Spatial motion-I: Points of inflection and the differential geometry of screws

Abstract

This paper examines infinitesimal spatial motion, introducing the concept of a “differential screw” to characterize the difference between two successive instantaneous screw axes (ISAs) on an axode. The locus of spatial inflection points, a twisted cubic curve common to three ruled quadric surfaces, is here derived in a coordinate system attached to the fixed axode, using the relationships of the ISA to velocity and the differential screw, as well as the ISA, to acceleration. The notable geometry is then discussed, and the more familiar instances of spherical and planar motion are described as special cases. The novel approach adopted in this paper, even though it does not yield any new results, is essential in laying the groundwork for Part II [ Mech. Mach. Theory 27, 17–35 (1992)] that deals with accelerations and the spatial equivalent of the Bresse circle.