the inversion

An inversion IP,r is a geometric transformation of the plane (mapping: point M → point M') defined by PM'×PM=r² (notation: PM is the distance between P and M) where the three points P, M and its image M' are collinear (lying on a same line), with M and M' on the same side of the pole P.
The point P doesn't have an image (when M comes close to P, its image M' speeds to infinity!).
The set of fixed points (such as M'=M) is obviously the circle with center P and radius r: it's the inversion circle.
The originality of this involution (bijection which is its own reciprocal: M → M' → M for every point M) is to transform a line not through P into a circle through P, and conversely. A line through P is invariant : as a whole. A circle not through P is transformed into a circle; beware, the centers of the two circles are NOT inverse!
In an inversion angles are preserved (conformal mapping), thus a circle orthogonal to the inversion circle is invariant as a whole.
Treating lines as circles of infinite radius, all circles invert to circles. Furthermore, any two nonintersecting circles can be inverted into themselves (each invariant as a whole), into concentric circles (with same center) or into two isometric circles (with same radius).

So a "line-grid" (white, deprived of its center P) is transformed into an "arc-grid" (red) composed of four half-lines and circles' arcs two by two orthogonal (the arcs belong to circles through P). The intersections of a segment and its image (arc) belong to the inversion circle (blue).

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grid and its inverse
inverse of a line
(drag the big white points A and B)
inverse of a circle
(drag the big white points C and M)
inverse of a square grid

A nice example: the two images below show a chessboard superimposed with its inverse.
The external cirle/square just limits the size of the image; indeed the four external zones are the infinite images of the four central squares (one of their vertices is the pole P). The central part is not coloured in green/blue; it's the image of the outside of the chessboard.

chess and its inverse (in green) chess and its inverse (in blue)

The properties that inversion transforms circles and lines to circles or lines and that it preserves the angles makes it an extremely important tool of plane geometry: by choosing a suitable inversion circle it is often possible to transform one geometric configuration into another simpler one in which a proof is more easy.

The definition and the properties of the plane inversion remain true in the space: it suffices to replace "line" by "plane" and "circle" by "sphere".
Two nice applications: the duality of the polyhedra and the "kissing circles/spheres".

references: •  inversion - MathWorld
•  Éléments de géométrie - L'inversion  by Arnaud Bodin, in French
•  Inversion géométrique - ChronoMath  by Serge Mehl, in French
•  L'inversion  (Catholic University of Louvain - Belgium), in French
•  Ceci n'est pas une géodésique ! (hyperbolic line)  in French - PDF version

August 2012
updated 02-09-2012