Table Of Content

Michèle Audin Geometry Michèle Audin Institut de RechercheMathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg cedex, France. E-mail : [email protected] Url : http://www-irma.u-strasbg.fr/~maudin 27th May 2002 Geometry Michèle Audin Contents Introduction ........................................................ 1 1. This is a book... ................................................ 1 2. How to use this book ............................................ 2 3. About the English edition ........................................ 3 4. Acknowledgements .............................................. 3 I. Affine Geometry .................................................. 7 1. Affine spaces .................................................... 7 2. Affine mappings .................................................. 14 3. Using affine mappings: three theorems in plane geometry ........ 23 4. Appendix: a few words on barycenters .......................... 26 5. Appendix: the notion of convexity ................................ 28 6. Appendix: Cartesian coordinates in affine geometry .............. 30 Exercises and problems ............................................ 32 II. Euclidean Geometry, Generalities ............................ 43 1. Euclidean vector spaces, Euclidean affine spaces .................. 43 2. The structure of isometries ...................................... 46 3. The group of linear isometries .................................... 52 Exercises and problems ............................................ 58 III. Euclidean Geometry in the Plane ............................ 65 1. Angles ............................................................ 65 2. Isometries and rigid motions in the plane ........................ 76 3. Plane similarities ................................................ 79 4. Inversionsand pencils of circles .................................. 83 Exercises and problems ............................................ 98 IV. Euclidean Geometry in Space ................................113 1. Isometries and rigid motions in space ............................113 2. The vector product, with area computations ....................116 3. Spheres, spherical triangles ......................................120 4. Polyhedra,Euler formula ........................................122 5. Regular polyhedra ................................................126 Exercises and problems ............................................130 V. Projective Geometry ............................................143 1. Projectivespaces ................................................143 2. Projectivesubspaces ..............................................145 ii Contents 3. Affine vs projective ..............................................147 4. Projectiveduality ................................................153 5. Projectivetransformations ......................................155 6. The cross-ratio ..................................................161 7. The complex projective line and the circular group ..............164 Exercises and problems ............................................170 VI. Conics and Quadrics ..........................................183 1. Affine quadrics and conics, generalities ..........................184 2. Classificationand properties of affine conics ......................189 3. Projectivequadrics and conics ....................................200 4. The cross-ratioof four points on a conic and Pascal’stheorem .. 208 5. Affine quadrics, via projective geometry ..........................210 6. Euclidean conics, via projective geometry ........................215 7. Circles, inversions,pencils of circles ..............................219 8. Appendix: a summary of quadratic forms ........................225 Exercises and problems ............................................233 VII. Curves, Envelopes, Evolutes ................................247 1. The envelope of a family of lines in the plane ....................248 2. The curvature of a plane curve ..................................254 3. Evolutes ..........................................................256 4. Appendix: a few words on parametrized curves ..................258 Exercises and problems ............................................261 VIII. Surfaces in 3-dimensional Space ............................269 1. Examples of surfaces in 3-dimensional space ......................269 2. Differential geometry of surfaces in space ........................271 3. Metric properties of surfaces in the Euclidean space ..............284 4. Appendix: a few formulas ........................................294 Exercises and problems ............................................296 A few Hints and Solutions to Exercises ..........................301 Chapter I ..........................................................301 Chapter II ..........................................................304 Chapter III ........................................................306 Chapter IV ........................................................314 Chapter V ..........................................................321 Chapter VI ........................................................326 Chapter VII ........................................................332 Chapter VIII ........................................................336 Bibliography ........................................................343 Index ................................................................347 Introduction I rememberthat I tried severaltimes to use a slide rule, andthat,severaltimesalso,Ibeganmodernmathstext- books, saying to myself that if I were going slowly, if I read all the lessons in order, doing the exercises and all, therewas noreason whyI should stall. Georges Perec,in [Per78]. 1. This is a book... This is a book written for students who havebeen taughta small amountof geometryatsecondaryschoolandsomelinearalgebraatuniversity. Itcomes from severalcourses I havetaught in Strasbourg. Two directing ideas. The first idea is to givea rigorousexposition,based on the definition of an affine space via linear algebra, but not hesitating to be elementary and down-to-earth. I have tried both to explain that linear algebra is useful for elementary geometry (after all, this is where it comes from) and to show“genuine”geometry: triangles, spheres, polyhedra, angles at the circumference, inversions,parabolas... It is indeed very satisfying for a mathematician to define an affine space asbeingasetactedonbyavectorspace(andthisiswhatIdohere)butthis formal approach, although elegant, must not hide the “phenomenological” aspect of elementary geometry, its own aesthetics: yes, Thales’ theorem ex- presses the fact that a projection is an affine mapping, no, you do not need to orient the plane before defining oriented angles... but this will prevent neither the Euler circle from being tangent to the incircle and excircles, nor the Simson lines from envelopinga three-cusped hypocycloid! This makes you repeat yourself or, more accurately, go back to look at certain topics in a different light. For instance, plane inversions, considered inana¨ıvewayinChapterIII,makeamoreabstractcomebackinthechapter on projective geometry and in that on quadrics. Similarly, the study of 2 Introduction projectiveconics in Chapter VI comes after that of affine conics... although it would have been simpler—at least for the author!—to deduce everything from the projective treatment. Thesecondideaistohaveanas-open-as-possibletext: textbooksareoften limited to the program of the course and do not give the impression that mathematics is a science in motion (nor in feast, actually!). Although the programtreatedhereis ratherlimited,I hopetointerestalsomoreadvanced readers. Finally, mathematics is a human activity and a large part of the contents of this book belongs to our most classicalcultural heritage, since are evoked therainbowaccordingtoNewton,theconicsectionsaccordingtoApollonius, the difficulty of drawing maps of the Earth, the geometry of Euclid and the parallelaxiom,themeasureoflatitudesandlongitudes,theperspectiveprob- lemsofthe paintersoftheRenaissance(1) andthePlatonicpolyhedra. Ihave triedtoshowthisinthewayofwritingthebook(2) andinthebibliographical references. 2. How to use this book Prerequisites. They consist of the basics of linear algebra and quadratic forms(3), a small amount of abstract algebra (groups, subgroups, group ac- tions...)(4) andoftopologyofRn andthedefinitionofa differentiablemap- pingand—forthelastchapteronly—theusualvariousavatarsoftheimplicit function theorem, and for one or two advanced exercises, a drop of complex analysis. Exercises. All the chapters end with exercises. It goes without saying (?) that youmust study and solve the exercises. They are of three kinds: – There are firstly proofs or complements to notions that appear in the maintext. Theseexercisesarenotdifficultanditisnecessarytosolve theminordertocheckthatyouhaveunderstoodthe text. They area complementtothe readingofthe maintext; they areoftenreferenced there and should be done as you go along reading the book. (1)The geometry book of Du¨rer [Du¨r95] was written for art amateurs, not for mathem- aticians. (2)Thewaytowritemathematicsisalsopartoftheculture. Comparethe“elevenproperties ofthesphere”in[HCV52]withthe“fourteenwaystodescribetherain”of[Eis41]. (3)There is a section reminding the readers of the properties of quadratic forms in the chapteronconicsandquadrics. (4)Transformationgroups arethe essence of geometry. I hope that this ideology is trans- parent in this text. To avoid hiding this essence, I have chosen not to write a section of generalnonsenseongroupactions. Thereadercanlookat[Per96],[Art90]or[Ber94]. 4. Acknowledgements 3 – There are also“just-exercises”, often quite nice: they contain most of the phenomena (of plane geometry, for instance) evokedabove. – There are also more theoretical exercises. They are not always more difficult to solve but they use more abstract notions (or the same notions, but considered from a more abstract viewpoint). They are especially meant for the more advanced students. Hints of solutions to many of these exercises are grouped at the end of the book. About the references. The main reason to have written this book is of coursethefactthatIwasnotcompletelysatisfiedwithotherbooks: thereare numerousgeometrybooks,the goodones being often too hardor too big for students (I am thinking especially of [Art57], [Fre73], [Ber77], [Ber94]). But there are many good geometry books... and I hope that this one will enticethereadertoread,inadditiontothethreebooksIhavejustmentioned [CG67], [Cox69], [Sam88], [Sid93] and the more recent [Sil01]. To write this book and more precisely the exercises, I have also raided (shamelessly, I must confess) quite a few French secondary school books of thelastfiftyyears,thatmightnotbeavailabletotheEnglish-speakingreaders but deserve to be mentioned: [DC51], [LH61], [LP67] and [Sau86]. 3. About the English edition This is essentially a translation of the French Géométrie published in 1998 by Belin and Espaces34. However, I have also corrected some of the errors of the French edition and added a few figures together with better explana- tions (ingeneraldue to discussionswith mystudentsinStrasbourg)in a few places,especiallyinthechapteronquadrics,eitherinthemaintextorinthe solutions to the exercises. ImustconfessthatIhavehadahardtimewiththeterminology. Although Iamalmostbilingualindifferentialoralgebraicgeometry,Iwasquiteamazed to realize that I did not know a single Englishworddealing with elementary geometry. Ihavelearntfrom[Cox69],from(theEnglishtranslation[Ber94] of) [Ber77] and from [Sil01]. 4. Acknowledgements I wish to thank first all the teachers, colleagues and students, who have contributed,for sucha long time, to myloveofthe mathematicsI presentin this book. 4 Introduction It was is Daniel Guin who made me write it. Then Nicole Bopp carefully read an early draft of the first three chapters. Both are responsible for the existence of this book. I thank them for this. ApreliminaryversionwastestedbytheStrasbourgstudents(5) duringthe academic year 1997–98. Many colleagues looked at it and made remarks, suggestions and criticisms, I am thinking mainly of Olivier Debarre, Paul Girault and Vilmos Komornik(6). The very latest corrections to the English edition were suggested by Ana Cannas da Silva and Mihai Damian. I thank them,togetherwithallthosewithwhomIhavehadtheopportunitytodiscuss the contentsofthis bookanditsstyle,especiallyMyriamAudinandJuliette Sabbah(7) for their help with the writing of the exercises about caustics. LaureBlascocarefullyreadthe preliminaryversionandcriticizedingreat detailthechapteronquadrics. Shehashelpedmetolookforabetterbalance between the algebraic presentation and the geometric properties. For her remarks and her discrete wayof insisting, I thank her. Pierre Baumann was friendly enough to spend much of his precious time reading this text. He explained to me his disagreements with tenacity and kindness in pleasant discussions. In addition to thousands of typographic and grammatical corrections, innumerable ameliorations are due to him, all convergingtomorerigorbutalsotoabetterappropriatenesstotheexpected audience. Forthetimehespent,forhishumorandhisspideryscrawl,Ithank him. I was very pleased that Daniel Perrin read the preliminary version with a lot of care and his sharp and expert eye, he explained me his many disagreements and has (almost always) convinced me that I was wrong. This bookowestohimbetterpresentationoftherelationlinearalgebra/geometry, a few arrows, a great principle, numerous insertions of“we have”, and a lot of (minor or not) corrections together with several statements and exercises (and probably even an original result, in Exercise V.38). Is it necessary to add that I am grateful to him? Finally, I am grateful to all the students who have suffered the lectures this book comes from and all those who have worked hard because of the errors and clumsiness of the preliminary versionand even of the Frenchedi- tion. I cannotname them all, but among them, I wantto mentionespecially Nadine Baldensperger, Régine Barthelmé, Martine Bourst, Sophie Gérardy, Catherine Goetz, Mathieu Hibou, E´tienne Mann, Nicolas Meyer, Myriam (5)Tobequitehonest,IshouldsaythatIhaveusedthesestudentsasguineapigs. (6)IwasalsoverypleasedtoincludehiselegantshortproofoftheErd˝os–Mordelltheorem (ExerciseIII.25)inthisedition. (7)Whohasalsodrawnsomeofthepictures.