Reciprocation is an incidence-preserving transformation in which points are transformed into their polars. A projective
geometry-like duality principle holds for
reciprocation which states that theorems for the original figure can be immediately
applied to the reciprocal figure after suitable modification
(Lachlan 1893, pp. 174-182). Reciprocation (or "polar reciprocation")
is the strictly proper term for duality. Brückner (1900) gave one the first
exact definitions of polar reciprocation for constructing dual
polyhedra, although the plane geometric version (inversion
pole, polar, and circle
power) was considered by none less than Euclid (Wenninger 1983, pp. 1-2).
Lachlan (1893, pp. 257-265) discusses another type of reciprocation he terms "circular reciprocation." However, the circular reciprocal figure is, in general, more complicated than the original, so the method is not as powerful as the usual polar reciprocation.
