All the transformations so far considered have taken points into points. The most characteristic feature of the “projective” plane is the principle of duality, which enables us to transform points into lines and lines into points. One such transformation, somewhat resembling inversion, is “reciprocation” with respect to a fixed circle. Every point except the center O is reciprocated into a line, every line not through O is reciprocated into a point, and every circle is reciprocated into a “conic” having O for a “focus”. After some discussion of the various kinds of conic, we shall close the chapter with a careful comparison of inversive geometry and projective geometry.

Reciprocation

For this variant of inversion, we use (as in Section 5.3, page 108) a circle ω with center O and radius K Each point P (different from O) determines a corresponding line p, called the polar of P; it is the line perpendicular to OP through the inverse of P (see Figure 6.1A).