Let  be a fixed circle and let P be any ordinary point other than the center O. Let P' be the inverse of P in circle . Then the line Q through P' and perpendicular to OPP' is called the polar of P for the circle c.  Note that when P is a line, then Q will be a point.

Note that this routine in particular, and the geometry package in general, does not encompass the extended plane, i.e., the polar of center O does not exist (though in the extended plane, it is the line at infinity) and the polar of an ideal point P does not exist either (it is the line through the center O perpendicular to the direction OP in the extended plane).

If line Q is the polar point P, then point P is called the pole of line Q.

The pole-polar transformation set up by circle  is called reciprocation in circle c

For a detailed description of Q (the object created), use the routine detail (i.e., detail(Q))

The command with(geometry,reciprocation) allows the use of the abbreviated form of this command.