Classification of the image curves of spherical four-bar linkages

The coupler motion of a spherical fourbar linkage is a continuous set of rotations in three dimensions. Rotations can be mapped to points in three dimensional projective space in which case the coupler motion becomes a curve in this space. In this paper we present the projective classification of the image curves of the coupler motion. This is a generalized eigenvalue problem and the categorization of possible eigenvalues of a certain pencil of quadrics identifies all the singular cases of coupler motion. The characteristic equation yields a condition for the rotatability of cracks which is analogous to Grashof criterion for planar four-bar linkages.