Inversion in Higher Dimensions
In the spirit of generalization to higher dimensions, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversion in an n-sphere:
where r is the radius of the inversion.
In 2 dimensions, with r = 1, this is circle inversion with respect to the unit circle.
As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is simply a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.
A remarkable fact about higher-dimensional conformal maps is that they arise strictly from inversions in n-spheres or hyperplanes and Euclidean motions: see Liouville's theorem (conformal mappings).
Read more about this topic: Inversive Geometry
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