The algebraic classification of the image curves of spherical four-bar motion

A rotational displacement of the coupler of a spherical four-bar linkage can be mapped to a point with coordinates given by the Euler parameters of the rotation. The set of rotational movements available to the coupler defines a curve in this three-dimensional projective space (four homogeneous coordinates). In this paper, we determine the generalized eigenvalues and eigenvectors of the pencil of quadrics that pass through this curve and examine their properties. The result is an algebraic classification of the image curves that parallels the well-known classification of spherical four-linkages. In addition, we find that the characteristic polynomial of the system yields Grashof’s criterion for the rotatability of cranks.