Projective Geometry and Algebraic Structures

223 Pages·1972·9.489 MB·English

by  R. J. Mihalek (Auth.)

PROJECTIVE GEOMETRY
AND ALGEBRAIC STRUCTURES
R. J. Mihalek
University of Wisconsin-Milwaukee
Academic Press
New York and London
COPYRIGHT © 1972, BY ACADEMIC PRESS, INC.
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PRINTED IN THE UNITED STATES OF AMERICA
Cover design by Jorge Hernandez
PREFACE
This book is intended as a text for an undergraduate course in pro-
jective geometry ordinarily taught at the junior-senior level. The study
of projective geometry is particularly suitable for introducing students
to the so-called mathematical method. We realize this in the present
setting through the consideration of several axiomatic systems, the
construction of numerous examples illustrating these systems, and a
study of interrelations among these systems.
Affine planes are studied first in the interest of staying close to the
students' probable past experience in euclidean geometry. Projective
planes are then considered. Next the connection between affine and
projective planes is carefully established. Here, the concepts of isomor-
phism and imbedding are presented, concepts fundamental in the study
of modern mathematics.
The main theme of this work introduces the student to the relation-
ship between geometry and algebra. Indeed, this may be considered as
an introduction to abstract algebra. For some students, algebraic con-
cepts appear less " abstract " when studied first in a geometric setting.
This study provides then another reason for abstracting in the algebraic
context. The study of this relationship is developed to include the real
projective plane and the system of real numbers.
Finally, an introduction to the study of projective spaces is taken up.
Euclidean geometry is used in motivating much of this study. A
modern abstract point of view is adopted, giving historical considera-
tions, to a large degree, a secondary role. At the outset, the details of
the motivation are kept to a minimum. This is done to expedite the
beginning of the theory. Then as the theory is developed and the need
ix
X PREFACE
arises, the motivational discussions are resumed. Hence for the most
part, the motivational discussions immediately precede the introduction
of new concepts.
Some fundamentals from set theory are used here. The usage, how-
ever, is from a very intuitive point of view. For an elaboration of these
set-theoretic ideas and for further discussions of the mathematical
method, references [6], [9], [13], and [20] are suggested reading.
(Numerals in brackets refer to the correspondingly numbered items in
the list of references.) Reference [10] is suggested for a more advanced
point of view concerning the set-theoretic ideas.
Each chapter and section is prerequisite to the units that follow it
with the following exceptions:
(a) Sections 3.4,6.6, and 7.3 are not necessary for subsequent material ;
(b) Chapters 10 and 11 can be studied after Section 5.2.
Experience has shown that the material can be covered in from one to
one and one-half semesters (meeting three hours per week). References
[2], [3], [4], [5], [6], [8], [14], [15], and [18] are suggested for further
reading and as sources for additional references.
ACKNOWLEDGMENTS
The author wishes to acknowledge his indebtedness to colleagues and
students for helpful suggestions and encouragement during the writing
of this book. In particular, Professor Karl Menger first introduced the
author to projective geometry and made lasting impressions which
influence the spirit of this book.
INTRODUCTION CHAPTER 1
We shall, ultimately, study projective geometry as an abstract theory,
self-contained and independent. Nevertheless, we should be remiss not
to take advantage, initially, of our earlier experiences in geometry. Such
experiences are generally realized in euclidean geometry, more specific-
ally, in the high school mathematics traditionally encountered during
the sophomore year. We are not interested in the study of euclidean
geometry per se, but merely in its use to motivate and to illustrate the
current theory. A less precise formulation of euclidean geometry will
generally suit our purpose. However, for a thorough treatment of the
subject, were the need to arise, we could refer to " The Foundations of
Geometry" [12], the notable work of David Hubert (1862-1943).
We shall assume, therefore, a familiarity with euclidean geometry,
something close to the high school experience. Presumably, then, we
share a common frame of reference as we begin our introductory
discussion.
1.1. EUCLIDEAN PLANES
In his theory of euclidean planes, Hubert uses point, line, incidence,
betweenness and congruence (of line segments and angles) as primitive
1
2 1. INTRODUCTION
(or undefined) concepts. (See Appendix A for the foundation of Hu-
bert's theory. This is based on a later version; the first appeared
in 1899.) The points and lines are the " objects " of the theory; incidence
is a "relation" between points and lines; betweenness is a "relation"
on triples of points; and congruence is a "relation" on pairs of line
segments and on pairs of angles. He also states 15 axioms that the
primitive concepts are assumed to satisfy. Then additional concepts are
defined and additional statements (theorems) are stated and proved.
This constitutes the theory.
We have, then, a collection of concepts (both primitive and defined)
and a body of statements (both assumed and proved) in the euclidean
theory. Now we can look for an alternate development of the theory by
starting with different primitive concepts (selected from our collection
of concepts) or different axioms (selected from our body of statements).
Then the remaining concepts would be recovered through definition
and the remaining statements through proof. Such an alternate develop-
ment of a theory might be undertaken to understand better the signifi-
cance of certain concepts and their properties, to connect the theory
with another apparently unrelated theory, to make the theory more
readily applicable to other situations, or merely to satisfy a personal
curiosity. In our study of projective geometry, we shall be undertaking,
in a sense, such an alternate development of euclidean geometry. Let us
make the following observations in this vein.
The concept of projection is of interest and importance in euclidean
geometry. Projections can be used to depict how an object casts a
shadow (see Figure 1.1). Also, projections are used in describing the
optics of a camera or a projector.
light source
plane surface
(a) (b)
Figure 1.1
1.1. EUCLIDEAN PLANES 3
Let se and se' be planes in euclidean space and O SL point not on the
planes (see Figure 1.2 where se and se' are parallel or Figure 1.3 where
se and se' intersect). Then under the projection from se to se' with
center O, point P' on se' is the correspondent of point P on se if O,
Figure 1.2
Figure 1.3
4 1. INTRODUCTION
P, and Ρ' are collinear. Also line /' on se' is the correspondent of line /
on se if at least two points on /' are correspondents of points on /.
Now we ask if any of the euclidean concepts are "invariant" under
projections. This means, for points, that each point on se has a corre-
spondent on se' and each point on se' is the correspondent of some
point on se'. A similar meaning is ascribed for the " invariance " of lines.
"Invariance of incidence" means that a point and line on si are inci-
dent if and only if their correspondents (when existent) on si' are inci-
dent. For parallelism to be invariant, we should want lines on si to be
parallel if and only if their correspondents (when existent) on si' are
parallel. The meaning of "invariant" for the other euclidean concepts
is formulated similarly.
In case the planes si and si' are parallel (as in Figure 1.2), the situa-
tion is readily described. The concepts of point, line, incidence, between-
ness, and parallelism, to name a few, are invariant while the congruence
of lines segments is not. (Why ?) This suggests studying concepts and
statements that depend only on these " invariants " or, for that matter,
an alternate development of the theory of euclidean planes using new
primitive concepts selected from among these invariants and new
axioms selected from among the statements involving only these con-
cepts. Indeed, we begin such a study in Chapter 2 with affine planes,
employing "point," "line," and "incidence" as primitive concepts.
However, in order to continue the study, we must bring some additional
considerations into play, in particular, some that arise when the planes
si and si' are not parallel.
Let the planes si and si' intersect as in Figure 1.3. Then in the figure,
points P, Ö, P, P, on si have correspondents P', Q\ R', T', respectively,
on si'. The point S on si is such that the line on O and S is parallel to
si'. Hence S has no correspondent on si'. Thus the concept of point is
not invariant under this projection. A similar situation holds for lines.
(What line on si has no correspondent on si' and what line on si' is
not the correspondent of any line on si ?) Incidence is invariant. (Why ?)
Parallelism is not; consider lines / and m on si in the figure. In addition,
betweenness is not invariant. Consider, in the figure, P between Q and R
while P' is not between Q' and R'. Congruence fails to be invariant for
this projection also.
1.1. EUCLIDEAN PLANES 5
When comparing the two projections considered, it appears that the
loss of invariance for some of the concepts relative to the latter pro-
jection is related to parallelism. Indeed, in Figure 1.3, point S has no
correspondent on s/' because the line on O and S is parallel to se'.
Early geometers recognized this situation and circumvented it by de-
vising a scheme through which they introduced additional objects
(called ideal points, ideal lines, and an ideal plane) to eliminate parallel-
ism. In other words, they enlarged euclidean space to a geometry with-
out parallelism. As might be expected in this enlarged geometry, many
of the analogs of euclidean concepts and statements undergo consider-
able alteration. For example, any two "lines" on a "plane" will
"intersect." Nevertheless, by design, the concepts in this enlarged
geometry that are the analogs of euclidean point, line, incidence, and
betweenness are invariant under projections. We shall undertake, then,
an alternate development of the theory of this enlarged geometry using
only these invariant concepts. We begin this development in Chapter 3
with the study of projective planes, employing "point," "line," and
"incidence" as primitive concepts. We do not introduce the analog of
betweenness, called separation, until Chapter 9. A considerable bulk of
the "planar" theory of this enlarged geometry can be developed before
separation is needed. We continue the alternate development of the
enlarged geometry with the study of projective spaces in Chapters 10
and 11.
We choose not to describe, at this time, the above " enlargement "
scheme. Consequently, we cannot know the details of the properties of
the enlarged geometry. However, this circumstance presents no deterrent
to our study. We already have an idea of what we need for affine and
projective planes. In Chapter 4, we describe the relation between these
two concepts, which is, in effect, the " enlargement " scheme for " planes."
Thereafter, we introduce and discuss properties of the enlarged geo-
metry as needed.
We are able to use the concept of invariance (under projections) to
classify geometric properties as descriptive or metric. Properties involv-
ing only the invariant concepts are said to be descriptive. They are the
properties concerned with positional relationships of geometric objects
such as incidence and betweenness. The metric properties involve the