Normal Distribution on the Rotation Group So(3)

We study the normal distribution on the rotation group SO(3). If we take as the normaldistribution on the rotation group the distribution defined by the central limit theorem inParthasarathy (1964) rather than the distribution with density analogous to the normaldistribution in Eucledian space, then its density will be different from the usual ( 1 / 2 π σ) exp ⁡ ( − ( x − m ) 2 / 2 σ 2 ) one. Nevertheless, many properties of this distribution will beanalogous to the normal distribution in the Eucledian space. It is possible to obtainexplicit expressions for density of normal distribution only for special cases. One of thesecases is the circular normal distribution. The connection of the circular normal distribution SO(3) group with the fundamentalsolution of the corresponding diffusion equation is shown. It is proved that convolutionof two circular normal distributions is again a distribution of the same type. Some projectionsof the normal distribution are obtained. These projections coincide with awrapped normal distribution on the unit circle and with the Perrin distribution on thetwo-dimensional sphere. In the general case, the normal distribution on SO(3) canbe found numerically. Some algorithms for numerical computations are given. Theseinvestigations were motivated by the orientation distribution function reproductionproblem described in the Appendix.