Table Of Contentmetric
affine
geometry
Ernst Snapper
Dartmouth College
Robert J. Troyer
Lake Forest College
Academic Press
New York and London
COPYRIGHT © 1971, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 76-154402
AMS (MOS) 1970 Subject Classification: 50A05,50A10,
50A15; 50B10, 50B35; 50C25; 50D05,50D10
PRINTED IN THE UNITED STATES OF AMERICA
To our parents
Preface
For a long time, the axioms of Euclid and Hilbert were the best for
Euclidean geometry, but not any more. The modern source for the axiom
systems of all affine and projective geometries, both metric and nonmetric, is
linear algebra. Examples of metric affine geometries are the Euclidean
geometries of various dimensions, the Lorentz plane, and Minkowski
space. Examples of metric projective geometries are the elliptic and hyper
bolic planes.
Linear algebra is the algebraic theory of vector spaces. How can this
provide us with axioms for all these different geometries ?
Nonmetric affine space of dimension « consists of a set X on which the
additive group of an «-dimensional vector space V acts (Chapter 1). The
elements of X are the points of the space and the action of a vector of V on
these points is a translation of the space. A "metric" is introduced in the
affine space X by choosing an inner product (metric) for V (Chapters 2 and 3).
Different choices of Kand inner product give rise to the different «-dimensional
metric affine spaces. For example, if V is a vector space over the real numbers
and the inner product is positive definite, X is «-dimensional Euclidean
space.
Nonmetric projective space of dimension « has the lines (one-dimensional
subspaces) of an (« + l)-dimensional vector space V as its points. The
geometry of these lines is «-dimensional projective geometry. If V has been
given an inner product, one obtains an «-dimensional metric projective
space. For instance, if V is a three-dimensional vector space over the real
numbers and the inner product is positive definite, one obtains the elliptic
plane as the projective plane.
In this way, the axioms for a vector space give rise to axiom systems for all
affine and projective, nonmetric and metric geometries. Such unity and
simplicity cannot be claimed by the classical axiom systems of Euclid and
Hilbert, nor by the Erlangen program of Felix Klein. Furthermore, the
axioms systems which come from linear algebra are eminently suited to
handle concrete problems in geometry, a virtue the axioms of Euclid and
Hilbert do not possess. For these reasons, our book makes no use of the
Euclid-Hilbert system or the Erlangen program and only mentions them to
give background material or historical tidbits. Nevertheless, it should be
realized that we are dealing with precisely the same geometries as are treated by
xv
xvi PREFACE
the classical systems. The geometries have not changed, only the choice of
axioms.
Chapter 1 develops nonmetric affine geometry, Chapter 2 studies inner
products of vector spaces, and Chapter 3 treats metric affine geometry.
Chapters 1 and 2 are independent and either can be used for a one-semester
(term) course. It will take two semesters (terms) to cover the entire book.
Our main disappointment in writing this book has been the omission of a
proper treatment of projective geometry because it would have made the book
too long. Consequently, we have completely omitted projective geometry,
except for background remarks for those readers who already have some
knowledge of the subject. Others will probably want to omit our remarks on
projective spaces.
The prerequisites for this book are a course in linear algebra, not neces
sarily including the study of inner products, and an elementary course in
modern algebra which includes the concepts of group, normal subgroup, and
quotient group. Although these prerequisites are light, we make heavy use
of them. We assume that the reader can think in terms of vector spaces over
an arbitrary field as scalar domain. We judge the level of our book to be
advanced undergraduate or beginning graduate.
The exercises have been placed in the text where we felt they were most
appropriate; they have not been held until the end of each section. A starred
exercise is one to which a specific reference is made in the text or in another
exercise and should not be omitted. Many exercises have hints and since text
material is often covered in the exercises, many of the hints are actually
short solutions. Of course, the reader should use the hints only in case of
"emergency"; with these hints most of the exercises can be worked by any
reader.
The numbering system for theorems, propositions, definitions, etc., is
based on the principle that the number should tell where the statement can be
found. Hence Proposition 378.2 means "the second proposition on page
378." Do not fear that 377 propositions precede it! Exercises are numbered
consecutively in each section and Exercise 12.46a means "Part a of Exercise
46 which begins on page 12."
Most of the second chapter of this book can be found in E. Artin's book,
Geometric Algebra. Our main effort has been to slow Artin's exposition and
thereby make the tremendous geometric wealth which his masterpiece
harbors available to the advanced undergraduate and beginning graduate
student. We hope that anyone who has worked through our second chapter
can read Geometric Algebra from the geometric viewpoint which Artin found
so important.
We fear the level of our book is too high for many high school teachers.
Nevertheless, this does not mean that discussions of questions related to
PREFACE xvii
teaching geometry in high school are out of place in a book such as ours. For
example, what does it imply for high school geometry that the proper axioms
for Euclidean geometry come from linear algebra? Several questions of this
kind have been discussed and we hope the reader finds these discussions of
interest. Both authors feel that the geometry curriculum for future high school
teachers should be based on linear algebra. However, the level should be
modified from that presented in this book.
A course along the lines of this book was taught by Snapper at the first
Cooperative Summer Seminar held at Cornell University in 1964. Excellent
notes ("Metric Geometry Over Affine Spaces") were written by Professor
J. T. Buckley of Western Michigan University and distributed by the Mathe
matical Association of America. A grant from the Course Content Improve
ment Section of the National Science Foundation made it possible for Troyer
to spend the academic year 1965-66 at Dartmouth College where Snapper
taught an improved version of the course and Troyer wrote the notes " Metric
Geometry of Vector Spaces." These notes were then taught by Professor
Sherwood Ebey, Mercer University; Jack Graver, Syracuse University;
Roger Lyndon, University of Michigan; E. R. Mullins, Grinnell College;
Paul Rygg, Western Washington State College; Jon Sicks, University of
Massachusetts; Seymour Schuster, University of Minnesota; and R. J.
Troyer, University of North Carolina. The present book is a rewrite of these
notes, based on the many fine suggestions by these teacher-friends and their
students, and on the careful reviews of the notes by Professors Jean Dieudonne
and Roger Lyndon.
Even with all this assistance, our book would never have seen the light of
day had it not been for the unselfish help and expert typing of Mrs. Ann
Hackney and Mrs. Mary Lou Troyer; the generous support from Dartmouth
College, Lake Forest College, the University of North Carolina, and the
National Science Foundation; and the infinite patience of Academic Press.
We are grateful to all who have assisted with the painful birth of our book.
SYMBOLS
0 2 origin of vector space
SU" 5 A7-dimensional affine space
V 6 vector space
X 6 nonempty set of points
e, xe X 6 x is an element of X
x,y 6 unique vector which maps x onto /
(X, V, k) 7 affine space
R 7 field of real numbers
Q 7 field of rational numbers
c 7 field of complex numbers
T 9 translation of X associated with the vector A
A
<Ü> 10 if and only if
s 10 permutation group of the set X
x
h 10 the identity mapping of X
T: X^X 11 mapping of X intoX
0Ί···} 11 all / such that ···
S(x, U) 11 affine subspace
dim 12 dimension
c, R <=S 13 R is a subset of 5
n, RnS 14 intersection of sets /? and S
0 14 empty set
n s 16 intersection of sets S, i e /
t t
iel
<At> 17 set of multiples of the vector At
(yi,-->y) 17 affine coordinates of point y
n
\w\ 18 cardinality of set W
ker/ 22 kernel of a linear function
im/ 22 image of a function
Θ 22 direct sum
S\\ S' 27 parallel affine subspaces
=> 29 implies
Λ:Χ v · · · v xd 35 affine space spanned by the points xlf...,xd
S1v -" v Sd 35 affine space spanned by the sets Slr.. .,Sd
M(c, r) 38 magnification with center c and magnification ratio r
k* 38 multiplicative group of nonzero elements of k
Di 40 group of dilations
*v 46 integers modulo a prime integer p
(p, q)l(x> y) 47 ratio of lengths of line segment (p, q) to (x, y)
r^s 52 isomorphism
Tr 52 the group of translations of X
XIX
XX SYMBOLS
X{c) 59 tangent space at c
Sa 71 the group of semiaffine transformations of X
Af 72 group of affine transformations of X
Sem 75,
90 the group of semilinear automorphisms of V
L(c, λ) 75 semilinear automorphism of X(c)
GL(n, k) 90 the general linear group
Aut k 93 group of automorphisms of k
±9A±B 115 vectors A and B are orthogonal (perpendicular)
Ul W 115 subspaces U and V are orthogonal (perpendicular)
u* 116,
148 orthogonal complement of U
det 122 determinant
PT 124 transpose of matrix P
k*2 125 group of squares of k*
disc V 125 discriminant of V
<Γι,...,Γ > 126 linear subspace of V generated by Y ...,Y
Μ lf m
rad V 142 radical of V
υ®···®υ 144 orthogonal sum of the subspaces U ,.. .,U
8 1 S
Art2d 193 Artinian space of dimension 2d
^φ···φσ 197 orthogonal sum of the isometries a...,a
5 lt s
a\S 197 the function σ restricted to S
det# 215 determinant of the linear transformation g
0(W) 215 group of isometries of W
-1κ 217 the function A -> — A
*A 219 reflection in the hyperplane </l>*
o+(v) 221 group of rotations of V
res 252,
317 restriction mapping
ext 252 extension mapping
0(V;l) 316 isometries of V with line / as axis
0 + (V;l) 316 rotations of V with line / as axis
Ω(Κ) 337 commutator subgroup of 0(V)
0 + (V)2 339 set of squares of 0+(V)
0(V2) 339 set of squares of 0(V)
n+(v) 340 commutator subgroup of 0+(V)
S(V) 359 group of similarities of V
7Ι®~'®7Η 363 orthogonal sum of the similarities γι,...,γη
(x9y)±(b,c) 376 perpendicular oriented line segments
{x, y} 1 {b, c} 376 perpendicular line segments
Mo 380 group of rigid motions of X
Sim 413 group of similarities of X
chapter 1
affine
geometry
1. INTUITIVE AFFINE GEOMETRY
Our subject is geometry and we shall begin with the study of affine
geometry. To follow a more historical order, we would begin with Euclidean
geometry. However, it seems useful to separate the axioms of affine geometry
from those of Euclidean geometry. Roughly speaking, affine geometry is
what remains after practically all ability to measure length, area, angles, etc.,
has been removed from Euclidean geometry. One might think that affine
geometry is a poverty-striken subject. On the contrary, affine geometry is
quite rich. Even after almost all ability to measure has been removed from
Euclidean geometry, there still remains the concept of parallelism. Conse
quently, the whole theory of homothetic figures lies within affine geometry.
The notions of translation and magnification (these are the dilations) are in
the domain of affine geometry and, more generally, as the name suggests,
affine transformations (one-to-one, onto functions which preserve parallelism)
constitute an affine notion.
2 1. AFFINE GEOMETRY
The question of how to obtain a rigorous approach to affine geometry
arises. We choose to base our axiom system on linear algebra, that is, on the
theory of vector spaces. First, let us discuss, on an intuitive level, the con
nection between plane affine geometry and vector spaces. What is the in
tuitive notion of the affine plane P over the field of real numbers? The points
of P can be viewed as the points of any physical plane. If we fix a particular
point 0 of P, we can construct a vector space in a natural way. The vectors
are the ordered pairs (0, A) where A e P and we picture such a vector as an
arrow from 0 to A (see Figure 2.1). We often write A for the vector (0, A),
D
Figure 2.1
but this can only be done if it is clear from the context which point is being
used as 0, that is, as the common initial point of the vectors. Vectors are
added in the usual way by "completing the parallelogram" and we perform
scalar multiplication by stretching or contracting the vector in the proper
direction. Both operations are pictured in Figure 2.2.
A A + B
Ο ^— ^B
— 71A ·* 1« I« M H o »I H H H ** 71A
-3A -2A -A-±A0 \A A 2A 3A
Figure 2.2
Thus we have associated a real, two-dimensional vector space with the
real affine plane. The main thing to realize is that the construction of this
vector space is based solely on notions of parallelism, that is, on affine
concepts, and not on Euclidean concepts such as distance. However, this
vector space is not unique because it depends on the choice of the point 0 in
P. It is true that a different point (V gives rise to an isomorphic vector space,
