In the ideal points page, we introduced the idea of ideal points for the
euclidean plane E.
These may be taken as directions in E, or as families
of parallel lines in E. The set consiting of the points of E, together with
the ideal points is the extended plane corresponding to E.
In the homogeneous coordinates page, we showed
that we can map
the RP2 model of projective geometry onto this extended plane. This
was achieved by taking the plane z=1 as (a model of) E. We could use
any plane Π not through the origin in place of z=1. Then each element
of RP2 either cuts Π in a single point or is parallel to Π and so defines
a diection in Π, i.e. an ideal point for Π. Again we have a mapping from
RP2 onto the extended plane corresponding to Π. This is called the
embedding of RP2 in Π, and is denoted by pΠ.
A figure F in RP2 is simply a set of elements of RP2. Then pΠ(F)
is a set
of points in the extended plane. If we ignore any ideal points which occur,
then we have a set of points of Π, i.e. a figure in the plane Π. Some of the
affine or euclidean features of this figure can be deduced from projective
results for F. In this context, any ideal points will give information about
parallel lines in Π.
In general, different choices of Π will give different plane figures. For example,
if F is a projective conic, then it may embed as an ellipse, a parabola or a
hyperbola. See plane section of a cone. However, if we chose two parallel
planes, then the embeddings will be scalings of one another. To see this,
consider embeddings on z=1 and z=k. A point A = [a,b,c] with c ≠ 0 embeds
in z=1 as the point (a/c,b/c,1), and in z=k as (ka/c,kb/c,k). This is just a
scaling with factor k. Thus, the important aspect of the choice is simply the
direction of the plane. This amounts to chosing a plane through O. Then
any embeddings on planes parallel to this give similar figures. The choice of
a plane through O is the choice of a line in RP2 to play the role of the ideal line
for the embedding.
To make use of the cross-ratio, we must consider how a cross-ratio in RP2
relates to the figure in the embedding on Π. As,we shall see, it turns out
to be related to signed ratios in the plane figure.
The Embedding Theorem
Suppose that A,B,C,D are distinct, collinear p-points, and that Π is a plane
not through O. Let A',B',C',D' denote the embeddings of A,B,C,D on Π. Then
either all of A',B',C',D' are ideal for Π or at most one is ideal.
(1) If none of A',B',C',D' is ideal, then (A,B,C,D) = (A'C'/C'B')/(A'D'/D'B')
(2) If only A' is ideal, then (A,B,C,D) = B'D'/B'C'.
(3) If only B' is ideal, then (A,B,C,D) = A'C'/A'D'.
(4) If only C' is ideal, then (A,B,C,D) = D'B'/D'A'.
(5) If only D' is ideal, then (A,B,C,D) = C'A'/C'B'.
Note that, in case (1), the cross-ratio is a ratio of signed ratios, hence the name
The Greeks actually studied cross-ratio in a euclidean setting. To see how
they did this, and to obtain some results in euclidean and affine geometry
we look at pencils of lines, and harmonic pencils in particular.