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

Chapter 28. Spherical Riemannian Geometry

We have already mentioned many times that the principal advantage of using unit quaternions to represent 3D orientation frames is the fact that the topological space of unit quaternions is the three-sphere S³ and that S³ is a simply-connected manifold with a natural and elegant distance measure or metric. We have exploited this property in a variety of ways to facilitate the study of particular applications. In this chapter we complete this story by writing down the standard elements of the Riemannian geometry of S³, so that when heuristic methods provide insufficient power or insight, the rigorous mathematics is in principle at our disposal. (For extensive treatments, see for example Gray [61], Lee