Orientation maps: Techniques for visualizing rotations

The set of possible orientations of a rigid three-dimensional object is a topological space with three degrees of freedom. This paper investigates the suitability of various techniques of visualizing this space. With a good technique the natural distance between orientations will be represented fairly accurately, and distortion to the "shape" of a collection of orientations induced by the change of reference orientation will be minor. The traditional Euler-angle parameterization fails on both counts. Less well-known techniques exploit the fact that there is a rotation that takes the reference orientation to a given one. The given orientation is represented as a point along the axis of this rotation. The distance of this point from the origin is determined by some scaling function of the magnitude of that rotation. Free natural scaling functions are studied. None is perfect, but several are satisfactory