# Projective Geometry

Eric Bainville - Oct 2007## Transformations

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^{-1}q, and then <u,S^{-1}q>=0,
equivalent to <S^{-T}u,q>=0. It means the image of u by S is a plane
represented by S^{-T}u.

A line A is the set of points p verifying Ap=0. If p=S^{-1}q as above, we have
AS^{-1}q=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^{-T}Ap. The matrix B=S^{-T}AS^{-1} is
antisymmetric of rank 2, and represents the image of line A by S.

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