“Normal” Orientation Distributions

Analogues of the normal distribution in Euclidean space for orientations represented by Rodriguesparameters are discussed. It is emphasized that different characterizations of the normal distributionin Euclidean space lead to different distributions in other spaces, none of which is mathematicallysuperior to any other one. Particular analogues of the normal distribution are the Binghamdistribution onS + 4for the purposes of mathematical statistics, and the Brownian motion distributiononS + 4in terms of probability theory and stochastic processes. It is reminded of the fact that a simpleanalogue of the central limit theorem in Euclidean space does not exist for the hyperspheres S P andprojective hyperplanesH P − 1 = S + 4 .