STUDENT MATHEMATICAL LIBRARY · Chapter 4. Hyperbolic geometry 167 §4.1. The axiomatic development of elementary geometry 167 §4.2. The Poincare model 174 §4.3. The disc model

STUDENT MATHEMATICAL LIBRARY Volume 43

Elementary Geometry Ilka Agricola Thomas Friedric h

Translated by Philip G . Spain

^ S ^ f c j

#AMS AMERICAN MATHEMATICA L SOCIET Y

Providence, Rhode Islan d

http://dx.doi.org/10.1090/stml/043

Editor ia l B o a r 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 & Soh n 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 & Soh n 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 /s tml-43

Library o f Congres s Cataloging-in-Publicatio n D a t a

Agricola, Ilka , 1973-[Elementargeometrie. English ] Elementary geometr y / Ilk a Agricola , Thoma s Friedrich .

p. cm . — (Studen t 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 chapte r fo r use in teachin g o r research . Permissio n i s granted t o quot e brie f passages fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t of the sourc e i s given.

Republication, systemati c copying , o r multiple reproductio n o f any materia l 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 th e 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 right s

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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 th e 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 plan e transformation s 12 7

§3.6. Elementar y transformation s o f th e spac e S 3 13 1

IV C o n t e n t s

§3.7. Discret e subgroup s o f th e plan e transformatio n grou p 13 9

§3.8. Finit e subgroup s o f th e 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 elementar y 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 th e 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 th e Englis h Edition

We than k th e America n Mathematica l Society , an d Edwar d G . Dunn e

in particular , fo r commissionin g a translatio n o f our origina l boo k 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 opportunit y t o mak e 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 than k al l ou r colleague s an d student s wh o

helped t o identif y them . Th e boo k i s intended fo r bot h undergraduat e

mathematics students , t o introduc e the m t o a n advance d poin t o f vie w

on geometry , an d fo r mathematic s teachers , a s a referenc e an d sourc e

book.

This edition , lik e th e original , ha s it s ow n homepage ,

h t tp : / /www.ams.org/bookpages/s tml-43

and an y furthe r correction s t o errors , mathematica l an d typographi -

cal, wil l be poste d ther e a s they com e t o ou r attention . W e als o 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 an d respon d t o an y comments . I n particular , an y studen t wh o

encounters difficultie s i n solvin g th e Exercise s i s invited t o outlin e th e

problem t o u s b y e-mail .

v

VI Preface t o t h e Engl i s h Ed i t i o 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 appeti te s fo r geom -

etry an d it s ramifications . I t i s no t a lis t o f prerequisites .

We than k ou r translator , Dr . Phili p G . Spain , fo r aidin g u s 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 ar e 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

A c k n o w l e d g m e n t

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 th e 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) , elementar y trans -

formations o f th e plane , an d surface s an d bodie s i n space . I n hig h

school on e come s t o analyti c 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 . Durin g th e stud y fo r

the teacher' s diplom a a t universit y th e syllabu s star t s wit h lecture s

on linea r algebr a an d analyti c geometr y i n th e first year , followe d b y

lectures o n elementar y geometr y i n the secon d year . Thi s i s to presen t

these geometrica l theme s t o th e prospectiv e teache r i n a mathemat -

ically systemati c form . I f on e consider s th e universit y educatio n i n

Germany ove r a longe r tim e span , i t i s eas y t o recogniz e tha t i n th e

lectures o n linea r algebr a th e geometri c theme s ar e reduce d ste p b y

step, ofte n almos t completel y maske d out .

vii

V l l l Preface t o t h e G e r m a n Ed i t i o n

In all , on e gain s th e impressio n tha t th e cours e o n elementar y ge -

ometry, wit h it s clearl y define d contents , form s th e mai n par t o f th e

geometric educatio n fo r teacher s i n training .

This boo k aros e fro m a one-semeste r lectur e cours e o n "elementar y

geometry" fo r futur e teacher s i n thei r secon d yea r o f stud y a t th e

Humboldt-Universitat i n Berlin . Th e student s ha d alread y at tende d

the first-year course s o n linea r algebr a an d calculus ; i n th e first chap -

ter w e presen t a summar y o f som e aspect s o f thes e lectures . Ou r

t reatment o f elementar y geometr y assume s thi s fundamenta l knowl -

edge, althoug h i n a larg e par t o f th e tex t the y wil l hardl y b e needed .

Accordingly, thi s tex t i s intende d a s a companio n boo k t o suc h a

course an d seminars . Further , w e hop e tha t th e boo k wil l b e use d b y

working teacher s a s a compendiu m o f th e curriculu m fo r elementar y

geometry. Selecte d part s o f th e tex t ar e als o suitabl e fo r goo d hig h

school pupil s an d ideall y migh t b e use d a s a foundatio n fo r indepen -

dent stud y o r projects .

Chapter 2 i s devote d t o th e elementar y geometri c figures an d thei r

properties. W e begi n wit h th e incidenc e theorem s fo r line s an d the n

tu rn ou r at tentio n t o th e triangle . Afte r th e congruenc e an d similar -

ity theorems , w e appl y i n particula r th e theorem s o f Menelau s an d

Ceva i n orde r t o trea t th e intersectio n point s o f th e specia l line s i n a

triangle. Further , w e discus s th e incircle , circumcircl e an d excircl e o f

the triangle , it s area , an d it s relatio n t o th e radi i o f th e circle . W e

treat th e circl e similarly , an d discus s i n particula r th e Feuerbac h cir -

cle, an d th e Simso n an d Steine r lines . Wi t h a vie w t o th e underlyin g

hyperbolic geometr y i n Chapte r 4 w e alread y introduc e a sectio n o n

inversion i n th e circl e here . Th e coni c section s follow , wit h th e deriva -

tion o f thei r genera l equation , thei r eccentricit y an d parameters , a s

well a s th e determinatio n o f th e focu s an d directrix . Som e strikin g

properties o f th e coni c section s ar e prove d directl y i n th e text ; th e

reader wil l find som e othe r propertie s i n th e Exercise s a t th e en d o f

Chapter 2 . The n w e tur n t o surface s an d bodie s i n space . W e deriv e

the formula e fo r th e surfac e are a o f a surfac e o f revolutio n an d als o

the formul a fo r th e volum e o f a bod y o f revolution : w e prov e Euler' s

polyhedron theore m (fo r conve x polyhedra ) an d finish Chapte r 2 wit h

the classificatio n o f th e Platoni c bodies .

Preface t o th e Germa n Editio n IX

Chapter 3 deals with the symmetrie s o f Euclidean space . W e describe afEne mapping s briefly , als o th e linea r mapping s correspondin g t o them, an d th e centroi d o f a finite weighte d poin t system . Paralle l projections ont o a plane along a line and ont o a line along a plane ar e the first example s o f affine mappings . The n w e treat centra l dilation s and translation s exhaustively . Firs t w e characteriz e the m throug h a commo n geometri c propert y an d deduc e tha t togethe r the y for m a nonabelia n grou p o f transformation s o f spac e t o itself . Then , fo r the plane , we determine i n detail their compositions an d discus s as an application th e dilation centers of two circles, with whose help one can construct th e common tangent s t o two circles. Nex t follow s th e stud y of isometrie s o f th e plane . Firs t com e example s o f axi s reflections , translations an d rotations , an d agai n w e stud y thei r compositions . Fixed point s ar e important : A n isometr y o f th e plan e wit h thre e noncollinear fixed point s i s the identity . Analogousl y w e characteriz e all isometrie s wit h exactl y tw o fixed points , wit h on e fixed point , and als o th e fixed poin t fre e isometries . Th e grou p generate d b y al l isometries an d centra l dilation s consist s o f th e similarit y transform s of the plane . I n a simila r wa y w e treat th e transformation s o f three -dimensional space . Firs t w e stud y th e compositio n o f distinc t suc h mappings and then turn again to the description of the fixed point set s of spatia l isometries . Thes e fixed poin t set s yiel d a classificatio n o f the isometrie s o f £3 . Th e las t tw o sections of this chapter ar e devote d to th e stud y o f th e discret e isometr y group s o f Euclidea n space . I n the cas e of the plan e w e treat th e cycli c rotation groups , th e dihedra l group an d lattice . W e deduc e a necessar y conditio n fo r th e poin t group o f a discret e isometr y grou p o f th e plan e an d finally obtai n a classification o f al l th e group s i n question . I n th e cas e o f spac e w e restrict ourselve s t o classifyin g th e finite isometr y groups . Thes e ar e the invarianc e group s o f th e Platoni c bodie s an d o f th e symmetr y group o f a pyramid o r o f a cylinder wit h regula r polygona l base . Th e tetrahedron group , th e cub e group , an d als o the dodecahedro n grou p are describe d completely .

We begin Chapte r 4 with th e axiomatic s o f elementary geometr y an d the significance o f the paralle l axiom . W e construct hyperboli c geom -etry i n the upper hal f plane , i n which lines are Euclidean circula r arc s or hal f lines . W e treat variou s expressions fo r th e hyperboli c distanc e

X Preface t o t h e G e r m a n E d i t i o n

of tw o points . Thi s ca n a s wel l b e represente d b y a cros s rati o a s b y

a direc t formula . I n particular , th e triangl e inequalit y holds , an d th e

hyperbolic plan e i s a metri c space . W e determin e it s isometr y grou p

and deriv e th e formula e fo r th e hyperboli c lengt h o f a curv e an d fo r

the hyperboli c are a o f a region . B y mean s o f th e Cayle y transfor m

we pas s t o th e dis c mode l o f hyperboli c geometry . W e the n trea t

selected propertie s o f geometri c figures i n th e hyperboli c plane . W e

compute th e perimete r o f a circle , it s hyperboli c area , an d deriv e th e

hyperbolic Pythagora s theore m a s wel l a s othe r formula e fro m trig -

onometry. Th e formul a fo r th e are a o f a triangl e an d it s angl e defec t

is prove d completely . I n th e Exercise s th e reade r wil l find a numbe r

of result s i n hyperboli c elementar y geometr y tha t ar e analogou s t o

those o f Euclidea n geometry . Thes e concer n pair s o f hyperboli c lines ,

triangles an d thei r notabl e points , th e incircl e an d circumcircl e o f a

triangle, an d als o th e horocycle . I n a furthe r sectio n w e presen t th e

classification o f th e isometrie s int o elliptic , paraboli c an d hyperboli c

transformations bot h b y mean s o f th e Jorda n norma l for m an d als o

through thei r fixed poin t sets . W e stud y i n detai l th e questio n o f

the typ e o f th e commutato r o f tw o isometries . Th e las t sectio n o f

this chapte r i s devote d t o Fuchsia n groups . Her e w e ar e dealin g wit h

discrete subgroup s o f th e isometr y grou p o f th e hyperboli c plane . A s

well a s a serie s o f examples o f such group s w e introduce thei r limi t set s

and prov e tha t thes e set s hav e eithe r 0 , 1 , 2 o r infinitel y man y points .

Fuchsian group s wit h n o mor e tha n tw o limi t point s ar e calle d ele -

mentary. W e classif y al l elementar y Fuchsia n groups .

Spherical geometr y i s t reate d i n th e las t chapte r i n imitatio n o f hy -

perbolic geometry . W e conside r th e se t o f al l point s o f th e two -

dimensional spher e § 2 . Th e grea t circle s pla y th e rol e o f spherica l

lines an d realiz e th e shortes t distanc e betwee n tw o point s i n spher -

ical space . W e determin e th e isometr y grou p an d als o th e grou p o f

all conforma l mapping s completely . T o conclud e w e prov e th e mos t

important formula e o f spherica l trigonometr y an d stud y th e pola r tri -

angle associate d t o eac h spherica l triangle . Fro m thi s w e obtai n th e

formulae fo r th e area s o f spherica l lune s an d triangle s an d variou s

inequalities betwee n th e sid e length s an d angles .

Preface t o t h e G e r m a n Edi t io n XI

At th e en d o f eac h chapte r th e reade r wil l find a selectio n o f Exer -

cises tha t hav e regularl y bee n assigne d t o ou r auditor s a s homework .

Any studen t wh o encounter s difficultie s i n solvin g thes e Exercise s

is warml y invite d t o outlin e hi s proble m t o u s b y e-mail . W e wil l

endeavor t o help . Fo r teacher s i n school s an d universitie s w e hav e

prepared a smal l volum e wit h hint s fo r solutions , whic h i s availabl e

from u s o n request . Moreover , th e Germa n editio n o f th e boo k ha s

its ow n Interne t page ,

http:/ /www-irm.mathematik.hu-berl in.de/^agricola/elemgeo.html

One wil l find ther e a lis t o f al l know n typographica l errors , an d pdf -

files o f al l th e page s o n whic h picture s appea r tha t ar e multicolore d

in th e origina l bu t ar e printe d her e i n black-and-whit e o n ground s o f

cost. Ther e i s als o a collectio n o f www-link s o n elementar y geometry ,

though i t make s n o clai m t o completeness .

We than k th e participant s i n ou r seminar s fo r numerou s suggestion s

tha t hav e le d t o extendin g an d improvin g th e text . Dr . sc. Huber t

Gollek an d Dr . Chris t of Puhl e hav e rea d throug h th e whol e man -

uscript an d hav e indicate d necessar y correction s i n man y chapters .

Not las t w e than k Fra u Schmickler-Hirzebruc h o f Viewe g Verla g fo r

her willingnes s t o prin t som e page s o f thi s boo k i n two-tone . W e ar e

aware tha t thi s i s a rar e (i f als o muc h desired ) privilege . W e hop e tha t

this wil l no t remai n a n isolate d cas e i n th e mathematica l li teratur e

and tha t th e reade r wil l appreciat e an d enjo y thi s enrichmen t o f th e

text, whic h wa s no t t o b e take n fo r granted .

Berlin, Decembe r 200 4

Ilka Agricol a

Thomas Friedric h

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Bibliography

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I. Agricola , T . Friedrich , Global Analysis, Graduat e Studie s i n Math -ematics, vol . 52 , America n Mathematica l Society , Providence , RI , 2002.

Anthony W. Knapp , Basic Real Analysis, Cornerstones , Birkhause r Boston, Inc. , Boston , MA , 2005 .

Serge Lang , Undergraduate Analysis, Springe r Verlag , Ne w York , 1997.

John McCleary , Geometry from a Differentiable Viewpoint, Cam -bridge Universit y Press , Cambridge , 1994 .

Frank Morgan , Real Analysis, America n Mathematica l Society , Prov -idence, RI , 2005 .

M. Helena Noronha , Euclidean and Non-Euclidean Geometries, Pren -tice Hall , Inc. , Ne w Jersey , 2002 .

MurrayH. Protter , CharlesB . Morrey , Jr. , A First Course in Real Analysis, 2nd Edition, Springe r Verlag , Ne w York, 1997 .

Walter Rudin , Principles of Mathematical Analysis, Third Edition, International Serie s i n Pur e an d Applie d Mathematics , McGraw-Hil l Book Co. , New York-Auckland-Diisseldorf , 1976 .

229

230 Bibliography


N. Altshiller-Court , College Geometry: An Introduction to the Mod-ern Geometry of the Triangle and the Circle (2n d edition) , Barne s & Noble, Ne w York , 1952 .

H. F. Baker , An Introduction to Plane Geometry, with Many Exam-ples, Reprin t o f 194 3 firs t edition , Chelse a Publishin g Co. , Bronx , NY, 1971.

Allan Berele , Jerr y Goldman , Geometry: Theorems and Construc-tions, Prentic e Hall , Inc. , Ne w Jersey , 2001.

George Davi d Birkhoff , Ralp h Beatley , Basic Geometry: Third Edi-tion, AM S Chelse a Publications , America n Mathematica l Society , Providence, RI , 1959 .

H. S.M. Coxeter , S.L . Greitzer , Geometry Revisited, Th e Mathemat -ical Associatio n o f America , Washington , DC , 1967 .

Robin Hartshorne , Geometry: Euclid and Beyond, Undergraduat e Texts i n Mathematics , Springe r Verlag , Ne w York , 2000 .

H. Knorrer, Geometric, Viewe g Verlag, Braunschweig and Wiesbaden , 1996.

Sebastian Montie l an d Antoni o Ros , Curves and Surfaces, Graduat e Studies i n Mathematics , vol . 69 , America n Mathematica l Society , Providence, RI , an d Rea l Socieda d Matematic a Espahola , Madrid , 2005.

S. Muller-Philipp , H.-J . Gorski , Leitfaden Geometric, Viewe g Verlag , Braunschweig an d Wiesbaden , 2004 .

H. Scheid , Elemente der Geometric, Spektru m Akademische r Verlag , Heidelberg-Berlin, 2001 .

H. Weyl, Symmetry, Princeto n Universit y Press , Princeton , NJ , 1952 .

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M. A. Armstrong, Groups and Symmetry, Springe r Verlag, Berlin-Hei -delberg-New York , 1988 .

Michael Artin , Algebra, Prentice-Hall , Englewoo d Cliffs , NJ , 1991 .

Gustave Choquet , Geometry in a Modern Setting, Herman n Publish -ers i n Art s an d Science , Paris , 1969 .

David W. Farmer , Groups and Symmetry: A Guide to Discovering Mathematics, Mathematica l World , vol . 5 , America n Mathematica l Society, Providence , RI , 1996 .

M. Klemm , Symmetrien von Ornamenten und Kristallen, Springe r Verlag, Berlin-Heidelberg-Ne w York , 1982 .

H. Scheid , Elemente der Geometric, Spektru m Akademische r Verlag , Heidelberg-Berlin, 2001 .


R. Baldus , Nichteuklidische Geometric, Sammlun g Goschen , Ban d 970 (2 . Aufl.) , Walte r d e Gruyte r & Co. , Berlin , 1944 .

A. F. Beardon , The Geometry of Discrete Groups, Springe r Verlag , Berlin-Heidelberg-New York , 1983 .


H. S. M. Coxeter , Non-Euclidean Geometry, 6t h Edition , Mathemati -cal Associatio n o f America , Washington , DC , 1998 .

R. Fricke , F . Klein , Vorlesungen iiber die Theorie automorpher Funk-tionen, Teil I und Teil II, Teubne r Verlag , Leipzig , 189 7 and 1912 .

D. Hilbert , Foundations of Geometry, 2n d edition , Ope n Cour t Pub -lishing Company , L a Salle , IL , 1971.

232 Bibliography

D. Hilbert , S . Cohn-Vossen , Geometry and the Imagination, AM S Chelsea Publishing , America n Mathematica l Society , Providence , RI , (reprinted) 1999 .

W. Klingenberg , Grundlagen der Geometrie, Bibliographische s Insti -tut AG , Mannheim , 1971 .

J. Lehner , Discontinous Groups and Automorphic Functions, Ameri -can Mathematica l Society , Providence , RI , 1964 .

George E. Martin, The Foundations of Geometry and the Non-Euclid-ean Plane, correcte d fourt h printing , Undergraduat e Text s i n Math -ematics, Springe r Verlag , Berlin-Heidelberg-Ne w York , 1998 .

D. Mumford , C . Series , D . Wright , Indra ps Pearls - The Vision of Felix Klein, Cambridg e Universit y Press , Ne w York , 2002 .


J. G. Ratcliffe , Foundations of Hyperbolic Manifolds, Springe r Verlag , Berlin-Heidelberg-New York , 1994 .



D.A. Brannan , M . F. Esplen , J .J . Gray , Geometry, Cambridg e Uni -versity Press , Cambridge , 1999 .

Julian Lowel l Coolidge, A Treatise on the Circle and the Sphere, AM S Chelsea Publishing , America n Mathematica l Society , Providence , RI , 2004.

R. Fenn , Geometry, Springe r Undergraduat e Mathematic s Series , Springer Verlag , Berlin-Heidelberg-Ne w York , 2001.

Marvin Ja y Greenberg , Euclidean and Non-Euclidean Geometry: De-velopment and History, W . H . Freeman , Ne w York, 1993 .

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G. A. Jennings , Modern Geometry with Applications, Springe r Uni -versitext, Springe r Verlag , Berlin-Heidelberg-Ne w York , 1994 .


B. A. Rosenfeld , Die Grundbegriffe der sphdrischen Geometrie und Trigonometric, i n "Enzyklopadi e der Elementarmathematik, Ban d IV: Geometrie", VEB Deutscher Verla g der Wissenschaften, Berlin , 1969 .

Further readin g

Arkady L. Onishchik, Rol f Sulanke , Projective and Cayley-Klein Geo-metries, Springe r Monograph s i n Mathematics , Springe r Verlag , Ber -lin, 2006 .

M. Berger, Geometry I, Springe r Verlag, Berlin-Heidelberg-New York , 1994; Geometry II, Springe r Verlag , Berlin-Heidelberg-Ne w York , 1996.

F. Klein , Lectures on the Icosahedron and the Solution of the Fifth Degree, 2nd revise d edition , Dove r Publications , Ne w York , 2003 .

C. J. Scriba , P . Schreiber , 5000 Jahre Geometrie, Springe r Verlag , Berlin-Heidelberg-New York , 2003 .

Marvin Ja y Greenberg , Euclidean and Non-Euclidean Geometry: De-velopment and History, W . H . Freeman , Ne w York , 1993 .

Audim Holme , Geometry: Our Cultural Heritage, Springe r Verlag , Berlin-Heidelberg-New York , 2002 .


List o f Symbol s

A4 - tetrahedro n group , 15 1

A5 - dodecahedro n group , 15 3

\AB\ - lengt h o f a segment , 1

AB : BC - divisio n ratio , 1 0

a,b,c - triangl e sides , 1 9

a,/3,7 - triangl e angles , 1 9

a x b - vecto r product , 5

B - grou p o f transformations , 108

CS(ri, r2 , h) - frustu m o f a cone, 79

C(Z1r) - circle , 4 5

C - extende d complex numbers, 59

Cn - cycli c group , 14 0

Conf0(S2) - Mobiu s group, 217

D2, 18 3

Dn - dihedra l group , 14 0

A(A,B,C) -triangle , 1 9

£n - affln e space , 1

F i ,F 2 -foci , 6 7

F(C) - surfac e o f revolution, 7 7

Fix(7) - fixed poin t set , 19 0

G - centroid , 10 3

[7,71] _ commutator , 19 1

H - hal f line , 11 3

MS2 - hemisphere , 22 4

H2 - hyperboli c plane , 17 4

H2 - extende d hyperbolic plane, 189

fyft,fc) ~ central dilation , 10 8

235

d(A, B) - distanc e between points, 1

236 List o f Symbol s

X - grou p o f isometries , 11 4

X+ - positiv e isometries , 12 3

X~ - negativ e isometries , 12 3

XQ - isometrie s with fixed point, 116

XQ - positiv e isometrie s wit h fixed point , 12 0

XQ - negativ e isometrie s wit h fixed point , 12 0

K(C) - bod y o f revolution , 7 7

L(A, B) - lin e through A and B, 6

L(C) - curv e length , 3

L(f) - vectoria l mapping , 10 1

A(r) - limi t set , 19 6

m(A) - area , 4

^A - poin t masses , 10 1

Afn - pol e o f grea t circle , 21 2

0(V(£n)) - orthogona l group , 115

Q ~ center o f dilation , 10 8

Pc - centroid , 3 8

Pcc - circumcenter , 3 8

Pic - incenter , 3 5

Poc - orthocenter , 3 8

7r, II - plane , 63 , 105

II(r) - paraboli c points , 19 6

PSL(2,C), 6 0

PSL(2,R), 17 7

PSL(2,Z), 19 4

R - extende d rea l line , 17 6

Rn - Euclidea n space , 1 r(Q,fc) ~ rotation, 11 8

S - similarit y transformatio n group, 12 4

<S+ - positiv e similarities , 12 4

S~ - negativ e similarities , 12 4

£4 - octahedro n group , 15 2

8L - reflectio n i n line , 11 7

sn - reflectio n i n plane , 13 2

Sc - circl e reflection , 5 9

SL(2,C), 6 0

SL(2,R), 17 7

§2 - spherica l space , 20 9

T - translatio n group , 11 5

t$ - translation , 11 0

vol(A) - volume , 4

V(Sn) - Euclidea n vector space, 1

ipA - exterio r angl e in triangle , 19

(zi : Z2 ; Z3 : Z4) - cros s ratio , 62

Index

affine mapping , 9 9 alternate angle , 2 2 altitude, 19 , 3 3

common, 20 5 in hyperboli c triangle , 20 6 theorem, 2 5

angle central, 5 1 inscribed, 5 1 peripheral, 5 1

angle bisectio n theore m fo r ellipse , 71

angle bisector s in Euclidea n triangle , 3 5 in hyperboli c triangle , 20 6

angle cosin e theorem , 22 0 angle su m

in Euclidea n triangle , 2 1 in hyperboli c triangle , 18 8 in spherica l triangle , 22 4

antipodal points , 21 1 antiprism, 8 7 Archimedean bodies , 8 7 area

Euclidean triangle , 4 0 hyperbolic, 18 2 hyperbolic triangle , 18 8

ASA theorem , 2 6 asymptotes o f th e hyperbola , 7 2 asymptotic triangle , 18 7

autopolar triangle , 22 6 axis reflection , 11 7

barycenter, 10 3 Beltrami, E . (1835-1899) , 17 3 Berger, Marce l (1927 - ) , 8 6 betweenness relation , 16 8 body o f revolution , 7 7

volume of , 7 8 Bolyai, J . (1802-1860) , 17 2 boundary circle , 18 7 boundary paralle l lines , 20 4

Caspar, Donal d L . D. (1927 - ) , 16 4 Cassini curve , 7 5 cathetus, 2 4

theorem, 2 4 Cavalieri's principle , 4 , 79 , 81 , 82 Cay ley transform , 18 3 central angle , 5 1 central dilation , 10 8 centroid, 34 , 38 , 91 , 101 , 10 4

in hyperboli c triangle , 20 6 weighted, 102 , 11 5

Ceva's theorem , 30 , 32 , 9 2 chord o f a circle , 4 6 chord theorem , 4 8 chordal quadrilateral , 5 6 circle, 4 5

chord theorem , 4 8

237

238 Index

Feuerbach, 40 , 5 3 generalized, 57 , 17 5 hyperbolic, 18 5 inversion in , 5 8 reflection in , 5 8 secant theorem , 4 7 spherical, 21 8 tangent theorem , 4 6

circular cone , 7 8 circumcenter

of Euclidea n triangle , 3 6 circumcircle

of Euclidea n triangle , 36 , 4 3 of hyperboli c triangle , 20 6

collinear points , 6 commutator, 19 1 complex numbers , 5 cone, 8 1

volume, 8 1 conformal

group, 21 7 mapping, 60 , 11 4

congruence theorems , 26 , 22 6 congruent sets , 3 conic section , 6 3

directrix, 6 7 eccentricity, 6 5 focus, 6 7 parameter, 6 5 polar equation , 6 5

conjugate matrices , 18 9 coordinates

polar, 6 3 spherical, 21 0

cosine la w Euclidean, 2 0 hyperbolic, 20 5 spherical, 22 0

cotangent formula , 22 1 cross ratio , 61 , 17 6 cube (hexahedron) , 86 , 15 2 cube group , 15 6 cuboctahedron, 88 , 9 7

large, 8 8 cyclic group , 140 , 15 6 cylinder, 8 2

volume of , 8 2

Delambre's equations , 22 6 Desargues' theorem , 1 8 diametrically opposit e points , 21 1 dihedral group , 140 , 15 6 dilation cente r (o f tw o circles) , 11 2 dilation factor , 10 8 direction vector , 6 directrix, 6 7 disc model , 18 3 distance

Euclidean, 1 hyperbolic, 17 6 in geometri c plane , 17 0 spherical, 21 4

divergent lines , 20 5 division ratio , 1 0 dodecahedron, 86 , 15 3

group, 153 , 15 6 truncated, 8 7

eccentricity, 6 5 Einstein, Alber t (1879-1955) , 17 3 ellipse, 6 6

angle bisectio n theorem , 7 1 length, 7 1 surface, 7 1

ellipsoid, 7 8 elliptic

isometry group , 21 5 line, 21 1 space, 20 9 transformation, 19 0

Euclidean cosine law , 2 0 distance, 1 isometry group , 2 length, 3 line, 6 plane, 5 sine law , 4 0 space, 1 triangle, 1 9

angle bisectors , 3 5 area, 4 0 circumcenter, 3 6 circumcircle, 4 3 excircle, 4 4 incenter, 3 6

Index 239

incircle, 36 , 4 3 Mollweide's equations , 9 1 Napier's equations , 9 1 orthocenter, 3 3 perpendicular bisectors , 3 6 side bisectors , 3 4

Euler line , 3 8 Euler's polyhedro n formula , 8 2 exact sequence , 11 6 excircle o f Euclidea n triangle , 4 4 extended

complex plane , 17 6 complex plane , 59 , 21 6 real line , 17 6

exterior angle , 2 0

Fermat problem , 2 4 Feuerbach circle , 40 , 5 3 first Steine r line , 5 5 five elemen t formula , 22 0 fixed poin t

of hyperboli c isometry , 19 0 focus, 6 7 focus-directrix pair , 6 7 football, 8 7 fractional linea r transformation , 6 0 frieze, 14 4

group, 146 , 16 4 Fuchsian group , 19 4

elementary, 20 1 first kind , 20 4 nonelementary, 20 1 second kind , 20 4

Fuller, Buckminste r (1895-1983) , 95

function (fractiona l linear) , 6 0

Gauss, C.F . (1777-1855) , 17 2 generalized circle , 57 , 17 5 geometric plane , 16 8 glide reflection , 11 8 golden

ratio, 5 0 rectangle, 89 , 15 4 section, 49 , 89 , 15 4

great circle , 21 1 pole, 21 2

group

cyclic, 140 , 15 6 dihedral, 140 , 15 6 dodecahedron, 153 , 15 6 Fuchsian, 19 4

elementary, 20 1 first kind , 20 4 nonelementary, 20 1 second kind , 20 4

octahedron, 152 , 15 6 orthogonal, 11 5 tetrahedron, 151 , 15 6

Haeckel, Erns t (1834-1919) , 8 6 half space , 8 2 hemisphere, 21 2 Heron's formula , 4 1 hexagonal lattice , 14 6 hexahedron (cube) , 86 , 15 2

snub, 8 8 truncated, 8 7

Hippocrates, lune s of , 9 0 homothety, 10 8 horocycle, 20 6 hyperbola, 6 6

asymptotes, 7 2 hyperbolic

area, 18 2 boundary circle , 18 7 circle, 18 5 cosine law , 20 5 distance, 17 6 isometry group , 18 0 length, 18 1 line, 17 5 lines

boundary parallel , 20 4 divergent, 20 5

plane, 17 4 disc model , 18 3 Poincare model , 17 4

sine law , 20 6 transformation, 19 0 triangle

altitude, 20 6 angle bisectors , 20 6 area, 18 8 asymptotic, 18 7 circumcircle, 20 6

240 Index

incircle, 20 6 orthocenter, 20 6 perpendicular bisector , 20 6 side bisector , 20 6

hyperboloid one-sheeted, 7 8 two-sheeted, 7 8

icosadeltahedron, 95 , 16 3 icosahedron, 86 , 15 4

truncated, 8 7 icosidodecahedron, 8 8

large, 8 8 incenter, 35 , 3 6 incidence theorem , 9

converse, 1 3 in space , 15 , 10 7 oriented, 11 , 31

incircle of Euclidea n triangle , 36 , 4 3 of hyperboli c triangle , 20 6

inscribed angle , 5 1 inversion i n circle , 5 8 isobarycenter, 10 3 isoceles triangle , 2 0 isometry, 1 , 114 , 17 1

group of ellipti c space , 21 5 of Euclidea n space , 2 of hyperboli c plane , 18 0

negatively/positively oriented , 123

isoperimetric problem , 4 2

Jordan measure , 4 Jordan norma l form , 19 0

Klein, Feli x (1849-1925) , 144 , 17 3 Klein fou r group , 14 4 Klug, Aaro n (1926 - ) , 16 4

Lambert projection , 21 0 lattice, 14 6 leg, 2 4 lemniscate, 7 6 length

Euclidean, 3 hyperbolic, 18 1 spherical, 21 9

lever law , 10 1 l'Huilier's equation , 22 7 limit circle , 20 6 limit set , 19 6 line

elliptic, 21 1 Euclidean, 6 Euler, 3 8 first Steiner , 5 5 hyperbolic, 17 5 second Steiner , 55 , 126 , 15 9 Simson, 54 , 126 , 15 9

lines hyperbolic

boundary parallel , 20 4 divergent, 20 5

parallel, 6 , 16 9 skew, 6

Lobachevsky, N.I . (1793-1856) , 17 2 lune, 21 2

of Hippocrates , 9 0

Mobius group , 21 7 mapping

affine, 9 9 angle-preserving ( = conformal) ,

60, 11 4 orientation-preserving/reversing,

123 vectorial, 10 1

mass, 10 2 median theorem , 34 , 10 4 Menelaus' theorem , 3 0 midtriangle, 3 8 modular group , 19 4 Mollweide's equations , 9 1 Morley's theorem , 9 1

Nagel point , 9 2 Napier's equation s

for Euclidea n triangle , 9 1 for spherica l triangle , 22 7

Napoleon's theorem , 13 0 nine poin t circle , 4 0

octahedron, 86 , 15 2 group, 152 , 15 6 truncated, 8 7

octant, 21 9

Index 241

one-sheeted hyperboloid , 7 8 order axioms , 16 8 orientation-preserving/reversing

mapping, 12 3 oriented

angle, 11 7 incidence theorem , 11 , 31

ornament group , 14 6 orthocenter, 9 1

in Euclidea n triangle , 3 3 in hyperboli c triangle , 20 6

orthogonal group , 11 5 orthogonal projection , 10 5

Pappus' theorem , 1 3 parabola, 6 6 parabolic

mirror, 9 4 point, 19 6 transformation, 19 0

parallel axiom , 17 2 parallelepiped, 5 parameter o f coni c section , 6 5 Pasch's Axiom , 16 9 pentagon, 50 , 16 1 perfect set , 20 4 peripheral angle , 5 1 perpendicular bisecto r

in Euclidea n triangle , 3 6 in hyperboli c triangle , 20 6 planes, 13 2

picornavirus, 16 3 plane

Euclidean, 5 geometric, 16 8 hyperbolic, 17 4

disc model , 18 3 Poincare model , 17 4

reflection, 13 2 Platonic body , 8 5 Poincare, Henr i (1854-1912) , 17 3 Poincare model , 17 4 point

group, 14 4 projection, 10 5 reflection, 108 , 13 6 space, 1 symmetry group , 14 4

polar, 21 2 coordinates, 63 , 18 5 equation o f coni c section , 6 5 triangle, 22 2

pole, 21 2 polyhedron, 8 2

dual, 9 6 polytope, 8 2

regular, quasiregular , 8 5 primitive pythagorea n triple , 2 3 principal congruenc e group , 19 4 prism, 8 7 problem

Fermat, 2 4 isoperimetric, 4 2

projection, 10 5 Lambert, 21 0 stereographic, 21 0

Ptolemy's theorem , 6 2 Pythagoras' theore m

Euclidean, 2 2 hyperbolic, 18 6 spherical, 22 1

Pythagoras triple , 2 3

quadratic lattice , 14 6

Radiolaria, 8 6 rectangular fac e centere d lattice ,

146 rectangular lattice , 14 6 reflection

glide, 11 8 in circle , 5 8 in grea t circle , 21 6 in hyperboli c line , 18 0 in plane , 13 2 in point , 108 , 136 , 21 6

relation (betweenness) , 16 8 revolution

ellipsoid of , 7 8 hyperboloid of , 7 8

rhombic lattice , 14 6 Riemann, Bernhar d (1826-1866) ,

173 rotation, 13 5

half, 13 3 rotation-dilation, 12 4

242 Index

SAS theorem , 2 6 secant o f circle , 4 5 secant theorem , 4 7 second Steine r line , 55 , 126 , 15 9 segment (i n absolut e geometry) ,

169 semiperimeter, 4 1 semiregular polyhedron , 8 7 sets

congruent, 3 , 17 1 Jordan measurable , 4 perfect, 20 4 similar, 3

side bisecto r in Euclidea n triangle , 34 , 10 4 in hyperboli c triangle , 20 6

side cosin e law , 22 0 similar sets , 3 similarity transform , 12 4 Simson line , 5 4 sine law , 4 0

hyperbolic, 20 6 spherical, 22 0

skew lattice , 14 6 skew reflection , 11 4 snub

hexahedron, 8 8 space, elliptic , 20 9 spherical

angle cosin e theorem , 22 0 circle, 21 8 coordinates, 21 0 cosine law , 22 0 cotangent formula , 22 1 distance, 21 4 excess, 22 6 five elemen t formula , 22 0 lune, 21 2 octant, 21 9 polar triangle , 22 2 sine law , 22 0 triangle, 22 0

area, 22 4 autopolar, 22 6 Delambre's equations , 22 6 l'Huilier's equation , 22 7 Napier's equations , 22 7

trigonometry, 22 0

star polygon , 9 6 Steiner, Jaco b (1796-1863) , 12 6 Steiner lin e

first, 5 5 second, 55 , 126 , 15 9

stereographic projection , 21 0 supplementary angle , 21 3 surface o f revolution , 7 7

surface are a of , 8 0 Sylvester an d Galla i theorem , 9 0 symmetry point , 20 5

tangent quadrilateral, 5 6 theorem, 4 6 to circle , 4 5 to tw o circles , 11 3

tetrahedron, 86 , 104 , 151 , 15 7 group, 151 , 15 6 truncated, 8 7

Thales' theorem , 15 , 50 , 10 7 theorem

alternate angle , 2 2 angle bisectors , 3 5 angle cosine , 22 0 central angle , 5 1 Ceva, 30 , 32 , 9 2 congruence, 22 6 Desargues, 1 8 ellipse

angle bisection , 7 1 exterior angle , 2 0 incidence, 9

in space , 15 , 10 7 median, 34 , 10 4 Menelaus, 3 0 Morley, 9 1 Napoleon, 13 0 oriented incidence , 1 1 Pappus, 1 3 Ptolemy, 6 2 Pythagoras

Euclidean, 2 2 hyperbolic, 18 6 spherical, 22 1

side cosine , 22 0 spherical

angle cosine , 22 0

Index 243

side cosine , 22 0 Sylvester an d Gallai , 9 0 Thales, 15 , 5 0

tiling, 14 4 torus, 7 8 transformation, 10 8

complex, 12 8 elliptic, 19 0 hyperbolic, 19 0 parabolic, 19 0

translation, 1 , 10 8 domain, 14 4 group,115 subgroup, 14 4

transversal, 3 0 vertex, 3 2

triangle asymptotic, 18 7 congruence theorems , 2 6 Euclidean, 1 9

angle bisectors , 3 5 angle sum , 2 1 area, 4 0 centroid, 3 8 circumcenter, 3 6 circumcircle, 4 3 excircle, 4 4 incenter, 35 , 3 6 incircle, 36 , 4 3 Mollweide's equations , 9 1 Napier's equations , 9 1 orthocenter, 3 3 perpendicular bisectors , 3 6 side bisectors , 3 4

hyperbolic angle bisectors , 20 6 area, 18 8 circumcircle, 20 6 incircle, 20 6 orthocenter, 20 6 perpendicular bisector , 20 6 side bisector , 20 6

isoceles, 2 0 leg, 2 0 right-angled

altitude theorem , 2 5 cathetus theorem , 2 4 Pythagoras' theorem , 2 2

semiperimeter, 4 1 similarity theorem , 3 0 spherical, 22 0

area, 22 4 autopolar, 22 6 Delambre's equations , 22 6 excess, 22 6 l'Huilier's equation , 22 7 Napier's equations , 22 7 polar, 22 2 surface, 22 0

transversal, 3 0 trigonometry (spherical) , 22 0 truncated

dodecahedron, 8 7 hexahedron, 8 7 icosahedron, 8 7 octahedron, 8 7 tetrahedron, 8 7

two-sheeted hyperboloid , 7 8

vector mapping, 10 1 product, 5 projection, 10 5 triple product , 5

vectorial isometry, 11 5 mapping, 10 1

vertex angle, 21 3 transversal, 3 2

weighted centroid , 102 , 11 5 Weyl, Herman n (1885-1955) , 8 6 Wiles, Andre w (1953 - ) , 2 4

Documents

STUDENT MATHEMATICAL LIBRARY · Chapter 4. Hyperbolic geometry 167 §4.1. The axiomatic development of elementary geometry 167 §4.2. The Poincare model 174 §4.3. The disc model