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In-Depth Information
CHAPTER 3
Projective Geometry
3.1
Overview
The last chapter outlined some of the basic elements of affine geometry. This chapter
looks at projective geometry. Some general references that look at the subject in more
detail than we are able to here are [Ayre67], [Gans69], and [PenP86].
Like in the last chapter, we shall start with dimension two (Sections 3.2-3.4) and
only get to higher dimensions in Section 3.5. In order to motivate the transition from
affine geometry to projective geometry we begin by studying projective transforma-
tions in affine space. Section 3.2 starts off by looking at central projections and leads
up to a definition of a projective transformation of the plane. We shall quickly see
that, in contrast to affine geometry, we have to deal with certain exceptional cases that
make the statement of definitions and theorems rather awkward. Mathematicians do
not like having to deal with results on a case-by-case basis. Furthermore, the exis-
tence of special cases often is a sign that one does not have a complete understand-
ing of what is going on and that there is still some underlying general principle left
to be discovered. In fact, it will become clear that Euclidean affine space is not the
appropriate space to look at when one wants to study projective transformations and
that one should really look at a larger space called projective space. This will allow
us to deal with our new geometric problems in a uniform way.
Projective space itself can be introduced in different ways. One can start with a syn-
thetic and axiomatic point of view or one using coordinates. Lack of space prevents us
from discussing both approaches and so we choose the latter because it is more prac-
tical. In Section 3.3 we introduce homogeneous coordinates after a new look at points
and lines that motivates the point of view that projective space is a natural coordinate
system extension of Euclidean space. This leads to a definition and discussion of the
projective plane
P
2
in Section 3.4. Some of its important analytic properties are
described in Section 3.4.1. Sections 3.4.2 and 3.4.3 define projective transformations
of
P
2
and show how affine transformations are just special cases if one uses homoge-
neous coordinates. We then generalize to higher dimensions in Section 3.5. The impor-
tant special case of 3-dimensional projective transformations is considered in Section
3.5.1. Next we study conics (Sections 3.6 and 3.6.1) and quadric surfaces (Section 3.7).
