Duality (projective Geometry) - Poles and Polars

Poles and Polars

In the Euclidean plane, fix a circle C with center O and radius r. For each point P other than O define an image point P so that OP • OP = r2. The mapping defined by P → P is called inversion with respect to circle C. The line through P which is perpendicular to the line OP is called the polar of the point P with respect to circle C. Let m be a line not passing through O. Drop a perpendicular from O to m, meeting m at the point Q (this is the point of m that is closest to O). The image of Q under inversion with respect to C is called the pole of m. If a point P (different from O) is on a line m (not passing through O) then the pole of m lies on the polar of P and viceversa. The incidence preserving process, in which points and lines are transformed into their polars and poles with respect to C is called reciprocation. In order to turn this process into a reciprocity, the Euclidean plane (which is not a projective plane) needs to be expanded to the extended euclidean plane by adding a line at infinity and points at infinity which lie on this line. In this expanded plane, we define the polar of the point O to be the line at infinity (and O is the pole of the line at infinity), and the poles of the lines through O are the points of infinity where, if a line has slope s (≠ 0) its pole is the infinite point associated to the parallel class of lines with slope -1/s. The pole of the x-axis is the point of infinity of the vertical lines and the pole of the y-axis is the point of infinity of the horizontal lines.

The construction of a reciprocity based on inversion in a circle given above can be generalized by using inversion in a conic section (in the extended real plane). The reciprocities constructed in this manner are projective correlations of order two, that is, polarities.