Pole and Polar

The polar of a point P with respect to a conic L is the set of points Q such that P, Q and the points of intersection of line PQ with L form a harmonic set. The polar is a straight line, and P is called its pole. The pole and polar of a plane with respect to a quadric surface are defined in a similar way.

Poles and polars satisfy the principle of duality—that is, if the polar of P passes through Q, then the polar of Q passes through P. If L is nondegenerate, then any line has a specific pole with respect to it, and to any pole there corresponds a specific polar. Thus, a one-to-one correspondence is established between points and lines; this correspondence is a special case of a correlation transformation. Poles and polars are used in projective geometry in the classification of conics and quadric surfaces.