Projective Geometry

Eric Bainville - Oct 2007

Polarities

A polarity is represented by an 4x4 invertible symmetric matrix H. By definition, the image of an element (point/line/plane) represented by a vector space E is represented by the orthogonal of HE: the vector space of all x verifying <Hp,x>=0 for all p∈E.

From this definition, we see that the image of the sum of vector spaces is the intersection of their images, and the image of the intersection is the sum of the images.

The image of a point p by H is the plane Hp, called the polar plane of p. The image of a plane u is the point H^-1u, called the pole of u. The image of a line by two points p and q is the line intersection of the two planes Hp and Hq. The image of a line intersection of two planes u and v is the line by the two points H^-1u and H^-1v.

Consider a line by two points p,q: A=(pq^T-qp^T)'. The image of A is B, the intersection of the two polar planes u=Hp and v=Hq, B=uv^Y-vu^T as seen in Lines II. Expanding u and v, we get B=HA'H. The image of B shall be A (up to a scalar, omitted here), so HB'H=A, or B'=H^-1AH^-1.

Consider now a line A, its image B=HA'H by polarity H, and a point p on A. The polar plane of p is Hp, intersecting A at point q=A'Hp. The polar plane of q is Hq=HA'Hp=Bp, i.e. the plane containing p and B. H defines on A an involutive transformation. This transformation is projective (restriction of A'H on line A).