On certain configurations of the cardinal points in plane kinematics

Abstract

In plane kinematics, the knowledge of the location of the instantaneous centre — deriving from the eighteenth century — affords information which is partial and very incomplete. But it is the case that there always exists an enumerable set of cardinal points, as they may be named, having the following characteristic: all the properties of the path of any, and every, element or series of elements, fixed in the moving plane, are completely determined by the configuration of this set of cardinal points. In this manner is given a very simple and complete synthesis of the whole realm of plane kinematics.

Essentially, in plane kinematics, there is a duality; for we deal with the relative coplanar motion of two planes, depending, in the usual mechanism or linkage, upon a single parameter. There is then a dual set of cardinal points, the configuration of each set depending upon the assigned relative motion, and also upon the parameter. And it is the case that, for an assigned relative motion the configuration of either set, for a single value of the parameter, determines the configurations of both sets, for all values of the parameter. These changing configurations, corresponding to varying values of the parameter, possess therefore certain properties which are conserved throughout, and express the underlying unity of the particular relative motion. But, in certain cases, there is a conservation of a more special kind — a conservation of the form of the set itself. In the present paper are considered some cases of this special conservation of form, and we are led to investigate sets of cardinal points which may be named rectangular, linear, spiral, circular, and orthogonal — together with the kinematical implications of such sets.