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Rational Families of Circles and Bicircular Quartics

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Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg DISSERTATION 2011 Thomas Rainer Werner Rational families of circles and bicircular quartics Rationale Kreisscharen und bizirkulare Quartiken Der Naturwissenschaftlichen Fakult¨at der Friedrich-Alexander-Universit¨atErlangen-N¨urnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Thomas Rainer Werner aus Lichtenfels Als Dissertation genehmigt von der Naturwissenschaftlichen Fakult¨at der Friedrich-Alexander-Universit¨atErlangen-N¨urnberg Tag der m¨undlichen Pr¨ufung:18. Juli 2012 Vorsitzender der Pr¨ufungskommission: Prof. Dr. Rainer Fink Erstberichterstatter: Prof. Dr. Wolf P. Barth Zweitberichterstatter: Prof. Dr. Wulf-Dieter Geyer Abstract This dissertation deals with special plane algebraic curves, with so called bicircular quar- tics. These are curves of degree four that have singularities in the circular points at infinity of the complex projective plane P2(C). The main focus lies on real curves, i.e. such curves that are invariant under complex conjugation. Many of the statements on bicircular quartics presented in this work are well known since the end of the 19th century, but the way of proving at that time did not fully employ the language of the then well developed projective geometry. The primary goal of this text is the formulation of the classical statements on bicircular quartics in modern language. In order to achieve this the theoretical framework is built beginning with the space of circles in the language of projective geometry. Within the space of circles are discussed at first linear and then quadratic families of circles. In the following the theorems on bicircular quartics and their degenerate form, the circular cu- bics, are proved by means of geometrical statements on the mentioned families of circles in the projective space of circles. An ancillary goal of this work is the provision of tools that facilitate an easy depiction of bicircular quartics and rational families of circles with the help of a computer. Zusammenfassung Diese Dissertation besch¨aftigtsich mit speziellen ebenen algebraischen Kurven, mit soge- nannten bizirkularen Quartiken. Das sind Kurven vierten Grades, die Singularit¨atenin den unendlich fernen Kreispunkten der komplex-projektiven Ebene P2(C) besitzen. Das Hauptaugenmerk liegt dabei auf reellen Kurven, d.h. solchen Kurven, die unter der kom- plexen Konjugation invariant sind. Viele der in dieser Arbeit vorgestellten Aussagen ¨uber bizirkulare Quartiken sind bereits seit Ende des 19. Jahrhunderts wohl bekannt, die Be- weisf¨uhrungvon damals nutzte jedoch nicht konsequent die Sprache der seinerzeit bereits entwickelten projektiven Geometrie aus. Das prim¨areZiel dieser Arbeit ist die Formulierung der klassischen Aussagen ¨uber bizirkulare Kurven in moderner Sprache. Zu diesem Zwecke wird das theoretische Ger¨ust beginnend beim Raum der Kreise in der Sprache der projektiven Geometrie aufgebaut. Im Raum der Kreise werden zun¨achst lineare, dann quadratische Kreisscharen diskutiert. Im Folgenden werden die Theoreme ¨uber bizirkulare Quartiken und ¨uber ihre Entartungs- form, die zirkularen Kubiken, mit der Hilfe von geometrischen Aussagen ¨uber die oben genannten Kreisscharen im projektiven Raum der Kreise bewiesen. Ein untergeordnetes Ziel dieser Arbeit ist die Bereitstellung von Werkzeugen, die eine einfache Darstellung von bizirkularen Kurven und von rationalen Kreisscharen am Rechner erm¨oglichen. Acknowledgement I thank my advisor Mr. Prof. Dr. Wolf Barth for his numerous precious suggestions and hints before and during the development of this text. Moreover I thank him for his patience and for the opportunity to work under his leadership as a scientific assistant. Above all I thank my wife S´arkaˇ for her love, her understanding and her support. Danksagung Ich bedanke mich bei meinem Betreuer Herrn Prof. Dr. Wolf Barth f¨ur seine zahlreichen wertvollen Ratschl¨ageund Hinweise vor und w¨ahrend der Entstehung dieses Textes. Weiterhin danke ich ihm f¨urseine Geduld und daf¨ur,dass ich unter seiner Leitung als wissenschaftliche Hilfskraft t¨atigsein konnte. Vor allem bedanke ich mich bei meiner Ehefrau S´arkaˇ f¨urihre Liebe, ihr Verst¨andnisund ihre Unterst¨utzung. Contents 1 Introduction 5 1.1 Main results . .5 Projective space of circles . .5 Solution of the general equation of degree four . .5 Geometric and computational results . .6 1.2 Motivation . .6 Quadratic families of circles . .6 Inversive geometry . .7 Systematic aspects . .8 1.3 History and sources . .9 Ancient mathematics . .9 From the Renaissance to the Industrial Revolution . .9 Irish mathematics in the 19th century . 10 Important sources . 10 2 Algebraic geometry 14 2.1 Basic terms . 14 Affine and projective space . 14 Algebraic curves . 16 2.2 Geometric tools . 18 Points on an algebraic curve . 18 Dual curve and Pl¨ucker equations . 19 3 Circles 21 3.1 Real and complex circles . 21 Classical definition . 21 ∼ 2 Circles in C = R ................................ 22 Description using a M¨obiustransformation . 23 Compliance with the classical definition . 23 2 Circles in C and P2(C)............................ 24 3.2 The space of circles . 25 Real and imaginary circles, nullcircles . 25 The space of circles . 26 Tangential planes of Π . 27 The polar plane of a representant C◦ ..................... 28 3.3 Elementary geometry . 29 Polarity and orthogonality . 29 1 Contents Associated circles . 30 Angle of intersection . 31 4 Transformations 33 4.1 Translations, rotations and dilations . 33 Translations . 33 Rotations . 34 Dilations . 35 4.2 Inversions . 36 The unit inversion " .............................. 36 General inversions . 37 Modification of the construction . 38 Inversion of algebraic curves . 38 4.3 Properties of the inversion . 40 Images of lines and circles under inversion . 40 Invariant circles . 42 Commuting inversions . 43 Angles . 43 5 Projective space of circles 46 5.1 The projective space of circles P(Circ)..................... 46 Motivation . 46 Definition . 47 Center and radius, infinitely large circles . 48 5.2 The action of transformations on P(Circ)................... 49 Translations, rotations and dilations . 49 The unit inversion . 51 General inversions . 51 5.3 Elementary geometry in P(Circ)........................ 53 The polar plane in P(Circ)........................... 53 Polarity and angles . 54 5.4 The inversive group . 55 Reflections and inversions . 55 Connection to M¨obiustransformations . 56 The transformation matrix M ......................... 57 6 Linear families of circles 59 6.1 Lines in P(Circ)................................. 59 Linear families of circles . 59 Base points . 61 Nullcircles . 62 6.2 The conjugated family . 62 Definition . 62 Regular configurations . 64 Singular configurations . 65 2 Contents 7 Inversion of a linear family 69 7.1 Inversion about a circle of the family . 69 Stabilizing inversions . 69 Representation of the inversion . 70 Fixed points { eigencircles . 71 7.2 Inversion about a general circle . 73 Projection onto a linear family . 73 Induced action on a linear family . 74 Eigencircles of the induced action . 75 Open questions concerning the induced action . 76 7.3 Normalized inversion . 77 Definition and basic properties . 77 Inversions between two given circles . 78 8 Bicircular Quartics 82 8.1 Inverse and pedal curve of a conic section . 82 Inverse of a conic section . 82 Pedal curve of a conic section . 84 8.2 Bicircular Quartics . 85 Definition . 85 Multicircular curves . 87 C-Q-form of a bicircular quartic . 87 8.3 Envelope of a rational family of circles . 89 Parametrization of a rational family of circles . 89 Envelope of a rational family of circles . 90 9 Inversion of Bicircular Quartics 95 9.1 Bicircular quartics as envelopes . 95 Geometric properties of the C-Q-form . 95 Degeneration of the quadric Qt ........................ 98 Determination of the rational family of circles . 100 9.2 Inversion of bicircular quartics . 102 The image under the unit inversion . 102 Circular cubics . 104 Inversions and the three circles form . 104.
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