Geometrical Foundations of Texture Analysis. Geodesic Curves and Motions in the group Space of Three‐Dimensional Rotations

Principal concepts and selected results relating to the inner geometry of the three-dimensional rotation group SO(3) are presented in a form which is appropriate for further applications to various problems of texture analysis. Starting from the basic concepts of regular and piecewise regular curves in the group space SO(3) we consider the functional of the angular length and introduce further geodesic curves. It is shown that the geodesics can be fully characterized, in the group-theoretical terms, as cosets of all possible one-parametric subgroups in the space SO(3). Two kinds of parallelism between geodesics in the group space are discussed as well as related congruences. Geodesic curves are characterized also in terms of their constitutive vectors. The related transformational rules under motions are obtained. The geometrical structure of general motions and non-euclidean rotations of the space SO(3) is described on the base.