Planar and Spatial Rigid Motion as Special Cases of Spherical and 3-Spherical Motion

Abstract

This paper examines spherical and 3-spherical rigid motions with instantaneous invariants approaching zero. It is shown that these motions may be identified with planar and spatial motion, respectively. The instantaneous invariants are ratios of arc-length along the surface of the sphere to its radius, thus the process of shrinking their value may be viewed as expanding the sphere while bounding the instantaneous displacements on the sphere. This allows a smooth transformation of the results of the curvature theory of spherical and 3-spherical motion into their planar and spatial counterparts.