Geometry - Audin PDF

Preview

Summary

Michèle Audin Geometry Michèle Audin Institut de Recherche Mathématique 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 Co...

Description

Mich` ele Audin

Geometry

Mich` ele Audin Institut de Recherche Math´ematique Avanc´ee, Universit´e Louis Pasteur et CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg cedex, France. E-mail : [email protected] Url : http://www-irma.u-strasbg.fr/~maudin

27th May 2002

Geometry Mich` ele Audin

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. This is a book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. How to use this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. About the English edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3

I. Affine Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Affine spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Affine mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Using affine mappings: three theorems in plane geometry . . . . . . . . 4. Appendix: a few words on barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Appendix: the notion of convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Appendix: Cartesian coordinates in affine geometry . . . . . . . . . . . . . . Exercises and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 14 23 26 28 30 32

II. Euclidean Geometry, Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Euclidean vector spaces, Euclidean affine spaces . . . . . . . . . . . . . . . . . . 2. The structure of isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The group of linear isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 46 52 58

III. Euclidean Geometry in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Isometries and rigid motions in the plane . . . . . . . . . . . . . . . . . . . . . . . . 3. Plane similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Inversions and pencils of circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 76 79 83 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. Projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2. Projective subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

ii

Contents

3. Affine vs projective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4. Projective duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5. Projective transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. Classification and properties of affine conics . . . . . . . . . . . . . . . . . . . . . . 189 3. Projective quadrics and conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4. The cross-ratio of four points on a conic and Pascal’s theorem . . 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 remember that I tried several times to use a slide rule, and that, several times also, I began modern maths textbooks, saying to myself that if I were going slowly, if I read all the lessons in order, doing the exercises and all, there was no reason why I should stall. Georges Perec, in [Per78].

1. This is a book. . . This is a book written for students who have been taught a small amount of geometry at secondary school and some linear algebra at university. It comes from several courses I have taught in Strasbourg. Two directing ideas. The first idea is to give a rigorous exposition, 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 as being a set acted on by a vector space (and this is what I do here) but this formal approach, although elegant, must not hide the “phenomenological” aspect of elementary geometry, its own aesthetics: yes, Thales’ theorem expresses 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 enveloping a 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 in a na¨ıve way in Chapter III, make a more abstract comeback in the chapter on projective geometry and in that on quadrics. Similarly, the study of

Introduction

2

projective conics 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. The second idea is to have an as-open-as-possible text: textbooks are often 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 program treated here is rather limited, I hope to interest also more advanced readers. Finally, mathematics is a human activity and a large part of the contents of this book belongs to our most classical cultural heritage, since are evoked the rainbow according to Newton, the conic sections according to Apollonius, the difficulty of drawing maps of the Earth, the geometry of Euclid and the parallel axiom, the measure of latitudes and longitudes, the perspective problems of the painters of the Renaissance(1) and the Platonic polyhedra. I have tried to show this in the way of writing the book(2) and in the bibliographical 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 actions. . . )(4) and of topology of Rn and the definition of a differentiable mapping and—for the last chapter only—the usual various avatars of the implicit 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 you must study and solve the exercises. They are of three kinds: – There are firstly proofs or complements to notions that appear in the main text. These exercises are not difficult and it is necessary to solve them in order to check that you have understood the text. They are a complement to the reading of the main text; they are often referenced there and should be done as you go along reading the book. (1) The

geometry book of D¨ urer [D¨ ur95] was written for art amateurs, not for mathematicians. (2) The way to write mathematics is also part of the culture. Compare the “eleven properties of the sphere” in [HCV52] with the “fourteen ways to describe the rain” of [Eis41]. (3) There is a section reminding the readers of the properties of quadratic forms in the chapter on conics and quadrics. (4) Transformation groups are the essence of geometry. I hope that this ideology is transparent in this text. To avoid hiding this essence, I have chosen not to write a section of general nonsense on group actions. The reader can look at [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) evoked above. – 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 course the fact that I was not completely satisfied with other books: there are numerous geometry books, the good ones being often too hard or 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 entice the reader to read, in addition to the three books I have just mentioned [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 the last fifty years, that might not be available to the English-speaking readers but deserve to be mentioned: [DC51], [LH61], [LP67] and [Sau86].

3. About the English edition This is essentially a translation of the French G´eom´etrie 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 explanations (in general due to discussions with my students in Strasbourg) in a few places, especially in the chapter on quadrics, either in the main text or in the solutions to the exercises. I must confess that I have had a hard time with the terminology. Although I am almost bilingual in differential or algebraic geometry, I was quite amazed to realize that I did not know a single English word dealing with elementary geometry. I have learnt from [Cox69], from (the English translation [Ber94] of) [Ber77] and from [Sil01].

4. Acknowledgements I wish to thank first all the teachers, colleagues and students, who have contributed, for such a long time, to my love of the mathematics I present in 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. A preliminary version was tested by the Strasbourg students(5) during the 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, together with all those with whom I have had the opportunity to discuss the contents of this book and its style, especially Myriam Audin and Juliette Sabbah(7) for their help with the writing of the exercises about caustics. Laure Blasco carefully read the preliminary version and criticized in great detail the chapter on quadrics. She has helped me to look for a better balance between the algebraic presentation and the geometric properties. For her remarks and her discrete way of 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 converging to more rigor but also to a better appropriateness to the expected audience. For the time he spent, for his humor and his spidery scrawl, I thank 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 book owes to him better presentation of the relation linear algebra/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 version and even of the French edition. I cannot name them all, but among them, I want to mention especially Nadine Baldensperger, R´egine Barthelm´e, Martine Bourst, Sophie G´erardy, ´ Catherine Goetz, Mathieu Hibou, Etienne Mann, Nicolas Meyer, Myriam (5) To

be quite honest, I should say that I have used these students as guinea pigs. was also very pleased to include his elegant short proof of the Erd˝ os–Mordell theorem (Exercise III.25) in this edition. (7) Who has also drawn some of the pictures.

(6) I

4. Acknowledgements

5

Oyono-Oyono, Magali Pointeaux, Sandrine Zitt and all those who have asked questions, raised criticisms and even made suggestions that were very useful, but, more importantly, have given some sense and life to the final writing of this version. ♦ For this book, I have used the LATEX 2ε packages of the Soci´et´e math´ematique de France. I cannot thank myself for writing, translating and typing this text, or for solving most of the three hundred and fifty-five exercises and “drawing” most of the one hundred and seventy-two pictures it contains, but I can thank Claude Sabbah for his singular, stylistic, etc. help. ♦

Chapter I Affine Geometry

An affine space is a set of points; it contains lines, etc. and affine geometry(1) deals, for instance, with the relations between these points and these lines (collinear points, parallel or concurrent lines. . . ). To define these objects and describe their relations, one can: – Either state a list of axioms, describing incidence properties, like “through two points passes a unique line”. This is the way followed by Euclid (and more recently by Hilbert). Even if the process and a fortiori the axioms themselves are not explicitly stated, this is the way used in secondary schools. – Or decide that the essential thing is that two points define a vector and define everything starting from linear algebra, namely from the axioms defining the vector spaces. I have chosen here to use the second method, because it is more abstract and neater, of course, but also, mainly, because I think that it is time, at this level, to prove to students that the linear algebra they were taught is “useful” for something!

1. Affine spaces Definition 1.1. A set E is endowed with the structure of an affine space by the data of a vector space(2) E and a mapping Θ that associates a vector of

(1)“Pure” affine

geometry, in the sense that there are no distances, angles, perpendiculars, these belonging to Euclidean geometry, which is the subject of the following chapters. (2) This is a vector space over a commutative field K of ...