Inversion

Inversion is the process of transforming points 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 .

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).

The general equation for the inverse of the point relative to the inversion circle with inversion center and inversion radius is given by

			(3)
			(4)

In vector form,

(5)

Note that a point on the circumference of the inversion circle is its own inverse point. In addition, any angle inverts to an opposite angle.

Treating lines as circles of infinite radius, 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.

The inverse of a circle of radius with center with respect to an inversion circle with inversion center and inversion radius is another circle with center

			(6)
			(7)

and radius

(8)

where

(9)

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).

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