Distributions of Rotation Axes for Randomly Oriented Symmetric Objects

The orientation of an object is determined by rotation angle and rotation axis. In the case of randomly oriented nonsymmetric objects, the distribution of rotation axes is uniform. But symmetry destroys uniformity. An orientation of a symmetric object can be represented by different but symmetrically equivalent sets of parameters. To obtain unique parameters of each orientation, possible rotations are confined (from among those that are symmetrically equivalent) to those that have the smallest rotation angle. The problem considered here is what is the distribution of rotation axes for such a set of rotations assuming the objects are oriented randomly. A similar situation occurs for misorientations and this issue is of importance for analyses of crystallite misorientations and of geometry of grain boundaries in polycrystalline materials. The paper gives the distributions of rotation axes for all proper symmetries in three-dimensional space.