The analogous notation of inversion can be performed in three-dimensional space with
respect to an inversion sphere.
If and are inverse points, then the line through and perpendicular to is sometimes called a "polar"
with respect to point ,
known as the "inversion pole". In addition,
the curve to which a given curve is transformed under inversion is called its inverse curve (or more simply, its "inverse").
This sort of inversion was first systematically investigated by Jakob Steiner.
From similar triangles, it immediately follows that the inverse points and
obey
(1)
or
(2)
(Coxeter 1969, p. 78), where the quantity is known as the circle power
(Coxeter 1969, p. 81).
Treating lines as circles of infiniteradius, all circles
invert to circles (Lachlan 1893, p. 221). Furthermore,
any two nonintersecting circles can be inverted into concentric circles by taking
the inversion center at one of the two so-called
limiting points of the two circles (Coxeter 1969),
and any two circles can be inverted into themselves or into two equal circles (Casey
1888, pp. 97-98). Orthogonal circles invert
to orthogonal circles (Coxeter 1969). The inversion circle itself, circles orthogonal to it,
and lines through the inversion center are invariant
under inversion. Furthermore, inversion is a conformal
mapping, so angles are preserved.
The property that inversion transforms circles and lines to circles or lines (and that inversion is conformal) makes it an extremely important tool of plane analytic geometry. By picking a suitable inversion circle, it is often possible to transform one geometric configuration into another simpler one in which a proof is more easily effected. The illustration above shows examples of the results of geometric inversion.
These equations can also be naturally extended to inversion with respect to a sphere in three-dimensional space.
The above plot shows a chessboard centered at (0, 0) and its inverse about a small circle also centered at (0, 0) (Gardner 1984, pp. 244-245;
Dixon 1991).
to a corresponding set of points 
known as their inverse points.
Two points 
and 
are said to be inverses with respect
to an inversion circle having inversion
center 
and inversion radius 
if 
is the perpendicular foot of the altitude
of 
, where 
is a point on the circle such that 
.
and 
are inverse points, then the line 
through 
and perpendicular to 
is sometimes called a "polar"
with respect to point 
,
known as the "inversion pole". In addition,
the curve to which a given curve is transformed under inversion is called its inverse curve (or more simply, its "inverse").
This sort of inversion was first systematically investigated by Jakob Steiner.
and 
obey
is known as the circle power
(Coxeter 1969, p. 81).
relative to the inversion
circle with inversion center 
and inversion radius 
is given by
with center 
with respect to an inversion circle
with inversion center 
and inversion radius 
is another circle
with center
