embedding planes

 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 cross-ratio. 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.