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Projective Line Geometry

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Projective Line Geometry Projective Line geometry By Dr. Robert Sauer Assoc. Professor at the Technischen Hochschule in Aachen Translated by D. H. Delphenich With 36 Figures Walter de Gruyter & Co. Formerly G. J. Göschen’s Publishing House J. Guttentag, Publisher-Bookstore – Georg Reimer – Karl J. Trübner – Veit & Co. Berlin W 35 and Leipzig 1937 Foreword The present book is an attempt to interest a broader circle of younger mathematicians in line geometry. It should not serve as a more or less complete summary of the old and new results in line geometry, but only a selection of the topics that promise to attract some lasting interest due to their simple and “intuitive” content. The main emphasis shall reside in the geometric content, not in the analytical tools; that viewpoint will serve especially for the comparison of models that are constructed from discrete lines. As the title of the book suggests, it will be essentially projective questions that are treated. This restriction was necessary in order to not exceed the scope of the book; however, it is also factually based, since metric line geometry finds its most natural representation in terms of entirely different analytical tools (viz., Study’s dual vectors). The larger part of the book relates to differential line geometry; the part on algebraic line geometry will present only the must essential basic concepts, and some examples that are occasionally interspersed throughout (e.g., third-degree ruled surface, third and fourth-order space curves, quadratic system of lines, complexes of lines, etc). The projective differential geometry of curves and surfaces will be employed in the treatment of torses and parabolic systems of lines, since line coordinates are better adapted to that problem than point and plane coordinates. In that sense, the book is an extension of volume 22 of this collection, in which E. Salkowski has presented the differential geometry of curves and surfaces. The analytical tools are the same ones that W. Blaschke applied to Lie’s sphere geometry in volume 3 of his Differentialgeometrie (Berlin, 1929), which was revised by G. Thomsen. One will find many of the theorems of differential line geometry that will be treated in what follows in that content-rich book (by way of analogy with sphere geometry). The same is true for Geometria proiettiva differenziale of G. Fubini-E. Čech (Bologna, 1926), which one might do better to read in the French version (Paris, 1931). The applications to mechanics (e.g., frameworks, Ball’s theorem of screws, stress distributions in membranes) take up a relatively sizable space. Many problems will be dealt with in them, and in the theory of infinitesimal surface bending, that are first posed in their metric formulation, but will nevertheless remarkably lead to projective aspects. Unfortunately, questions of integral geometry must be passed over. I would like to thank Herren Dr. O. Baier and Prof. Dr. J. Lense, as well as Herrn Dr. Lennertz, for their friendly assistance in the correcting and for many worthwhile suggested improvements. ________ Table of contents Page Introduction………………………………………………………………………. 1 § 1. Basic features of analytic geometry……………………………………… 2 § 2. Basic features of the theory of curves and surfaces ………..……………. 7 Chapter I: Basic concepts of algebraic line geometry. § 3. Line coordinates…………………………………………………………… 11 § 4. Projective maps…………………………………………………………… 18 § 5. Projective force transformation and projective kinematic transformation… 24 § 6. Linear complex……………………………………………………………. 27 § 7. Invariants of n six-vectors…………………………………………………. 32 § 8. Linear manifolds of complexes……………………………………………. 38 § 9. Simplest projective invariants of linear complexes and straight lines…….. 48 § 10. Force screws and motion screws………………………………………….. 52 § 11. Unsteady frameworks and rectangle nets…………………………………. 55 Chapter II: Families of lines. § 12. Definition of a family of lines…………………………………………….. 62 § 13. Contact structures…………………………………………………………. 67 § 14. Invariants of a family of lines…………………………………………….. 74 § 15. Hyperbolic ruled families…………………………………………………. 76 § 16. Parabolic ruled families…………………………………………………… 83 § 17. Torses……………………………………………………………………. 86 § 18. Establishing a family of lines by invariants; self-projective families of lines……………………………………………………………………….. 88 Chapter III: Line systems. § 19. Definition of a line system………………………………………………. 100 § 20. Tensors…………………………………………………………………… 103 § 21. Invariants of a line system ………………………………………………. 106 § 22. Contact structures for hyperbolic line systems ………………………….. 111 § 23. Differential equations of hyperbolic line systems ………………………. 117 § 24. Contact structures for parabolic line systems ……………………………. 123 § 25. Differential equations of parabolic line systems ………………………… 128 § 26. Projective differential geometry of negatively-curved surfaces………….. 133 Chapter IV: Special line systems. § 27. Self-projective line systems ……………………………………………… 139 § 28. Special parabolic line systems ..…………………………………………. 144 ii Table of contents Page § 29. Special hyperbolic line systems. W-systems……………………..……… 152 Chapter V: Infinitesimal bending of surfaces. § 30. Screw cracks…………………………………………………………….. 161 § 31. Infinitesimal bending of mutually-projective surfaces…………………... 165 § 32. Connection with the theory of W-systems……………………………….. 169 § 33. Torsion-fixed and curvature-fixed nets of curves………………………… 172 § 34. Stress distributions in membranes……………………………………….. 181 Chapter VI: Line complexes. § 35. Definition of a line complex……………………………………………… 184 § 36. Contact structures…………………………………………………………. 187 § 37. Invariants of a line complex………………………………………………. 195 § 38. Differential equations…………………………………………………….. 200 § 39. Tetrahedral complexes……………………………………………………. 204 § 40. Tetrahedral line systems of class 2 (order 2, resp.)………………………. 214 ___________ Introduction If one employs a line as the basic element of spatial geometry in place of the point or the plane following the process of J. Plücker (1801-1868) then one can speak of line geometry (1). We will be concerned with that kind of geometry here. We will thus mainly investigate the relationships that remain preserved under the group of projective maps, which implies projective line geometry. All considerations will relate to real, three-dimensional projective space – i.e., to the space of Euclidian geometry, when it is extended by the imaginary structures of projective geometry. Analytically, we will then be dealing with the development of the invariant theory of projective transformations in line coordinates in a manner that is similar to the way that one presents the motion invariants in Cartesian point coordinates in elementary analytical geometry. After we have learned about the necessary basic concepts of algebraic line geometry, we will turn to differential projective line geometry. We will then successively treat the differential geometry of 1, 2, and 3-parameter sets of lines ( families, systems, and complexes of lines, resp.) in the space of lines, which depends upon four parameters, just as one examines 1 and 2-parameter point-sets (viz., curves and surfaces) in a point space that depends upon three parameters in point geometry. It will be shown in that way that the projective differential geometry of curves and surfaces will be handled at the same time. As S. Lie has remarked, line geometry is closely connected with a geometry that employs the sphere as the basic element. Whoever would wish to learn about this very intriguing relationship could confer, e.g., W. Blaschke-G. Thomsen: Differentialgeometrie III (Berlin, 1929) ( 2). Before we take up our line-geometric investigations, we shall summarize the necessary basic notions of analytic geometry and the differential geometry of curves and surfaces (or at least, the main terms) in the first two paragraphs. In addition, in order to simplify the analytical tools, we shall assume once and for all that: All functions will be assumed to be regular; i.e., they can be developed in convergent power series in the domains in question of the independent variables. (1) Let some of the following older presentations of line geometry be mentioned: G. Koenigs , La géométrie reglée et ses applications , Paris, 1895. K. Zindler : Liniengeometrie I, II, Leipzig, 1902, 1906. F. Klein: Volume 1 of his Gesammelte Abhandlungen , Berlin, 1921. Metric differential line geometry is treated by e.g., W. Blaschke : Differentialgeometrie I , Berlin, 1930; one finds many details about projective differential line geometry in Fubini-Čech : Geometria proiettiva differenziale , Bologna, 1926 (or better yet, Paris, 1931) and W. Blaschke and G. Thomsen: Differentialgeometrie III , Berlin, 1929. Among the large number of individual papers on the subject, let us mention: G. Sannia : Ann. di mat. (3) 17 (1910). G. Thomsen: Math. Zeit. (1924) and Hamburger Abhandlungen (1925). W. Haack: Monatshefte für Math. u. Phys. 36 (1929); 44 (1936); Math. Zeit. 33 (1931); 35 (1932); 40 (1935); 41 (1936). The algebraic questions were treated by E. A. Weiss in the sense of E. Study in Einführung in die Lineiengeometrie und Kinematik , Leipzig-Berlin, 1935. (2) Cf., also, L. Bieberbach: Einführung in die höhere Geometrie , Leipzig-Berlin, 1933. 2 Introduction § 1. Basic features of analytic geometry. 1. Vectors. Rectangular point and plane coordinates. We start with three- dimensional Euclidian space and assume that the concept of a vector is known, as well as the laws of vector addition and subtraction. We shall denote vectors by large German
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