Graphics Reference
In-Depth Information
CHAPTER 2
Affine Geometry
2.1
Overview
The next two chapters deal with the analytic and geometric properties of some impor-
tant transformations of
R
n
. This chapter discusses the group of affine maps and its
two important subgroups, the group of similarities and the group of motions. Affine
maps are the transformations that preserve parallelism. Similarities are the affine
transformations that preserve angles. Motions are the distance-preserving similarities
and their study is equivalent to the study of metric properties of Euclidean space. As
a historical note, this reduction of geometric problems to algebra (namely the study
of certain groups in our case) was initiated by the German mathematician Felix Klein
at the end of the 19
th
century.
Except for some definitions and a few basic facts, the first part of the chapter (Sec-
tions 2.2-2.4) concentrates on the important special case of the plane
R
2
. Presenting
a lot of details in the planar case where it is easier to draw pictures should make it
easier to understand what happens in higher dimensions since the generalizations are,
by and large, straightforward.
Motions are probably the most well-known affine maps and we analyze planar
motions in quite some detail in Section 2.2. Section 2.2.8 introduces the concept of a
frame. Frames are an extremely useful way to deal with motions and changing from
one coordinate system to another. It is not an overstatement to say that a person who
understands frames will find working with motions a triviality. There is a brief dis-
cussion of similarities in Section 2.3 and affine maps in Section 2.4. Parallel projec-
tions are defined in Section 2.4.1. Section 2.5 extends the main ideas from the plane
to higher dimensions. The important case of motions in
R
3
is treated separately in
Sections 2.5.1 and 2.5.2.
There is not enough space to prove everything in this chapter and it will be up to
the reader to fill in missing details or to look them up in the references. Hopefully,
the details we do provide in conjunction with what we did in Chapter 1 will make
filling in missing details easy in most cases. Unproved facts are included because it
was felt that they were worth knowing about and help as motivation for the next
