Points, also called polar reciprocals, which are transformed into each other through inversion about a given inversion
circle 
(or inversion sphere). The points 
and 
are inverse points with respect to the inversion
circle if
 |
(Wenninger 1983, p. 2). In this case, 
is called the inversion pole
and the line 
through 
and perpendicular to 
is called the polar. In the above figure, the quantity
is called the circle
power of the point 
relative to the circle 
.
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 plane to three-dimensional space.
