangents of

generator x0 of Σ. According to the number 2, 1, 0 of its flecnode tangents we call a

generator x0 a hyperbolic, parabolic or elliptic generator of the ruled surface respectively. To be of “hyperbolic” or “elliptic” type is a local property of a surface Σ ,

while a “parabolic” generator x0 may be an isolated one. In case of Rank(,,,)Rank(,,)3XXXX0000= XXX 000= we call a regular non-torsal

generator x()u0 a reguloid generator of Σ.

()4 – The osculating 4-space S (uXX )=∨∨∨ ...... Xu ( ) : 00( ) (4) (4) Γ−1 Assuming that dimSu (0 )= 4, then is Qu():00=⎯⎯→ S () uμ kQ0 the accompanying

linear complex of Σ at u0. It defines an accompanying null-polarity γ 0 , the restriction of which to the pencil of planes through x0 is a projectivity β, i.e. βγ::=→|EEx00xx P 0 . (7.7)

2 (4) The Γ-transform of γ 0 is a harmonic homology of M 4 with center Su()00μ = :(), Qu (4) and Su()0 consists of points Y fulfilling (4) S( u00 )== : Qμ ( u ) (qq , ), y ⋅+⋅ q y q = det( XXXXXY , , , , , ). (7.8) In case of Rank (XXXXX , , , , )( u00 )= Rank ( XXXX , , , )( u )= 4, we call a regular

generator x()u0 a congruence-type generator of Σ .

7.1.2 The flecnodes and the LIE-curves of a ruled surface

Let the PLÜCKER coordinate representation Xxu:(= α ()), uI∈ ⊂ , α =1,… , 6, be given.

It facilitates calculations, if we split PLÜCKER coordinate sextuples ()xα into two triplets 33 3 (,xx )∈× , as well as we use pairs (,),p0 p (,)a0 a ∈ × instead of coordinate 3 quadruplets (),pi ()ai of points and planes of Π respectively. 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 186

∗) Then the null-point S of a plane α = (,)a0 a with respect to a regular or singular null- polarity γ = (,)qq can be written as

Ss==⋅−+×(,)00saqqaq ( , a ), (7.9) the null-plane α = (,)a0 a of a point Ss= (,)0 s with respect to a regular or singular null- polarity γ = (,)qq can be written as (cf. WEISS [72], POTTMANN-WALLNER [53] p.139/140)

α ==⋅−+×(,)as00asqqsq ( , ). (7.10)

If the origin of the coordinate system is not incident with the common plane of two intersecting lines g and h , then the intersection point S and their connecting plane σ is determined by

Ss==⋅×(,)0 sghgh ( , ),

σ ==⋅×(,)s0 s (g h ,g h ), (7.11)

()ghαα,0.=⋅+⋅=gh gh

Qμ PP′ = χχ

2 M 4

Pχ

Q Xxμμ ()QX∨ μ P

Pχ

X

Fig. 7.3

∗) In order to distinguish plane coordinates ( or line coordinates) from point coordinates, we write α=(,)aa0 with the symbol at the left hand side of the coordinate vector. 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 187

As we already have declared in the introductory Chapter 1, the LIE-points Lui () of a LEIN −1 generator x0 are the fixed points of the so called K -involution λββ():uPP0 =→ :x00x with β and β from § 7.1.1. The fact that the projectivity ββ −1 is indeed involutoric can easily be seen in Π5 : The projectivities β and β are restrictions of null-polarities γ and γ , −−112 which map to harmonic homologies χ :,= ΓΓγχ : =ΓΓ γ of M 4 with μ -conjugate centers X and Q respectively. In analogy to the product of two (planar, Euclidean) reflexions at orthogonal axes, the product of such homologies then must be an axial automophic collineation of M 2 with axis (QX∨ ) and center (QX∨ )μ and thus involutoric, cf. Fig. 7.3. 4

(A) The LIE-points Li ()u0 of xu()0 In the following we assume Σ to be an arbitrarily given ruled surface not belonging to a linear congruence and (7.10) to be solved such that the PLÜCKER coordinate functions

(())quα of the accompanying linear complexes Qu ( ) are given.

S (4)

(4) QS:= μ E1 X E 2

X

ΣΓ 2 M 4

Fig. 7.4

We want to construct (real or imaginary) the pair of LIE-points Lui ()0 of a non torsal generator xu()0 ∈Σ , i.e. the pair of fixed points of the KLEIN-involution κ()u0 along x(),u0 2 by using the KLEIN model: The line X ()uQu00∨ () intersects M 4 in a (real, coinciding or −−11 imaginary) pair of points EE12, , which suit into both homologies χ :,= ΓΓγχ : =ΓΓ γ and thus EEiiχχ = are fixed by the product χχ of these homologies. The plane (2) 2 P():( u00= XXQ )() u therefore intersects M 4 in pair of lines EXi (), u0 i =1, 2, which also remains fixed by χχ and represents the Γ-images of two pencils of lines, which are fixed by the KLEIN-involution κ().u0 Note that the two pencils do not consist of tangents of Σ , as (2) (2) EXi () u0 is not conjugate to X ()!u0 It is the μ-conjugate plane P of P that intersects 2 M 4 in a pair of lines EXi (), u0 which are on one hand the Γ-images of two tangent pencils of Σ and on the other hand are crossed complementary to the former pair of pencils having the same pair of planes and vertices Lui ().0 Therewith follows that Lui ( 0 ) can be calculated as −1 the intersection points of generator xu( 0 )∈Σ with e.g. two lines euii()0 =Γ E , 2 Eui ()004∈∨() X Qu () ∩ M , cf. Fig. 7.4 . 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 188

2 The points {,EE12 }=∩ QXM 4 have coordinates Eeii= (),ααα=+ ( qtx i ), i =1, 2, whereby the coefficients ti ∈ fulfil the quadratic equation

eeii,,αααααα, =+ qtxqtx i, + i =0. (7.12)

This results in

2 QQ, 2 t =− , tttt121:,==− :. (7.13) X, X

Applying (7.9) to Eui ( ) and Xu ( ) we receive the coordinate representation of the LIE-points of x()u as Luiii()=+ (xq t xx , x ×+× q t ( x x )) (7.14) with ti from (7.13), i =1, 2.

The “dual” formula delivers the tangential planes λij (ueuxuj ) := ( )∨= ( ), 2,1, as λiii()utt=− (xq xx , x ×−× q ( x x )) (7.15) with ti from (7.13), i =1, 2.

Remark 1:

To construct the asymptotic tangent lui ()0 of Σ in Lxuii ∈ (),0 = 1,2, we could intersect the nullpolar planes Lijγ = λ , (j=2,1), and Liγ, (γ being the regular or singular nullpolarity to X ()u ).

U2 e2 L1 L1 l1 V2 l2 L1γ l2 e 2 l1 l2 e1 l l2 λ1γ 1 λ2γ l1 L2γ U1 V L L2 1 2 λ1 λ λ 2 1 U x()u λ2 e1 0 Q E1 X E Fig. 7.5a 2 ()QXX C 1 C 2 ()QXX μ Cl=Γ 22 (2) Sμ lCΓ = 11

Fig. 7.5b 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 189

To facilitate further calculations, we think of Π3 as a projectively extended affine 3-space with the ideal plane ω = (1,o ) and x(),u0 Lui ()0 such that they are not contained in ω.

Then UuUuUu:(0,(==xe0110220 )), :(0,( )), :(0,( = e )) are ideal points of x(),u0 eui ()0 and we can span li by Li and the ideal point VUiji= +==λ Ui ; 1,2; j 2,1. Therewith, we may put lCiiiΓ= =((exexxeex ⋅ ).( ji +λ ), ( × iji ) × ( +λ )) , i = 1,2, j = 2,1, (7.16) and find λi such that CXi ,0= yields. 2 A point CXEMii∈∨() ⊂4 is the Γ - image of a line li in Liγ, if Ci fulfills CXi ,0,= what finally results in

⎛⎞XX,, XX Cu ()=+⎜⎟ X X + Q , t from (7.13), uIi∈ , = 1,2, (7.17) i ⎜⎟ i ⎝⎠XX,, ti XX whereby, for the moment, the denominators in (7.17) are assumed not to vanish for all uI∈ .

Remark 2: The lengthy calculations above suggest to label a point with PLÜCKER-coordinates (,)−xx by X ⊂Π5.

The mapping ρ : Π→Π55 with X X is the harmonic axial collineation with fixed spaces ωΓ={(ox , ) } and OΓ = {(xo , ) }, (ω the field in the ideal plane and O the bundle of orianted lines with the origin of the coordinate frame as vertex in the original space Π3 ). The mapping ρ is therefore an analogue of the so called CLIFFORD-translation in the elliptic 3- space (cf. E. A. WEISS [71] p.110). Using the interpretation ρ ∗∗: Π→Π55 with XX==−(,xx )(4) ( xx , ) we find a 5 canonically defined null polarity in Π described by the matrix T6 and an inner product , as

∗∗∗55 ρ :,Π→Π XTX =6 ,

⎛⎞100 ⎜⎟0010 ⎜⎟001 T6 = ⎜⎟ (7.18) ⎜⎟−100 ⎜⎟010− 0 ⎜⎟ ⎝⎠001−

XY,,:,,=− XY = XY =xy −xy X:(,). = −xx

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 190

According to the non symmetric settings of (7.16), the non symmetric conditions for λi read as follows

QQ, CX1,,()= XX xx XX, ++++tQX , (xx ) QX , (xx ) XX , ( qx )λ EXXX , , = 0, 111( ) (7.19) QQ, CX2 ,,(),()=−() XXxx XX xx XX, −−−+tQX , (xx ) QX , (xx ) XX , ( qx )λ EXXX , , = 0. 222( )

With λi from (7.19) and Ei from (7.12), resp. Ei from (7.18), the asymptotic tangents li and their Γ - images Ci have PLÜCKER-coordinates, cf. (7.16),

lC11211122122Γ= =()()(ex EE + )()( + ex EEEEE − ) − , − (,)2 0x − λ 11 EXX , , (7.20) lC22211122122Γ= =()()(ex EE − )()( + ex EEEEE + ) − , + (,)2 0x − λ 22 EXX , .

When replacing inner products of vectors of 3 by

xy =+1 X ,,,YXY xy =−1 X ,,,YXY 2 ()2 ( ) the expressions (7.20) describe the asymptotic LIE-tangents of a ruled surface Σ in PLÜCKER- coordinates alone, independantely of normings and special parameters.

(B) The flecnodes Fi0()u of xu()0 To find the flecnodes of Σ we use a quite similar calculation as for the LIE-curves. The (real, coinciding or imaginary) flecnode tangents fi ()u0 of a generator x()u0 have Γ-images (3) 2 (3) (1) (1) Tui ()004∈∩ Sμ () u M , whereby SQμ =: may be spanned by two points Ri ()uQu00∈ ().

Let e.g. U1 := (1,0,0,0,0,0) be fixed unit point and Σ such that in a certain interval I yields dim(span(,,,,)())4,XXXXU1 u =

(4) and let R111= (,)rr be the spanned hyperplane and R111=(rr , ) its μ-pole. Then (4) R21:= span (XXXXR , , , , ) would be a hyperplane with a μ-pole R2 , which is μ-conjugate (3) to R1 as well as it is contained in Sμ , and thus TRtRii= (),12+ i =1, 2, is given by

22RR11, ttttt=−, 121 : = , =− . (7.21) RR22,

(1) If we, in the construction above, replace U1 by Qu()0 ∈ Q , we notice that (,TT12 ,, QQμ ) is a harmonic quadruplet, from which follows immediately by Γ−1 that the LIE points and flecnodes (- if distinct -) form a harmonic quadruplet on a generator xu()0 ∈Σ (cf. Fig.7.6 ).

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 191

SQ(4) := SS(4)∩ (2) μ μ := R 1 C1

VM22⊂ f1Γ 14

R2

f2Γ C 2 S (3) μ

(3) (2) SS∩ μ

S (2) μ

Fig. 7.6

7.1.3 LIE-curves as BERTRAND-curves

In the following we endow the projective 3-space Π3 with an Euclidean structure by using the plane ω = (1,0 ) as the ideal plane at infinity and the induced Euclidean inner product in Π()6

, :66×→ ; xy , : = (xx , ),(yy , ) =⋅x y. (7.22) eeeαα

−1 Thus the condition for Cui ()0 Γ (7.17), both being Euclidean orthogonal to generator

xu()0 ∈Σ , reads as follows,

2 2 ()ex21+=λ x 0, and ()ex12+=λ x 0. (7.23)

From now on we will admit a special norming of the PLÜCKER-coordinates (())xuα of Σ by x()uu=∀∈ 1 I. Then (7.23) gives

2()(qx++λ12λλλ )0, = 2()(t 1 xx +− 12 )0, = and finally

λλ==−=−()qx QX , . (7.24) 12 e Thus we can state that

Result: A surface Σ in the Euclidean space with the property, that the asymptotic tangents in the LIE- points of each generator xu()∈Σ are orthogonal to x(),u fulfills the differential equations QQ,()xx+−−= X , X QX , ()( qx t1 qx xx )()0, qx (7.25) ()()()()()()()qx xx −++ qx xx qx xx qxXX , ()0, xx =

with t1 from (7.13) and x()uu=∀∈⊂ 1 I .

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 192

Remark 1:

In general, at any LIE-point Lui (),0 we have four distinguished tangents: the generator x(),u0 the asymptotic tangent lui (),0 the tangent rui ()0 to the LIE-curve at Lui ()0 and the generator

sui ()0 of the LIE-strip along the LIE-curve, all four lines forming a harmonic quadruple.

We may ask as well for the ruled surfaces Σ in the Euclidean 3-space, which have orthogonal trajectories of the generators x (u ) as LIE-curves. Rewriting (7.14) as

Luiii()=× (ex , x e ) , with ei from (7.12), it follows Luiiiii()=+ (exexxexe , ×+× ) , and the tangents rui () of the LIE-curves have the direction vectors rexxexeexexxeiiiiiii(u )=×+×−+× ( )( ) (( ) ( ))( ).

From the conditions rxi (uu )⋅== ( ) 0, i1,2, follows, after some lengthly calculations, the differential equations (,xxq , )+− (, xxq , )tt11 (, xxx , ) − (, xxx , ) = 0, (7.26a)

((qxxxxqxxqxxxxxx++++ttt111 ))(,() , ) (, , ) (, , ) (, , ) ++QX , (xxq , , ) t1 X , X ( xxq , , ) (7.26b)

i + tQX−+ 2(xx )( x , x , q ) , ( x , x , x ) = 0, 1 ( ) with t1 from (7.13) and (,,)x y z symboloizing the determinant of the three vectors.

The “dual” question concerning the generators su()i of the LIE-strips along the LIE-curves would start with λiii()u =× (ex , x e ) and λiiii=+×+×(,exexxe xe i ), and we would end up with similar complicated differential equations as conditions for sui (), both being orthogonal to x().u Up to now, in Remark 1, we did not use special normings and parameterisation.

In the following we consider a surface Σ , which is contained in a linear complex K, i.e. the accompanying linear complex is fixed, and if the euclidean orthogonality condition (7.24) holds for all u ∈I, then the LIE-curves of Σ form a pair of BERTRAND curves for which Σ is the surface of common main normals. For such a pair of curves a linear natural equation in curvature κ()s and torsion τ ()s holds, ( cf. e.g. KRUPPA [39]). So the idea of interpreting BERTRAND curves as asymptotic LIE-curves on their common main normal surface Σ , under the condition that Σ is contained in a fixed linear complex might occur. A positive answer could give a hint to find explicit examples of such curves without integration, if one could manage to adapt Σ∈Π3 by a fixed collineation of Π3 such that (7.24) holds. 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 193

A pair of BERTRAND-curves bii , =1,2, is characterised by common principal normals and, as it is already known, they have the additional property that the distance ds ( ) of corresponding points B1122()sbBsb∈∈ , () is constant and ds ( )= A . Also the angle ϕ (s ) between the tangents of bi at Bi ()s is constant and cotϕ (sBA )= / , if b1 has the natural functions κτ(ss ), ( ) with Asκτ ( )+= Bs ( ) 1.

Therewith we aim at calculating the distance du()0 of the corresponding LIE-points Lui ()0 of

xu()0 ∈Σ : 2 2 11 du()012=⋅×−⋅× (xe ) ( xe ), (7.27) ()ex12 () ex with eeii, from (7.12). This equation can be transformed to −⋅××−×⋅×4(qq )() x22 ( x ( q x )) (( q x ) ( x x )) 2 du2 ()= (7.28) 0 ()()xx qx22− ()() qq xx

Now let us specialize the coordinate frame and use the canonical TAYLOR-development of one of the LIE-curves {L1()s } in arc length parameter s and with curvature κ()s and torsion τ ():s ⎛⎞10 ⎛⎞ ⎛⎞ 0 ⎛⎞00 ⎛ ⎞ ⎛⎞1010⎜⎟ ⎜⎟ ⎜⎟ss23⎜⎟−κ 2 ⎜−3κκ ⎟ s 4 Ls1()==+++⎜⎟⎜⎟ ⎜⎟ s ⎜⎟ ⎜⎟ + ⎜32 ⎟ +… (7.29) ⎝⎠ls1() 0 0 κ 26κκκτκ−+() − 24 ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜ ⎟ ⎝⎠00 ⎝⎠ ⎝⎠ 0 ⎝⎠κτ ⎝2 τκ + τκ ⎠

Then the principal normals x (s ) at Ls1() form a ruled surface Σ , which we assume to belong to a fixed linear complex K , such that this LIE-curve is an asymptotic curve on Σ . Let KΓ have the fixed coordinates Qq==(,)qq (α ) ; then Σ has the following PLÜCKER- coordinates development

⎛⎞−3κ ⎛⎞0 ⎛⎞−κ ⎜⎟22κ ⎜⎟1 ⎜⎟κ −+−()κτ ⎜⎟κ ⎜⎟κ ⎜⎟0 ⎜⎟κ 2 xs()Γ=⎜⎟ +⎜⎟τ s +2ττ − s +… (7.30) 0 ⎜⎟⎜⎟κ 2 ⎜⎟ 0 ⎜⎟0 ⎜⎟0 ⎜⎟0 ⎜⎟ ⎜⎟⎜⎟2τ ⎝⎠0 ⎜⎟1 ⎜⎟ ⎝⎠⎝⎠2κ

Replacing xx, ,… in (7.28) with the corresponding developments of (7.30) and putting qq⋅=:a = const., then follows

2 2 4()aq⋅+⋅5 ds()=− 2 , (7.31) ()()()qa3 −⋅τ s +⋅⋅ thereby ()⋅ and ()⋅⋅ are expressions depending on s of at least 2nd order.

So, at any ss= 0 the torsion τ is stationary and therefore has to be constant, if we assume 2 ds() to be fixed. But then, as {Ls11()} = : b is a BERTRAND-curve, too, we find also the curvature κ to be constant, b1 is a helix and Σ necessarily belongs to a linear hyperbolic congruence, such that K is not uniquely determined. 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 194

Result: Euclidean helices are the only BERTRAND curves, which are complex curves on surfaces belonging to a fixed linear complex.

Remark 2:

To study projective geometric properties of the LIE-curves lLuii= { ()} ⊂Σ is more or less a matter of discussing (7.14): E.g. the condition for l1 being planar, namely Rank(,,,)()Rank(,,)()3,LLLL1111 u= LLL 111 u= uI∈ , can be calculated without specialised parameters and normings of the PLÜCKER coordinates of Σ from det(,,,)()0LLLL1111 u= . (7.32)

7.1.3 Examples

We apply the theory presented above to the following surfaces Σ :

2543 (A) Σ=A {()xu ∈ G x() u Γ=− (2,, uu , u , u , u ) , u ∈ }. This surface has two torsal generators of 3rd order at u = 0 and u = ∞, and its accompanying complexes K()u are described by the curve {()}{(2,Qu=− 10,20,,10,20)} u − u25 − u − u 4 u 3 ∈Π 5 .

The LIE-curves li of ΣA are the imaginary pair of curves

57 89 5789 luuuuiuuuui … ()(12−−−±⋅−−− , 30 , 21 , 9 ) 49,8 (3 , , 8 , ) . (7.33)

th We notice that the torsal generators of ΣA are “LIE-singular” of 5 order. For this new concept we give the following

Definition 7.1.3.1:

Let x()u0 be a generator of a ruled surface Σ not contained in a linear line congruence and which is represented by the TAYLOR-development at parameter value u0 till the order k +1. Then, by (7.14), it is possible to calculate the TAYLOR-development of the LIE-curves th k lLuiii=={(), 1,2} at u0. We call x()u0 LIE-singular of k order, if ()u− u0 is a common k +1 factor of the coordinate functions of Lui () and ()u− u0 is not.

Remark: Obviously, LIE-singular generators are all those generators x of Σ , where span (XXXXX , , , , ) is not a hyperplane in Π5.

th For the algebraic surface ΣA the proper (imaginary) LIE-curves are of 4 order.

The flecnodal tangents of ΣA form a pair of real surfaces Φ1, Φ2 with the PLÜCKER- coordinates representation

253 Φ−++iiiii…{( (1vv ), (1 2 )uuu , 2(3 + v ) , (2 + v ) , (7.34) 43 (−+ 11 7vvii )uu , − (14 + 8 ))} , 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 195

with vi solutions of a quadratic equation resp. with the values

2 6vvii++= 187 193 0, v1 =−1.0687… , v 2 = −30.0979… (7.35)

Therewith the flecnodal curves ci of ΣA are described by

vvvv23431 25 cuuuii…{()3(2+−+ ), 3(1 5 i ) , (22 + 9 i ) , ( − 6 i )} , (7.36) with vi from (7.35). th Again the torsal generators of ΣA are singular of 5 order with respect to the construction of their flecnodes such that the proper (real) flecnode curves are of fourth order.

⎧ ⎛⎞⎛⎞⎛⎞(1)uu+ 222α ⎪ ⎜⎟⎜⎟⎜⎟22 (B) Σ=B ⎨xu() ∈G | xu () Γ=⎜⎟⎜⎟⎜⎟ ( u + 1)( u ++ uαα ) , − uu ( ++ u α ) , ⎪ ⎜⎟⎜⎟⎜⎟uu222(1)(++ααα u ++ u ) uu (1) + ⎩ ⎝⎠⎝⎠⎝⎠ ⎫ ⎪ u ∈ , αα=≠−const., 0, 1⎬ . ⎪ ⎭ th −1 These rational 4 order surfaces ΣB ()α are contained in a regular linear complex K =ΓQ with Q =−( 1, 1, 2ααα − 1, − 2 , , 0) and possess two torsal generators at the parameters 1 values uu12,114.=−2 () ± + α

nd 1 1 Thus ΣB ()α has real resp. imaginary torsal generators of 2 order for α >−4 resp. α < − 4 . They are surfaces of STURM’s type III, cf. KRAMES [37] and ANZBÖCK [1], and their asymptotic LIE-curve is of order 6, according to Voss [70]. With

xx, =−2α ( u22 + u −α ) , QQ, =+2αα ( 1), xx,2(21)(),=−α u + u2 + u −α the asymptotic LIE-curves (7.14) are described as

⎛⎞⎛⎞(1)(1uu++−α ) ⎛⎞uu(1)+−32α ⎜⎟⎜⎟⎜⎟ α(1)u + 2 ⎜⎟2 ⎜⎟⎜⎟(1)u +−α Lui ()=− ( u +− u αα ) ±− ( + 1) . (7.37) ⎜⎟⎜⎟α 2 (1)u + ⎜⎟α ⎜⎟⎜⎟ ⎜⎟⎜⎟22 ⎜⎟2 ⎝⎠⎝⎠αα()u + ⎝⎠αα()u −

th The curves lLuii()α = { ()} turn out to be of 4 order, in contradiction to KRAMES ([37], p. 255), and they are real – i.e. the KLEIN-involutions λ()u = ββ −1 are hyperbolic – if α < −1.

For ΣB , the parameter ti from (7.21) is found to be tuuuuu22=−αα( + − ) 2432 [ + 22 + (131 − 2 αααα ) − 22 u ( − 5) + ( − 35)] (7.38)

2 where tttt121==−, .

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 196

Let ψ =+uuu43222 + (131 − 2αααα ) − 22 u ( −+ 5) ( − 35), (7.39) then the solutions of the equation ψ = 0 are found to be

u =−−1 11 101 + 4α − 20 1 +α , 1 2 ( ) u =−++−+1 11 101 4α 20 1α , 2 2 ( ) (7.40) u =−−1 11 101 + 4α + 20 1 +α , 3 2 ( ) u =−++−+1 11 101 4α 20 1α , 4 2 ( )

which are the factors of ψ. Moreover, for α = −1, the function ψ takes the form ψ =++(116).uu22 (7.41)

(3) Now, since SRQtμ =+ i with ti from (7.38), then from (7.11) we have Fx=∩()(, RQt + =xf ⋅ x × f) iiii (7.42) =⋅+ (xrq (ttii ), x ×+ ( rq ))

with ti from (7.38), and xq⋅=−+(1)(uu −+α 1), xr⋅=−αα26[uuu + 15 5 − 4 ( − 55) + u 3 (85 − 10 ααα ) + u 2 ( 2 − 32 + 60) +−+u (7αα2 29 16)], (7.43) xq×=−()αα(1),uuu + −222(1), + − αα(), + ⎛⎞−++−−−+−−ααααα25(uu 13 4 30 uu 3 2 ( 34) u ( 2 8 16) 6 ), ⎜⎟25 4 32 2 xr×=⎜⎟ αα ((112)uu + − +−+uu(10 20ααα ) (2 −+−+ 33) u αα (7 16) 6 α ), ⎜⎟26 5 4 3 22 2 3 ⎝⎠ αα (uu−−+−−− ( 11) u (10 7 αααααα ) 5 u 12 u + u ( −+ 16) 6 ).

Remark 1:

From (7.40), for ujjij== 0, 1, 2,… ,6 ⇒ tu ( ) = 0, thus for these 6 values u j the flecnode 2 (3) projectivity along x (u j ) is parabolic, that means FF12= and f12Γ=fRM Γ= ∈ 4, i.e. Sμ 2 is tangent to M 4 .

7.2 The method of ANZBÖCK

Following an idea of BARNER [4], ANZBÖCK [1] constructs a canonical moving frame of ()k reference to Xu()0 ⊂ΣΓ , based on the osculating k -spaces Su()0 and the polarity μ of 2 M 4 described in § 7.1.1. The ANZBÖCK-frame to a non torsal generator xu()0 ∈Σ of r 5 differentiation class C , r ≥ 5, is a simplex {PP05,,… } ⊂Π with

(2) 2 PXu00:(),= PXu10:(),= PSuMP20401:()\=∩( { }) ∩ Pμ ,

(3) (2) (4) (3) (4) PSu30:()(),=∩( Su 0μ ) PSu40:()(),=∩( Su 0μ ) PSu50:();= μ 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 197

whereby we have to assume that x()u0 is neither a reguloid nor a congruence-type generator of Σ. Note that {PPP345, , } form a polar triangle with respect to the conic 22(2) VM14:.=∩() Sμ By special norming of the PLÜCKER-coordinates one can achieve PP, ≠∈⊂0, ∀ u I .

This norming leads to the following FRENET-type equations of the simplex vertices Pk (see [1]) r−2 PPPauPP011==, ( ) 02 + , 2 au ( ): =−Ω∈( PPC 11 ,) , PauPbuP2113=+()ε () , r−3 (PPPbuPPbuC333132 is normed by ,=∈+−ε { 1, 1}, ( ) : = , > 0, ( ) ∈ )

PbuPcuP =+()ε () , 3024 (7.44) r−4 (PPPcuP4442 is normed by ,=∈+−ε { 1, 1}, ( ) : =43,0,PcuC>∈ ()) P413=−ε cuP() + duP () 5 , r−5 (PPPduPPC50554 is normed by det() ,… ,==∈ 1, ( ) : , ) PduP52343=−εε() , ε ∈ {1,1}. + −

In (7.44) only three combinations of the signs ε1, ε 2 , ε3 make sense:

()()()εεε123, ,∈−−−−{ 1, 1, 1 , 1, 1, 1 ,( − 1, 1, − 1)} . (7.45)

The relative positions of the simplex points Pk are described by the μ-product values

PPjk, in the following table PPjk, P01 P P 2 P 3 P 4 P 5 P 5

P0 00− 10 0 0 0

P1 01 00 0 0 0

P2 −10 00 0 0 0 (7.46)

P3100 0ε 0 0 0

Pd42300 00ε 0− ε

P5 00 00 0ε3 0.

Surfaces of “ANZBÖCK type 4” are contained in a fixed, non-singular accompanying linear complex KK()uu= (0 ) with an arbitrary uI0 ∈ , and they are characterised by identically vanishing function d(u ). For those surfaces Σ we already know that the LIE-curves ll12, are asymptotic, as stated by S. LIE in [43] (cf. Fig. 7.7a,b).

Based on two contour curves of Σ (with respect to two central projections), NEUDORFER [51] 3 gave a ruler-and-compass construction of the LIE-points Luii()∈ l of a generator x()u in Π . For surfaces of “ANZBÖCK type 1” Σ , the accompanying linear complexes K()u are not stationary, i.e. Rank(PP55 , )( u )= 2 in I .

The pencil of tangents eui ()0 to Σ in the LIE-points Lui () of x()u0 fulfil

EEii,,,,0,==== EP i015 EP i EP i (7.47)

with uI0 ∈ , Eii:,=Γe PXu00:(),= PQu50:()= (cf. (7.12)). 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 198

P 3 P4 PP45() P3

P5 PP54 (2) ( ) (2) Sμ Sμ

P 0 P0 (2) (2) S P1 S P2 P2 P1 2 M 2 M 4 4

εε=−1, =1 − 1 , ε =− 1 1 ε =1, εε==−1 12() 3( ) 123 Fig. 7.7a Fig. 7.7b

From (7.46) follows

22 Ei =++pP00 pP 33 pP 44, p j ∈ , pp34::,=−ε 21ε (7.48)

and Ei are real only if (εεε123,,){∈−−− (1,1,1),(1, 1,1). −} (1) (1) 2 The set of points Ei (7.47) form the pair of lines {GG1,() 2} =∩ M 4 PPP 034 representing the

Γ-images of two pencils of tangents {},e1 {}e2 in planes λ10(),u λ20()u touching Σ in

L10()u resp. L20().u

Varying uI0 ∈ , the pairs ()Luii(),()λ u run through the so called LIE-strips along the LIE- curves li ⊂Σ, while the manifold of tangent pencils {eui ()} is a pair of line congruences with li as one focal curve. The Γ -images of these line congruences are 2-dimensional ruled (1) (1) 2 surfaces {Gu124(),} { Gu()} ⊂ M , which intersect along cPu=ΣΓ={ 0 ().}

To construct the asymptotic tangents eui ()0 in the LIE-points Li ()u0 of x()u0 we consider their Γ-images: The points euii()00Γ= Eu () on one hand belong to the plane (,,)PPP034 from 2(2)2 (7.48), and on the other hand they are points of Vu10()=∩ Sμ M 4 with (2) Suμ ()(,,).0345= PPP Thus it follows

Ei ()uPP034=± . (7.49)

−1 Varying uI0 ∈ , the asymptotic tangents Euii()Γ= eu () describe ruled surfaces Σi , and these surfaces are torsal if, and only if, EEii,0.= By the FRENET-equations (7.44) and the table (7.46) follows therewith

2 bP024135+±+±=±ε cP∓∓εεεε cP dP, bP 024135 cP cP dP d 3, (7.50)

such that li turn to be asymptotic curves, iff d(u )≡ 0, according to the theorem of LIE.

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 199

Remark 1:

Within each of the LIE-strip-congruences {eui ()} there are two special torsal surfaces {tui ()} and {tui (),} i=1,2; the first consisting of the set of tangents tuiii ( ) = L∨ L of li , the second consisting of the limit lines tuiii():=∩λ λ of the strip. A short calculation shows that, with

Ttii=Γ, Ttii=Γ,

Tui ()d()0=+± u 0034 P P P , Tui ()00034= −+± d() u P P P . (7.51)

Now we have the possibility to compare ANZBÖCK’s invariants with those of BOL of a general ruled surface Σ . According to BOL, a strip along a curve on a ruled surface Σ is a LIE-strip, if the strip tangent t and the strip limit line t and their derivatives have the following , -products:

⋅−−αα hTggg2 0 0 − 2 − 8 α ⋅

αα −⋅hTgg201 002

−−2010000α Tgg −− 2 3 0100T 0 0 0 0 0 TTTT , T T T T T (7.52) 0000T 0 0 1 0 2α −−−−−2000010gT 2αα 3 h gg 200 T 102αα − h ⋅ −−8200gggαααα T −−+⋅2 h 42 gg

From this table follows that the ANZBÖCK’s invariants a , b, c, d may be converted into BOL’s invariants α, g, g, h by

2 ae1 ⎛⎞2 d αεεε=+⎜⎟edec + +22 ⋅, : = 2 − , 24⎝⎠ 2 d 2 ii h =+++2,ab eb e b b dd2 22 ( ) (7.53) i 1 ⎡⎤bb 2 F2 g =−α ⎢⎥ + , V1 P 2 ⎣⎦( dd) ( ) 3 ga =+−22α g . E2 P4

E 1 F Remark 2: 1 Using ANZBÖCK’s moving frame, the flecnodal tangents fi result from the intersection points (3) 2 Fi of PP45= Sμ with M 4 (cf. Fig.7.8). P5 Fig. 7.8

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 200

ε 2 Putting FPtPii=±45, follows from FFii,0= with (7.46) that ti =± . ε3

So the flecnodes Fui ()0 of a generator x()u0 are real if ε 2 and ε3 have the same sign. Additionally, we have received the same result as given in (7.16), (7.17), when specializing

R1 , R2 to PQu50= () and P4.

The question arises when the ruled surfaces Σ fi , i = 1,2 consisting of flecnodal tangents of Σ are torsal. An analogue calculations as (7.50) shows that Tuii(),() Tu=⇔ 0 cu () = 0, (7.54) such that Σ is not of ANZBÖCK-type 1 as we assumed in this chapter.

7.3 The method of KANITANI

7.3.1 KANITANI’s fundamental quantities

Also KANITANI (cf. [28], [29] ) uses the KLEIN model of the line space to treat ruled surfaces 3 2 Σ⊂Π from a projective geometric point of view. He uses a moving frame of ΣΓ =cM ⊂ 4 based on the following normings of the PLÜCKER-coordinates of x()uXuuI000Γ =∈ (), . Assume that Σ is non torsal, then we can put

XX, :=− 2 h2 ≠ 0 in I and P : =1 X , 1 h PP, :==− 8 hk and P :1 P k X , (7.55) 11 22hh 1 PPkP:=−1 ( ), and hkI , , λ : → . 321λ

KANITANI chooses two points P4 , P5 according to “main types” of surfaces Σ , namely surfaces ΣI with distinct flecnode curves ( in algebraic sense) and surfaces ΣII with coinciding flecnode curves.

(A) The surfaces ΣI of KANITANI-type I:

A ruled surface ΣI is never a regulus and thus ΣIIΓ=c not contained in a fixed plane (2) SXPPu=∨∨()(),120 as we assumed ΣI to have well defined flecnode curves. Thus (3) (2) λ()u ≠ 0 and SuuI(),00 ∈ , is three dimensional and intersects Sμ in a point P3 and we 2 (3) (2) assume PM34∈ in case of ΣI. Then the μ -polar line SSμμ⊂ intersects the conic 2(2)2 VS14=∩μ M in two different points PP45, (in algebraic sense). The PLÜCKER-coordinates of these points can be normed such that PP45, =1 in I. A further norming of P3 can make

PP33, = 2 constantly in I (cf. Fig. 7.9 und Fig. 7.10). Hence, putting PP34, = 2η , PP35, = 2η 1 , PP45, =ν , we have

PXuPhPPhPkPPPkP:===+=+ ( ), , 2 2 , λ , 0001120231 (7.56) P3145=+−222,ηηλ PPPPPPPPP 044355131 =+ νη , =−+⋅≠ νηηη , ( 0).

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 201

Thereby the values PPjk, can be expressed as shown in the following table

PPjk, P012345 P P P P P

P0 0010 . .

P1 0200− . .

P2 1000 . . (7.57)

P3 000200

P4 000001−

P5 000010− .

P 2 4 V1 P3 (2) p A2 Sμ 2 P5 A1

p5 p31− p P0 P p4 1 (2) 2 S V1 p31+ p P2

2 p F1 F M 4 2

Fig. 7.9 Fig. 7.10

3 −1 In Π , the lines pPjj=Γ, j =0, 2, 4,5 form a skew quadrilateral with vertices FF12, (the flecnodes on p00= xu()) and AA12, ∈ p 2(cf. Fig.7.10).

The tangents qi of the flecnode curves in Fi can be expressed by qpp1104=+ρ resp.

qpp2205=+ρ . Therefore the derivatives q1, q2 are lines meeting p = ppp043, , + p 1 resp.

p =−ppp053, , p 1 . (Thereby we determine λ by special norming such that p31+ p resp. p31− p coincide with the diagonals FA12 and FA21). From QPhPPP110113=+ρρ ++ ηv 4 and QP10431,== 0 for P P , P , P + P follows η = ρ1h and

QPhP104=+η .

Similarly, the tangent q2 in Fu20() is expressed by

QPhP2105=−η .

2 If η ==η1 0, then P4 , P5 will be fixed points on V1 and ΣI belongs to a linear hyperbolic or −1 −1 elliptic congruence with directrices pP44= Γ , pP55= Γ . In this case the functions

hk,,λ : I→ are the only fundamental quantities of ΣI , as ν ≡ 0.

Assuming that just η or η1 vanishes, the ruled surface ΣI belongs to a special linear complex.

We choose the index s ∈ {4,5} and consider Ps as Γ -image of that fixed complex. Then for

ΣI yields η1 ==ν 0 and the fundamental quantities are the remaining functions hk,,,λ η : I→ . 7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 202

If neither η nor η1 vanishes, then by a special norming of P4 with a factor σ (and therefore 1 of P5 with factor σ ), we can make η =η1. Now hk,,,λ ν : I→ are non trivial and represent the fundamental quantities of such a ΣI. The hyperplane spanned by the points (4) PPPPP01234,,,, + P 5 represents the osculating hyperplane Su()0 of c = ΣΓI . The derivative of that hyperplane contains the point ()()2.PP45+=ν PP 45 −+η P 3

Hence, In order that the ruled surface Σ with distinct flecnode curves be belong to a regular linear complex, it is necessary and sufficient that ν = 0. If η ≠η1, this condition takes the form

⎛⎞ηη 2.ν =−⎜⎟1 (7.58) ⎝⎠η η1

(B) The surfaces ΣII of KANITANI-type II: (3) (2) 2(2)2 In this case PP33, = 0 and SSμμ⊂ is the tangent of VS14=∩μ M at P3. There are two subclasses:

a) PP33,0= in I and P3 is fixed. That means ΣII belongs to a parabolic linear

congruence. In this case λ (u ) can be chosen such that the coordinates of P3 are constant.

The functions hk , , λ : I→ can be considered as the fundamental quantities of ΣII . (3) Now, let us take an arbitrary point PS43∈ μ \{ P} which shall be fixed for all uI∈ , and 2 which can be normed such the PP44,2.= The μ -polar hyperplane of P4 intersects V1

in P3 and in a second point PP53≠ , which is also fixed ∀uI0 ∈ and it can be normed

such that PP35,1.= − Hence we get the FRENET-equations

P==+=+ hP, P 2 hP 2 kP , P λ P kP , 0112 0231 (7.59) PP34=={}, {} , PP 50 =λ .

5 Note that P5 is fixed as a point in Π , but its PLÜCKER-coordinates might depend on a non constant factor σ ():uI→ . The scheme of PPjk,-values take the form

PPjk, P012345 P P P P P

P0 0010 . .

P1 0200− . .

P2 1000 . . (7.60)

P3 000001−

P4 000020

P5 000100− .

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 203

P5 2 V1 P4

S (2) A2 μ P p2 3 A1

p41− p p5 P0 p3 P1 (2) V 2 S 1 p41+ p P2 2 p M 4 F1 F2

Fig. 7.11 Fig. 7.12

2 b) PP33,0≠ in I. Then one can determine λ such that PP33,:20.= h> Hence we put 1 P4 coincident with P3 as PP43= h . Next, we take P5 on PP34∨ as 22,hP54=− Pν P 3 whereby ν (u ) comes from PP45,2.= ν Then the FRENET-equations take the form PhPP==+=+, 2 hPkPP 2 , λ PkP , 01 1 2 0 2 31 (7.61) PhPP34==+=+, 4 2 hPP 5 2ν 3 , P 5λν P 04 P .

Again we have labelled the fundamental quantities of ΣII with hk,,,λ ν : I→ .

The not constant point P3 represents the coinciding flecnodal tangents of ΣII touching −1 ΣII in Fu()00∈ xu () of ΣII . Thus the tangent qQ= Γ of the flecnode curve f in Fu()0 can be expressed by

QPP=+ρ 03, and the set {qu()} is torsal if, and only if, QQ,0.= With Q=+ρρ P014 hP + hP ,

−1 follows ρ(u )=≠ 1 0, so the ruled surface Σ f :()=Γ{Pu3 } is never torsal. (5) 5 From (7.61) follows that PP00,,,,… PP 00∈Π are projectively independent, that is, Su(4)() is not a constant hyperplane. So we can state:

If a ruled surface ΣII has coincident, non-rectilinear flecnode curves, then it never belongs to a fixed linear complex.

Remark: In case of ν ≡ h , the flecnode curve tangents qu ( ) fulfil the following 3rd order differential equation 1d⎧⎫ 1d⎛⎞ 1dQ d ννd Q ⎨⎬⎜⎟=++QQλ , (7.62) 2duhuhu⎩⎭ d⎝⎠ d d uh( ) hu d what says that Q is always a linear combination of Q and Q .

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 204

So we state the result (KANITANI [28]),

The flecnode curve f of a ruled surface ΣII with coinciding, non rectilinear flecnode curves is a plane curve if, and only if, the KANITANI-functions ν ()u and h() u are identical.

7.3.2 The KANITANI-invariants of ΣI

KANITANI’s fundamental quantities depend on normings of PLÜCKER-coordinates and of a special parameter. KANITANI himself therefore studied the behaviour of hk,,,,λ ηη1 and ν under parameter transformations ufu= () and renormings X ()uuXu=⋅ρ () (), and he found

ααα11⎧ 2 ⎫ PPP110=+ε , P 2 =⎨ P 2 + P 1 + P 0⎬ , ( hhh) ρ ⎩⎭24( ) λσPP===λ , PP , PP1 , 33ρ f 44 55σ

2 (7.63) ερ ⎪⎧ ⋅ ()α ⎪⎫ hh==+−, kεα k11 , ⎨ 24( hh) ⎬ ffρ ⎩⎭⎪ ⎪ ε εσ λλ= 1 , ηηνν==+1 , 1 σ , ρ f ff( σ ) with

ρ ε =±1, α = , ε =±1. ρ 1

Furthermore, by combining these expressions (7.63), he found a system of invariant functions connected with ΣI :

224222 Kud, (ληηθ h )d, u 1 d, u d, u ⋅⋅⋅ 2 (7.64) ⎧⎫⎧⎫11λλh ⎛⎞η′ η1′ Khkh=+⎨⎬⎨⎬) −) , θν =−⎜⎟ − . ⎩⎭⎩⎭44λ ( hhληη( 2⎝⎠1

For these invariant functions there yields a main theorem:

Up to projective transformations, a ruled surface is uniquely determined by the set of invariant functions (7.64), depending on a known parameter.

KANITANI gave a geometric meaning of these invariant functions by interpreting them as anharmonic ratios, in which appear mainly the following surface elements: The intersection points of the osculating quadric along a generator p0 with the neighbouring generator that infinitely near to p0 , the asymptotic tangents, the flecnode tangents and the flecnodes themselves (cf. KANITANI [28]).

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 205

7.3.3 The flecnode transformation

In another research of KANITANI [29], the flecnode transformation of a ruled surface is studied. He gave the necessary and sufficient conditions for the transformation to be a projective transformation. The author studied the different cases of the flecnode curve on a ruled surface, and he stated the following results:

(1) Let a non reguloid generator xu()00I= p ⊂Σ varies along ΣI , then the envelope of the 21− set of osculating quadrics Vu1 ()Γ consists of ΣI itself and the two flecnode tangent surfaces.

(2) In order that the flecnode transformation of a ruled surface ΣII , which have just one and only one non-rectilinear flecnode curve, to be a projective transformation, it is necessary and sufficient that the flecnode curve be a plane curve. (3) In order that the flecnode transformation of a ruled surface which belongs to a special linear complex, without to belong to a linear congruence, to be a projective transformation, it is necessary and sufficient that Σ admits four systems of ∞1 projective transformations in itself. When this condition is verified, then the flecnode transformation will be a correlation.

(4) In order that a flecnode transformation of a ruled surface ΣI to be a correlation, it is

necessary and sufficient that K =ηη1, θ23θ = 0, where η ()λh η1 ()λh θν2 =−+ , θν3 =+ − . η 2λh η1 2λh (5) In order that the transformation which let one of the flecnode surfaces of a ruled surface

ΣI to be transformed in the other to be a correlation, it is necessary and sufficient that all

the generators of ΣI belong to a regular linear complex.

(6) Let ΣI be a ruled surface which have two distinct non-rectilinear flecnode curves, and whose generators don’t belong to a linear complex. In order that the transformation which let one of the flecnode surfaces to be transformed in the other, to be a homology, it is 1 1 necessary and sufficient that ΣI admits ∞ homologies and ∞ correlations in itself.

(7) Let p0 be a generator of a ruled surface ΣI , f1 and f2 are the flecnode tangents at the

flecnode points F1 and F2 on p0 respectively, A1 the flecnode point different from F1 on

f1, A2 is the flecnode point different from F2 on f2 , B1 is the point of intersection of f1

with the osculating plane of the flecnode curve at F2 , B2 is the point of intersection of f2

with the osculating plane of the flecnode curve at F1. If one of the lines A12∨ A and

B12∨ B is a generator of the regulus osculating ΣI along p0 , then it is the same for the

other. That occurred in the case where all the generators of ΣI belong to a regular linear complex, and only in that case.

(8) Let p0 be a generator of a ruled surface ΣI , f1 and f2 are the flecnode tangents at the

flecnode points on p0 , Φ1 and Φ2 are the flecnode surfaces described respectively by f1

and by f2 , QΦ1 is the osculating quadric of Φ1 along f1, QΦ2 is the osculating quadric

of Φ2 along f2. Then, QΦ1 and QΦ2 intersect along the generator p0 , and along two

other distinct lines, each of them touches ΣI at a point of contact of an asymptotic

tangent of ΣI , which belongs to the osculating linear complex of ΣI along p0.

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 206

7.3.4 Applications of KANITANI’s calculus of ruled surfaces

3 In Π the flecnodes F1, F2 of a surface ΣI , and on the flecnode tangent surfaces Φ1, Φ2 the second flecnodes AfuAfu1102∈∈(), 20() form a quadrangle, because if a ruled surface ΣI is of KANITANI type I, then the flecnode tangent surfaces Φi are of the same type (cf. § 7.3.2). 3 We use the points F1, F2 , AA12, as a fundamental moving frame in Π and for the unit point, the point of intersection of the two lines with Γ-images

1 1 PP04++2 () PP 31 + and PP25++2 (). PP 31 + (7.65)

5 A point X ∈Π with coordinates ()xx05… represents a line, if the PLÜCKER-coordinates are q===+x , qxxx , q , 10 24 331 (7.66) q43152=−xx , q = x , q 65 = x .

By normalizing the coordinates of p0 and the parameter u , one makes h = const., λ = const., (7.67) then p2 , which is a generator of the osculating quadric of ΣI , describes the principal ruled surface, as p0 moves on ΣI (cf. § 2.4 of chapter 2). By means of (7.56), we get PPP443=+νη, 2 PP40=−22,λη +()()() η + ην PPP 3 + η 54 + (7.68) PPhPPPP401534=−22λ( η′ + ην ) − 2 λη + 6 ηη + () + () , and PPP5513=−νη + , 2 PP510113145=−22,λη +()()() η − ην P + η PP + (7.69) PPhPPPP51101111435=−22ληην() − − 2 λη + 6 ηη + ()() + ,

−1 so that the flecnode tangents fFii=Γ of the flecnodal surfaces Φi are expressed by

2 ηη⎛⎞ ηη ⎛⎞ η FPPPP10124=−⎜⎟ −νν + ⎜⎟ − − + , 4λ ()h 2 ⎝⎠ηληλ2 h ⎝⎠ (7.70) 2 ηη11⎛⎞ ηη 11 ⎛⎞ η 1 FPPPP20125=−2 ⎜⎟ +νν + ⎜⎟ + − + . 4λ ()h ⎝⎠ηληλ112 h ⎝⎠

5 respectively. In Π the five points PPPF0451, , , and F 2 are independent, and accordingly they determine a linear complex K1 whose pole is P3. M. TSUBOKO [66] investigated the relation between K1 and the osculating linear complex K of ΣI , the osculating regulus QΣ of

ΣI , and etc., and also he succeeded to determine the envelope of K1 as p0 moves on ΣI.

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 207

If we denote by Ψ the developable surface generated by the tangents along an asymptotic curve on ΣI , then we have the following results:

(1) The linear complex K osculating ΣI along p0 and the linear complex K2 osculating Ψ

along an asymptotic tangent (as a generator of Ψ ) at a point on p0 of ΣI have in

common the linear congruence osculating ΣI along p0 , whose directrices are the

flecnode tangents f1 and f2 of ΣI.

(2) The linear complexes K1 and K2 have in common a linear congruence whose directrices

are the asymptotic tangents of ΣI issued from two points, harmonic conjugate with

respect to the flecnodes F1 and F2.

(3) The linear complex K1 intersects the osculating linear complex K of ΣI in a linear

congruence whose directrices are the asymptotic tangents at the involute-points I1 and I2

on p0 of ΣI , where I1 and I2 are described by

⎛⎞η1 ⎛⎞η1 Ii1 = ⎜⎟, 1, 0, 0 , Ii2 =−⎜⎟, 1, 0, 0 . ⎝⎠η ⎝⎠η

(4) Let t1 be a line through a point F on p0 , QΣ be the quadric osculating ΣI along p0 , and

t2 be the reciprocal polar of t1 with respect to QΣ. When t1 and t2 are conjugate with

respect to the linear complex K1, the point F is a flecnode on p0 of ΣI and t1 intersects

the flecnode tangent at another flecnode on p0 of ΣI. Accordingly t2 passes through the

latter flecnode and intersects the flecnode tangent at F of ΣI.

(5) Let F1 and F2 be the flecnodes on p0 of ΣI which describe the flecnode curves c1 and

c2 respectively, and f14 ()≡ p and f25()≡ p the flecnode tangents at F1 and F2 of ΣI. Then there exists ∞1 developable surfaces in the congruence which is generated by the

pencil of lines intersecting f2 (or f1 ) with the centre at F1 (or F2 ), as p0 displaces on

ΣI. The anharmonic ratio of the four lines, along which the tangent plane at any point on

c2 (or c1 ) of ΣI intersects four fixed developable surfaces in the congruence, is constant.

(6) In the congruence generated by the pencil of lines intersecting f2 (or f1 ) with the centre

at F1 (or F2 ), as p0 moves on ΣI , the focal surface is the developable surface which is

enveloped by the tangent planes of ΣI at all points on c2 (or c1 ), and whose generator is

the harmonic conjugate of the tangent at a flecnode F of c2 (or c1 ) with respect to the

generator of ΣI and the flecnode tangent of ΣI , passing through the same flecnode F. We have to notice that, this developable is explained before by WILCZYNSKI (cf. § 2.5) , who called it the secondary developable of the flecnode curve, where the primary

developable is the one formed by the tangents of c2 (or c1 ).

(7) Let g1 and g2 be the generators of the developable surfaces described by the planes

p02∨ f and p01∨ f respectively, when a generator p0 displaces on ΣI , and let t1 and t2

be the lines which intersect p0 , and are reciprocally polar with respect to the quadric QΣ

osculating ΣI along p0 and conjugate with respect to the linear complex K1 in the same

time. As t1 and t2 generate two flat pencils of lines, the line joining the point of

intersection of t1 with g1 to that of t2 with g2 describes a regulus belonging to K1.

7 The method of treating ruled surfaces as curves in the KLEIN-model of line space 208

The quadric containing this regulus touches QΣ along p0 and has in common with QΣ

the asymptotic tangents at the LIE-points L1 and L2 on p0 , where L1 and L2 are described by

⎛⎞η1 ⎛⎞η1 L1 = ⎜⎟, 1, 0, 0 , L2 =−⎜⎟, 1, 0, 0 . ⎝⎠η ⎝⎠η

(8) As p0 moves on ΣI , the linear complex K1 envelops a general complex having for the characteristic the linear congruence whose directrices are the two common generators of

the reguli osculating the flecnode surfaces Φ1 and Φ2 along f1 and f2 respectively.

These directrices touch ΣI at the LIE-points on p0.

7.4 Final remarks

Chapter 7 showed a possibility to treat ruled surfaces in a somehow symmetric manner using lines as fundamental elements and it showed furthermore that there is not only one optimal norming, parameterisation and moving frame, which suits to any problem and question. While KANITANI’s calculus is well suited to treat the problem of flecnodes and flecnodal curves of ruled surfaces, it is less suited to investigate LIE-points, LIE-curves and LIE-strips. ANZBÖCK’s calculus is better suited to the latter problems, but has to distinguish more types of ruled surfaces than KANITANI. It seems that my own, in collaboration with G. WEISS, approach in § 7.1 avoids some of the disadvantages of both, KANITANI and ANZBÖCK-BOL.

There still many open problems, which I believe are worthy to be treated:

If h, k, λ, η, η1, ν are KANITANI fundamental functions of a ruled surface ΣI , what are the fundamental quantities of the flecnode tangent surfaces Φ1, Φ2 ? Are there “BERTRAND-like” conditions for the invariants of Φ1, Φ2 ?

Analogues questions can be formulated for Σ and the LIE-strip surfaces of the asymptotic tangents along the LIE-curves of Σ . Appendix A

The method of FUBINI

FUBINI (1879 – 1943) became interested in projective differential geometry about 1914. He undertook to define a surface in ordinary space, except for a projective transformation, by means of differential forms. By 1916 he had perfected an analytical method in projective differential geometry, namely, the method of differential forms. His name is associated with that of one of the distinguished followers of his method, EDOUARD CECH, through their collaboration in publishing the well-known treatise [21]. FUBINI’s method is employed in their treatise, and we are going to give a summary of this method.

The theory of differential forms and the absolute calculus of RICCI are very useful in this theory. In particular, the process of covariant differentiation with respect to a fundamental quadratic differential form is frequently employed. The formulas are very much simplified by use of the summing convention of tensor analysis, summation being understood with respect to any index that appears twice in the same term.

To define a surface in ordinary space by means of differential forms, FUBINI considered a fundamental binary quadratic differential form G,

ij G = aij du du . (1)

1 2 The coefficients aij are functions of the two variables u , u in a certain region, and the descriminant A of the form is supposed not to vanish in this region. Let us also consider in space Π3 a surface S whose parametric vector equation in projective homogeneous coordinates is x = xu(1,u2) (2)

Let us now define two differential forms F2 , Φ3 by the equations

2 −12 3 −12 Fx21= (,,xx2,dx)A, Φ=31( xx,,x2,dx) A (3) wherein numerical subscripts of x denote covariant differentiation with respect to the form G, parenthesis indicate determinants of the fourth order of which only a typical row is written in each case, and vertical bars indicate absolute value. The forms F2 , Φ3 are of the first and 1 2 second orders respectively, since F2 is independent of the second differentials of u , u , and

Φ3 is independent of the third differentials.

Both forms F2 , Φ3 can be shown by direct calculations to be absolutely invariant under all proper transformations of parameters,

uu11= (,u1u2), uu22= (,u1u2) (4)

To express this invariantive property, the forms F2 , Φ3 are called intrinsic. Appendix 210

One undesirable feature of the present situation is that the forms F2 , Φ3 are of different orders. It is possible to replace Φ3 by an intrinsic form of the first order. In fact, a little calculation suffices to show that the form f3 defined by

f33=Φ2−3dF2 (5) is actually of the first order, being independent of the second differentials of u1, u2. We now have two binary differential forms F2 , f3 one quadratic and the other cubic. These forms are independent of the parametric representation of the surface S, but still depend on the form G and on the proportionality factor λ of the homogenous coordinates x.

We wish to find precisely how the forms F2 , f3 , and incidentally Φ3 , depend on the form G and the factor λ. Replacing G by another form G′ and indicating the new expressions by accents, we find

F2′ = RF2, Φ=′3RΦ3, f33′ = Rf − 3F2dR (6) where RA= A′ 12. (7)

Moreover, multiplying each coordinate x by a factor λ, i.e., making the transformation x = λx, (8) and distinguishing the new expressions by dashes, we find

4 43 43 F22= λ F , Φ=33λ Φ+3,F2λdλ f33=−λ fF6.2λdλ (9)

Since the form f3 is not transformed in the same way as the form F2 under these transformations, we inquire whether it is possible to replace f3 by another form having all the desirable properties possessed by f3 and having also the property of being cogredient to F2 under these transformations. The answer is in the affirmative. Let us denote the descriminant of F2 by D, and define a form F3 by the equation

F33=Φ23−dF2+()34F2d log(D A). (10)

Now it is easy to verify that under the transformations and with the notations of the preceding paragraph we have

4 Fi′= RFi, Fi= λ Fi (i = 2,3) . (11)

Thus we reach the conclusion:

The forms F2 , F3 are of the first order and of degrees indicated by the subscripts, are intrinsic, and are cogredient and relatively invariant under change of fundamental form G and the transformations (8) of proportionality factor.

We next choose the proportionality factor λ in a special way. Precisely, we choose λ so that the descriminant of the form F3 shall be a non-zero constant times the cube of the descriminant of F2. That is possible to make this choice is evident when one considers that the descriminant of F3 is of the fourth degree in the coefficients of F3 while the discriminant of F2 is quadratic in the coefficients of F2.

Appendix 211

Denoting these discriminants by ∆, D respectively we find

∆=λ16∆, D= λ 8 D. (12)

Then in order to have ∆ =const.⋅D3 it is sufficient to choose λ so that

∆=const.⋅λ 8D3. (13)

We are supposing D ≠ 0. It follows that ∆ ≠ 0, and in particular that F3 is not the cube of a linear factor. We denote by ϕ2 , ϕ3 the two forms F3 , F2 given by the second of equations (11) with λ satisfying (13).

The fundamental form G has hitherto been arbitrary. We now choose G to be the form ϕ2.

With this choice of G, direct calculation shows that the hessian of the form ϕ3 is proportional to ϕ2 , so that ϕ3 is apolar to ϕ2. The coordinates x as now normalized are FUBINI’s normal coordinates. The forms ϕ2 , ϕ3 are not only intrinsic but are covariant to the surface.

The question now is whether these two forms are sufficient to determine the surface S up to a projective transformation. The answer is in the negative. It is not difficult to show that the coordinates x satisfy equations of the form x =+px α x +β x , uu u v (14) xvv =+qx γδxu +xv wherein the coefficients are scalar functions of u, v, and subscripts indicate partial differentiation. But that the surface is now determined only up to a projective applicability, the coefficients p, q of (14) being not yet determined. If we adjoin to the forms ϕ2 , ϕ3 the form ψ 2 defined by

22 ψ 2 =−pdu qdv , (15) these three forms suffice to determine the surface S up to a projective transformation, since now all the coefficients of (14) are determined.

FUBINI’s method, which we have just explained for surfaces in ordinary space, has also been used by SANNIA in studying plane and space curves; by CECH in his theory of ruled surfaces; and by FUBINI in studying congruences and complexes in ordinary space, surfaces in space Π4 , and hypersurfaces in space Π n. This method is available for any configuration that has a quadratic form covariantly connected with it.

References

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A Chasles-correlation, 184 circular ruled surface, 158 affine Clifford-translation 189 – binormal, 113 concordant, 11, 121, 185 – curvature, 93, 111 cone of Sergre, 58 – normal, 111 conic, – principal normal, 113 – absolute, 127 – principal tetrahedron, 94 – derivative, 14 – rectifying plane, 96 – direction, 178 – torsion, 93, 113 – osculating, 21, 185 anharmonic ratio, 13 ff – residual, 171 Anzböck, Friedrich, 182, 196 ff congruence, arc length, – axial, 163 – projective, 64, 120 – bisector, 163 – centro-affine, 111 – concordant, 139 automorphic collineation, 183 – elliptic, 3 – flecnode, 11 ff B – hyperbolic, 3 – osculating, 124 Barner, Martin, 139 – parabolic, 3, 184 Bertrand-curves, 4, 191 – surface, 3 ff binormal, 113 – W-, 13, 78, 123 Blaschke, Wilhelm, 102 coordinates, Bol, Gerrit, 1, 117 ff – affine, 93 Brianchon’s theorem, 178 – Cartesian, 110 bundle, – curvilinear, 53 – of planes, 67 – homogenous, 184 – of quadrics, 66 – line, 186 – normal, 63, 68 C – Plücker-, 22, 24, 182 – projective, 182 Carpenter, Allen Fuller, 27 correlation, 205 Cartan’s parameter, 123 covariants, 5 ff Cayley, Arthur, 2 cubic, Cayley’s ruled cubic, 26, 69 – derivative, 14 – osculating, 21 – primary flecnode, 36 centro-affine – primary Lie, 36 – arc length, 111 curvature, 192 – group, 110 curve, – invariants, 113 – asymptotic, 3 ff – principal normal, 111 – bi-conjugate, 89 Cech, Eduard, 1, 51 ff – characteristic, 127 central illumination, 157 – conjugate, 89 characteristic, 105 – Darboux-, 53 ff Index 217

– double, 160 fundamental – flecnodal, 2 ff – mapping, 159 ff – focal, 198 – quantities, 201 – fundamental, 105 – theorem, 8, 54 – harmonic, 3, 65, 136 – integral, 7 G – involute, 3 – Lie-, 2 ff generator, – median, 142 – congruence-type, 185, 197 – osculating, 143 – elliptic, 185 – pangeodesic, 57 – hyperbolic, 130, 185 – principal, 13 – Lie-singular, 194 – rational, 163 – parabolic, 185 – Tzitzeica-, 93, 96 – reguloid, 185, 197 – torsal, 117, 174, 184 D G-surface, 108

Darboux, Jean Gaston, 53 H developable, – primary, 15, 16 harmonic – secondary, 15, 16 – axial collineation, 189 discriminant, 103 – homology, 183 DV-, – transformation, 103 – conjugate, 139 harmonisant, 92 – curves, 135 Hedley, James Walter, 45 – isogonal, 139 helix, 193 Hesse, Otto, 126 E Hlavaty, Vaclav, 182 hyperbolic, edge of regression, 20 – arc length, 131 envelope, 205 – centre of curvature, 131 evolute, 131 – curvature, 131 – evolute, 131 F – image, 126 flecnode, – transformation, 134 – vertex, 131 – curve, 2 ff hyperplane, 80 – congruence, 11, 121 – polar, 183 – form, 54 – osculating, 202 – Lie-tetrahedron, 38 hypersphere, 81 – sequel, 47, 162 – strip, 146 – surface, 2ff I – transformation, 205 Frenet-equations, 197 improperly intrinsic, 58, 62 Fubini, Guido, 51 index of quadric, 182 intrinsic, 53 ff invariant, 5 ff Index 218 involution, 13 P – Desargues, 167 – principal, 108 pencil, – of linear complexes, 184 J – crossed complementary, 187 – harmonic, 31 Jacobian, 72, 103 – parabolic, 184 – tangent, 187 K plane, – null, 186 Kanitani, Joyo, 200 ff – torsal, 184 Kanitani-functions, 208 Plücker, Julius, 22 Klapka, Jiri, 90, 163 ff point, Klein, Felix, 2, 182 ff – cuspidal, 178, 184 Klein, – fundamental, 89 – bijection, 182 – harmonic, 3, 65 – hyperquadric, 182 ff – ideal, 189 – image, 4, 182 – inflexion, 128, 178 – involution, 2, 187 – involute, 3 – Model, 182 – Lie-, 2 ff Krames, Josef, 163 – median, 136 Kruppa, Erwin, 192 – parametric, 165 – rectifying, 99 L polarity, 80, 183

Laguerre-Forsyth, 19, 21 Q Lasley, John Wayne Jr., 47 Lie, Sophus, 3 quadric, Lie-curves, 2 ff – complex-permute-, 43 Line, – flecnode-Lie-, 40 – asymptotic, 3 ff – flecnode-reciprocal-, 44 – complex, 2 ff – Klein, 182 – congruence, 3 ff – Lie-, 54, 185 – rectifying, 99 – osculating, 120 – striction, 49 – tangent, 135 quadriderative, 6 M quadrature, 56, 101 quadrilateral, 201 MacLean, Neil Bruce, 38 Maeda, Jusaku, 96, 116 R mapping, 158 ff Mayer, Octave, 93 Reaves, Samuel Watson, 48 regulus, 11 N resultant, 65 Riccati-equation, 40, 101 Neudorfer, Hans, 3, 197 null-polarity, 182 ff null-system, 15, 183 Index 219 ruled, T – cubic, 26 – helicoid, 48 tangent, – surface, 1 ff – asymptotic, 13 ff R-curves, 93 – conjugate, 144 R-family, 93 ff – cuspidal, 174 – quadratic, 106 – Darboux, 54 – axial, 106 – flecnode, 2 ff – inflexion, 178 S – Lie-, 190 – osculating, 141 Salmon, George, 2 tangential image, 102 Sauer, Robert, 182 Taylor’s theorem, 21, 193 Schwarzian, 12 Terracini, Alessandro, 89 self-dual, 171 torse, 163 semi-invariants, 6, 145 torsion, 192 sextacktic, 145 transversal, 11, 121, 185 simplex, 196 Tsuboko, Matsuji, 206 space, Tzitzeica, Georges, 48, 50, 96 – affine, 93 T-curves, 96 – dual, 32 – elliptic, 189 V – Euclidean, 191 – osculating, 184 vertex of cone, 169 – projective, 182, 191 Voss, Aurel, 2, 163 stratified set of curves, 139 strip, W – bi-flecnode, 147 – coincidence, 145, 151 weight, 7 – flecnode, 146 Weiss, Gunter, 182 – Lie-, 198 Weyr, Emil, 178 – pangeodetic, 147 Wiener, Christian, 173 surface, Wilczynski, Ernst Julius, 5 ff – accompanying, 155 Winternitz binormal, 113 – affine binormal, 115 – algebraic, 164 – complex, 4 – conjugate, 48 – cubic, 26 – derivative, 11 – developable, 7 – flecnode, 2 ff – focal, 13, 121 – principal, 13 – rational, 163 – torsal, 199 – Winternitz, 115 *-transformation, 119 Curriculum Vitae

Family name : Khattab First name : Ashraf Date of Birth : 12 March 1971 – Alexandria Nationality : Egyptian Marital status : Single

September 1977 – Mai 1988 : Primary and Secondary School in Cairo

Mai 1988 : General Secondary Education Certificate from “Madinet Nasr” Experimental School in Cairo

September 1988 – June 1993 : B.Sc. studies in the Faculty of Engineering, Ain Shams University in Cairo

16 August 1993 : B.Sc. Degree in Civil Engineering from the Public Works Dept.

October 1993 – November 1994 : Military service

December 1994 – July 2000 : Assistant for Descriptive Geometry in the Engineering Physics and Mathematics Dept., Faculty of Engineering, Ain Shams University, Cairo

November 1994 – October 1995 : The final examination for the qualifying year in the Department of Engineering Physics and Mathematics

November 1995 – October 1997 : The final examination in the qualifying subjects for the M.Sc. studies

January 1998 – July 2000 : Research work for the M.Sc. with topic of research: “A geometrical treatment for establishment of panoramic perspective from successive photographs“

24 July 2000 - : M.Sc. Degree in Engineering Mathematics from the Engineering Physics and Mathematics Dept., Faculty of Engineering, Ain Shams University, Cairo

August 2000 - : Assistant Lecturer for Descriptive Geometry in the Engineering Physics and Mathematics Dept., Faculty of Engineering, Ain Shams University, Cairo

April 2002 – December 2005 : Ph.D. student at the Institute for Geometry – TU Dresden.

Versicherung

Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt.

Erklärung

Die Dissertation mit dem Thema „Flecnodal and Lie-curves of ruled surfaces“ wurde an der Technischen Universität Dresden an der Fakultät für Mathematik und Naturwissenschaften im Institut für Geometrie unter der Betreuung von Prof. Dr. G. Weiß angefertigt. Ich erkenne die Promotionsordnung der Fakultät Mathematik und Naturwissenschaften in der Fassung vom 20.03.2000 an.

Dresden, den Unterschrift