We study the normal distribution on the rotation group SO(3). If we take as the normal
distribution on the rotation group the distribution defined by the central limit theorem in
Parthasarathy (1964) rather than the distribution with density analogous to the normal
distribution in Eucledian space, then its density will be different from the usual (1/2πσ)
exp(−(x−m)2/2σ2) one. Nevertheless, many properties of this distribution will be
analogous to the normal distribution in the Eucledian space. It is possible to obtain
explicit expressions for density of normal distribution only for special cases. One of these
cases is the circular normal distribution.The connection of the circular normal distribution SO(3) group with the fundamental
solution of the corresponding diffusion equation is shown. It is proved that convolution
of two circular normal distributions is again a distribution of the same type. Some projections
of the normal distribution are obtained. These projections coincide with a
wrapped normal distribution on the unit circle and with the Perrin distribution on the
two-dimensional sphere. In the general case, the normal distribution on SO(3) can
be found numerically. Some algorithms for numerical computations are given. These
investigations were motivated by the orientation distribution function reproduction
problem described in the Appendix.
