Inversive Geometry - Inversion in Higher Dimensions

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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