Projective Geometry

Eric Bainville - Oct 2007


A projective transformation is represented by a 4x4 invertible matrix S. By definition, the image of a point p by S is the point Sp. Since S is invertible, Sp always represents a point when x represents a point (i.e. when p is non 0). For any scalar λ≠0, S and λS represent the same transformation.

From our definition it is obvious that any projective transformation S has an inverse, represented by S-1. The product of two transformations is represented by the product of their matrices.

Projective transformations preserve dimensions: the image of a point/line/plane is a point/line/plane.

Projective transformations preserve point/line/plane incidences: the image of an intersection is the intersection of the images, and the image of an element included in another element e is included in the image of e.

A plane u is the set of points p verifying <u,p>=0. If q=Sp is the image of p by a transformation S, we have p=S-1q, and then <u,S-1q>=0, equivalent to <S-Tu,q>=0. It means the image of u by S is a plane represented by S-Tu.

A line A is the set of points p verifying Ap=0. If p=S-1q as above, we have AS-1q=0: q is in the kernel of AS-1. For any point p, Ap is a plane containing the line A. Its image is S-TAp. The matrix B=S-TAS-1 is antisymmetric of rank 2, and represents the image of line A by S.