In geometry, the terms pole and polar are used to describe a point and a line that have a unique reciprocal relationship with respect to a given conic section. If the point lies on the conic section, its polar is the tangent line to the conic section at that point.

For a given circle, the operation of reciprocation in a circle corresponds to transforming each point in the plane into its polar line and each line in the plane into its pole.

The pole of a line L in a circle C is a point P that is the inversion in C of the point Q on L that is closest to the center of the circle. Conversely, the polar line (or polar) of a point P in a circle C is the line L such that its closest point Q to the circle is the inversion of P in C.

The relationship between poles and polars is reciprocal. Thus, if a point Q is on the polar line A of a point P, then the point P must lie on the polar line B of the point Q. The two polar lines A and B need not be parallel.

There is another description of the polar line of a point P in the case that it lies outside the circle C. In this case, there are two lines through P which are tangent to the circle, and the polar of P is the line joining the two points of tangency (not shown here). This shows that pole and polar line are concepts in the projective geometry of the plane and generalize with any nonsingular conic in the place of the circle C.