February 2006
Intermediate to advanced
600 pages
8h 57m
English
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