Projective Geometry - Computer Graphics and Geometric Modeling - page 139

Recall that equations for lines are not unique. Also, if L is the line defined by equa-
tion (3.14), then the family of lines parallel to L is obtained as solutions to equations
where we fix a and b in (3.14) but let c vary (Exercise 1.5.7(b)). Looking at this another
way, what we have just shown is that there is a one-to-one correspondence between
families of parallel lines in the plane and ideal points, namely, to the family of lines
parallel to L we associate the ideal point that is the class of solutions to (3.15) deter-
mined by (-b,a,0). That such a correspondence exists was foreshadowed in the dis-
cussion at the end of Section 3.2.
Finally, the equations in (3.14) give rise to all of the equations in (3.15) except
the equation cZ = 0, that is, Z = 0. Thus there is only one equation in (3.15) not arising
from a line in the plane, but this is precisely the equation that defines the ideal points.
It should not be surprising if, as we shall do in the next section, one defines a “line”
in the projective plane to mean a set of points determined by the solutions to a linear
equation of the form shown in (3.15). Then lines in the projective plane will corre-
spond to solutions to linear equations just like in the Euclidean plane.

3.4 The Projective Plane

The informal discussion of linear equations and their solutions in the previous section
led to homogeneous coordinates and suggested a new way of looking at points in the
plane. We shall now develop these observations more rigorously. Although we are only
interested in the projective line and plane for a while, we start off with some general
definitions so that we do not have to repeat them for each dimension.

The relation ~ defined on the points p of R n+1

3.4.1. Lemma. - 0 by p ~ c p , for c π

0, is an equivalence relation.

Proof. This is an easy exercise.

Definition. The set of equivalence classes of R n+1 - 0 with respect to the relation ~
defined in Lemma 3.4.1 is called the n-dimensional (real) projective space P n . In more
compact notation (see Section 5.4 and the definition of a quotient space),

( )

n n +1

PR 0 = - ~.

The special cases P 1 and P 2 are called the projective line and projective plane , respec-
tively. If P ΠP n and P = [x 1 , x 2 ,..., x n+1 ], then the numbers x 1 , x 2 ,..., x n+1 are called
homogeneous coordinates of P . One again typically uses the expression “(x 1 ,x 2 ,..., x n+1 )
are homogeneous coordinates for P ” in that case.

Note that P 0 consists of the single point [1]. We can think of points in P 1 or P 2 as
equivalence classes of solutions to (3.13) or (3.15), or alternatively, as the set of lines
through the origin in R 2 or R 3 , respectively. Other characterizations of the abstract
spaces P n will be given in Section 5.9. There are actually many ways to introduce coor-
dinates for their points. In the next section we shall see how this can be done for P 1
and P 2 .
It is easy to check that the maps