The Generalization of Line Trajectories in Spatial Kinematics to Trajectories of Great Circles on a Hypersphere

Abstract

This paper examines the instantaneous kinematics of motion defined by a parameterized set of 4 × 4 orthogonal matrices, termed hyperspherical motion. The transformation equation defining the trajectories traced by planes through the origin of the moving hypersphere is presented. These planes intersect the hypersphere in great circles. This transformation takes the convenient form of “double spherical motion” with the introduction of matrices and vectors with double number elements. A limit process exhibits the kinematics of line trajectories in space and its dual number formulation as a special case of the kinematics of the hyperspherical trajectories of great circles.