http://dx.doi.org/10.1090/stml/043 STUDENT MATHEMATICAL LIBRARY Volume 43 Elementary Geometry Ilka Agricola Thomas Friedric h Translated by Philip G . Spain #AM^S^fcj S AMERICAN MATHEMATICA L SOCIET Y Providence, Rhode Islan d Editorial Boar d Gerald B . Follan d Bra d G . Osgoo d Robin Forma n (Chair ) Michae l Starbir d 2000 Mathematics Subject Classification. Primar y 51M04 , 51M09 , 51M15 . Originally publishe d i n Germa n b y Friedr . Viewe g & Sohn Verlag , 65189 Wiesbaden, Germany , a s "Ilk a Agricol a un d Thoma s Friedrich : Elementargeometrie. 1 . Auflag e (1s t edition)" . © Friedr . Viewe g & Sohn Verlag/GW V Fachverlag e GmbH , Wiesbaden , 2005 Translated b y Phili p G . Spai n For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/stml-43 Library o f Congress Cataloging-in-Publicatio n Dat a Agricola, Ilka , 1973- [Elementargeometrie. English ] Elementary geometr y / Ilk a Agricola , Thoma s Friedrich . p. cm . — (Student mathematica l librar y ; v. 43) Includes bibliographica l reference s an d index. ISBN-13 : 978-0-8218-4347- 5 (alk . paper ) ISBN-10 : 0-8218-4347- 8 (alk . paper ) 1. Geometry. I . Friedrich, Thomas , 1949 - II . Title. QA453.A37 200 7 516—dc22 200706084 4 Copying an d reprinting. Individua l reader s o f this publication , an d nonprofi t libraries actin g fo r them, ar e permitted t o mak e fai r us e of the material, suc h a s to copy a chapter fo r us e in teaching o r research. Permissio n i s granted t o quote brie f passages fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the sourc e i s given. Republication, systemati c copying , o r multiple reproductio n o f any material i n this publication i s permitted onl y unde r licens e fro m th e American Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o the Acquisition s Department , American Mathematica l Society , 20 1 Charles Street , Providence , Rhod e Islan d 02904 - 2294, USA . Request s ca n also be made b y e-mail t o [email protected] . © 200 8 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l rights except thos e grante d to the United State s Government . Printed i n the United State s o f America. @ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guidelines established t o ensure permanenc e an d durability. Visit th e AMS home pag e at http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 0 8 Contents Preface t o th e Englis h Editio n v Preface t o th e Germa n Editio n vi i Chapter 1 . Introduction : Euclidea n spac e 1 Exercises 6 Chapter 2 . Elementar y geometrica l figures an d thei r propertie s 9 §2.1. Th e lin e 9 §2.2. Th e triangl e 1 9 §2.3. Th e circl e 4 5 §2.4. Th e coni c section s 6 3 §2.5. Surface s an d bodie s 7 7 Exercises 8 9 Chapter 3 . Symmetrie s o f the plan e an d o f spac e 9 9 §3.1. Affin e mapping s an d centroid s 9 9 §3.2. Projection s an d thei r propertie s 10 5 §3.3. Centra l dilation s an d translation s 10 8 §3.4. Plan e isometrie s an d similarit y transform s 11 4 §3.5. Comple x descriptio n o f plane transformation s 12 7 §3.6. Elementar y transformation s o f the spac e S 3 13 1 IV Contents §3.7. Discret e subgroup s o f the plan e transformatio n grou p 13 9 §3.8. Finit e subgroup s o f the spatia l transformatio n grou p 15 1 Exercises 15 6 Chapter 4 . Hyperboli c geometr y 16 7 §4.1. Th e axiomati c developmen t o f elementary geometr y 16 7 §4.2. Th e Poincar e mode l 17 4 §4.3. Th e dis c mode l 18 3 §4.4. Selecte d propertie s o f the hyperboli c plan e 18 5 §4.5. Thre e type s o f hyperboli c isometrie s 18 9 §4.6. Fuchsia n group s 19 4 Exercises 20 4 Chapter 5 . Spherica l geometr y 20 9 §5.1. Th e spac e § 2 20 9 §5.2. Grea t circle s i n S 2 21 1 §5.3. Th e isometr y grou p o f §2 21 5 §5.4. Th e Mobiu s grou p o f §2 21 6 §5.5. Selecte d topic s i n spherica l geometr y 21 8 Exercises 22 6 Bibliography 22 9 List o f Symbol s 23 5 Index 23 7 Preface t o the Englis h Edition We thank the American Mathematical Society , and Edward G . Dunn e in particular, fo r commissioning a translation o f our original book an d for hi s willingnes s t o prin t th e whol e boo k i n four-color . W e seize d the opportunity t o make infinitel y man y infinitesima l correction s tha t have bee n observe d sinc e th e first Germa n edition , al l o f whic h ar e not wort h listin g here . W e thank al l our colleague s an d student s wh o helped to identify them . Th e book i s intended fo r both undergraduat e mathematics students, to introduce them to an advanced point o f view on geometry, an d fo r mathematic s teachers , a s a reference an d sourc e book. This edition , lik e the original , ha s it s ow n homepage , http://www.ams.org/bookpages/stml-43 and an y furthe r correction s t o errors , mathematica l an d typographi - cal, will be posted there as they come to our attention. W e also inten d to presen t ther e additiona l materia l an d a collectio n o f relate d We b links tha t w e hop e th e reade r ma y find useful . W e wil l b e happ y t o receive and respon d t o any comments. I n particular, an y student wh o encounters difficultie s i n solving the Exercise s i s invited to outline th e problem t o u s b y e-mail . v VI Preface t o th e Englis h Editio n For teacher s i n school s an d universitie s w e hav e prepare d a smal l volume wit h hint s fo r solutions , availabl e fro m u s o n request . The Bibliograph y ha s bee n chose n t o signpos t othe r materia l whic h may b e helpfu l t o ou r readers , an d t o whe t thei r appetite s fo r geom - etry an d it s ramifications . I t i s not a lis t o f prerequisites . We thank ou r translator, Dr . Phili p G . Spain , fo r aidin g us i n makin g our boo k availabl e t o a wide r public . I n it s presen t for m th e boo k owes a lo t t o hi s expertise, an d w e are ver y muc h indebte d t o hi m fo r an exceptionall y pleasan t an d intensiv e collaboration . Berlin, Augus t 200 7 Ilka Agricol a Thomas Friedric h Acknowledgment The impression s o f th e ornament s o n page s 14 5 an d 16 5 ar e printe d from Owe n Jones , Grammatik der Ornamente (unchange d reprin t from th e Firs t Editio n o f 1856) , 1987 , b y graciou s permissio n o f Greno Verlag , Nordlingen , Germany . Th e origina l Englis h editio n appeared unde r th e titl e The Grammar of Ornament an d ha s ofte n been reprinted . Preface t o the Germa n Edition for Juliu s Geometry occupie s a n extensiv e par t o f th e mathematica l syllabu s in Germa n schools . I n middl e school , on e start s wit h propertie s o f the elementar y plan e figures (line , triangle, circle) , elementary trans - formations o f th e plane , an d surface s an d bodie s i n space . I n hig h school one comes to analytic geometry , trigonometry , advance d trans - formations, specia l curve s an d th e coni c sections . Element s o f non - Euclidean geometr y ca n b e covere d i n furthe r optiona l courses . Al - together w e hav e a broa d spectru m o f geometrica l theme s tha t th e mathematics teache r ca n presen t t o hi s pupils .