Abstract In the elementary theory of geometrical optics all the non-aberration properties of the symmetrical optical system may be derived from the three pairs of cardinal points, due to Gauss, of which two pairs only are independent. And it is the case, in the general theory of plane kinematics, that there are certain points playing a somewhat similar role. For, associated with any relative co-planar motion of two planes, there are two enumerable sets of points, from the configuration of either of which may be derived all the properties of the relative path of any, and every, point, or series of points, fixed in either plane. The configuration of each of these sets of points is uniquely characteristic of the particular relative motion of the two planes, and conversely; and gives a very simple and compact synthesis of the whole realm of plane kinematics. The purpose of the following investigation is to establish the existence of these ‘cardinal points’, as we may name them, in plane kinematics, and to examine some of their properties. In addition, various new curves and configurations are obtained, relating to the generation of the Burmester points, and similar points of higher orders; together with a generalization of certain kinematical results which have emerged, stage by stage, in the writings of Tschebycheff, Burmester, Muller and others.