Inverse points, also called polar reciprocals, are points transformed into each other through inversion about
a given inversion circle 
(or inversion sphere).
Let 
be the inversion center and 
the inversion radius.
The points 
and 
, neither equal to 
, are inverse points with respect to 
if they lie on the same ray from 
and satisfy
 |
(Wenninger 1983, p. 2). In the figure, 
is the inversion pole
of the line 
through 
and perpendicular to 
, so 
is the polar of 
. The circle power
of the point 
relative to 
is 
, where 
.
Inverse points with respect to a triangle are generally understood to use the triangle's circumcircle as the inversion circle (Gallatly 1913).
The point 
which is the inverse point of a given point 
with respect to an inversion
circle 
may be constructed geometrically using a compass only
(Coxeter 1969, p. 78; Courant and Robbins 1996, pp. 144-145).
Inverse points can also be taken with respect to an inversion sphere, which is a natural extension of geometric inversion from the Euclidean plane to three-dimensional Euclidean space.
