Isotropic probabilistic models for directions, planes and referential systems

Although simple probabilistic models for directions in two–dimensional (2D) and three–dimensional (3D) spaces are well known (respectively, the von Mises and Fisher distributions), probabilistic models for directions, planes and referential systems in an n–dimensional (nD) space are either not popular or not available. The results for directions in 2D and 3D can be generalized for (i) arbitrary directions; (ii) planes; (iii) referential systems; and (iv) symmetric random tensors, and in turn, all these results can be generalized to nD spaces. In this paper we will focus on and build up isotropic probabilistic models for directions, planes and referential systems in nD space.