Introduction to Classical Geometries - A. Galarza, J. Seade (Birkhauser, 2002) WW.pdf - Geometry - Kuya

Ana Irene Ramírez Galarza

José Seade

Introduction to

Classical Geometries

Birkhäuser

Basel · Boston · Berlin

Authors:

Ana Irene Ramírez Galarza

Departamento de Matemáticas

Facultad de Ciencias

Universidad Nacional Autónoma de

México

Ciudad Universitaria, Circuito Exterior

México 04510 D. F.

México

e-mail: anai@matematicas.unam.mx

José Seade

Instituto de Matemáticas

Unidad Cuernavaca

Universidad Nacional Autónoma de México

Av. Universidad s/n

Ciudad Universitaria,

Lomas de Chamilpa

Cuernavaca, Morelos

México

e-mail: jseade@matem.unam.mx

Translated from the Spanish original “Introducción a la Geometría Avanzada”

Figures by Juan Pablo Romero Méndez

2000 Mathematical Subject Classi cation: 51, 53, 14, 22, 83

Library of Congress Control Number: 2007922251

Bibliographic information published by Die Deutsche Bibliothek

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliogra e; detailed

bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-7517-1 Birkhäuser Verlag AG, Basel – Boston – Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the mate-

rial is concerned, speci cally the rights of translation, reprinting, re-use of illustrations, recita-

tion, broadcasting, reproduction on micro lms or in other ways, and storage in data banks. For

any kind of use permission of the copyright owner must be obtained.

Basel • Boston • Berlin

P.O. Box 133, CH-4010 Basel, Switzerland

Part of Springer Science+Business Media

Printed on acid-free paper produced from chlorine-free pulp. TCF∞

Printed in Germany

ISBN-10: 3-7643-7517-5

e-ISBN-10: 3-7643-7518-3

ISBN-13: 978-3-7643-7517-1

e-ISBN-13: 978-3-7643-7518-8

9 8 7 6 5 4 3 2 1

www.birkhauser.ch

Goran Peskir

School of Mathematics

The University of Manchester

Sackville Street

Manchester

M60 1QD

United Kingdom

e-mail: goran@maths.man.ac.uk

Contents

Preface

vii

List of Symbols

1 Euclidean geometry 1

1.1 Symmetries ............................... 2

1.2 Rigidtransformations ......................... 15

1.3 Invariants under rigid transformations . . . . . . . . . . . . . . . . 28

1.4 Cylindersandtori ........................... 37

1.5 Finite subgroups of E (2) and E (3) .................. 46

1.6 Friezepatternsandtessellations.................... 58

2 A ne geometry 75

2.1 The line at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.2 A netransformationsandtheirinvariants.............. 83

3 Projective geometry 91

3.1 Therealprojectiveplane ....................... 92

3.2 TheDualityPrinciple ......................... 99

3.3 The shape of P 2 ( ) .......................... 103

3.4 Coordinate charts for P 2 ( ) (and for P 1 ( ))............ 109

3.5 Theprojectivegroup.......................... 113

3.6 Invarianceofthecrossratio...................... 121

3.7 Thespaceofconics........................... 126

3.8 Projectivepropertiesoftheconics .................. 129

3.9 Polesandpolars ............................ 134

3.10 Elliptic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4 Hyperbolic geometry 149

4.1 Modelsofthehyperbolicplane .................... 149

4.2 Transformationsofthehyperbolicplane ............... 157

4.3 Steinernetwork............................. 164

Introduction to Classical Geometries - A. Galarza, J. Seade (Birkhauser, 2002) WW.pdf

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