The Morgan Kaufmann Series in Interactive 3D Technology: Visualizing Quaternions

Appendix I. Hyperspherical Geometry

Our quaternion analysis has made essential use of the properties of the three-sphere S³ and its spherical geometry. Spheres themselves exist in all dimensions, and have a list of similar properties that may help to answer some general questions about the special case S³ that has been so important to us, and to place it in a useful general context.

Definitions

A sphere S^N is most naturally defined as the embedding of the constant-radius equation in (N + 1)-dimensional Euclidean space ^N+1:

(x1)2 + ··· + (xN)2 + (xN+1)2 = r2.

A hypersurface element, or solid-angle integral, of a hypersphere S^N is [47,166]

The volume of ...