The Curvature Theory of Line Trajectories in Spatial Kinematics

TLDR

Lines on ruled surfaces can be used to synthesize spatial motions in the same way that points on inflection circles and curvature cubics synthesize planar motion. This paper develops the differential properties of ruled surfaces in a form applicable to spatial kinematics. The authors derive three curvature parameters and related quantities for ruled surfaces, use them to characterize lines that instantaneously generate special trajectories, and illustrate the method with loci of lines producing helicoids and right hyperboloids.

Abstract

This paper develops the differential properties of ruled surfaces in a form which is applicable to spatial kinematics. Derivations are presented for the three curvature parameters which define the local shape of a ruled surface. Related parameters are also developed which allow a physical representation of this shape as generated by a cylindric-cylindric crank. These curvature parameters are then used to define all the lines in the moving body which instantaneously generate speciality shaped trajectories. Such lines may be used in the synthesis of spatial motions in the same way that the points on the inflection circle and cubic of stationary curvature are used to synthesize planar motion. As an example of this application several special sets of lines are defined: the locus of all lines which for a general spatial motion instantaneously generate helicoids to the second order and the locus of lines generating right hyperboloids to the third order.