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In higher dimensions

In a real n -dimensional Euclidean space, an inversion in the sphere of radius r centered at the point O = (o 1,..., o n) {\displaystyle O=(o_{1},...,o_{n})} is a map of an arbitrary point P = (p 1,..., p n) {\displaystyle P=(p_{1},...,p_{n})} found by inverting the length of the displacement vector P − O {\displaystyle P-O} and multiplying by r 2 {\displaystyle r^{2}}:

The transformation by inversion in hyperplanes or hyperspheres in E can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a dilation or homothety about the hyperspheres' center.

When two parallel hyperplanes are used to produce successive reflections, the result is a translation. When two hyperplanes intersect in an (n –2)- flat, successive reflections produce a rotation where every point of the (n –2)-flat is a fixed point of each reflection and thus of the composition.

Any combination of reflections, translations, and rotations is called an isometry. Any combination of reflections, dilations, translations, and rotations is a similarity.

All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry.

The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an n -sphere as the base space. The transformations of inversive geometry are often referred to as Möbius transformations. Inversive geometry has been applied to the study of colorings, or partitionings, of an n -sphere.