A DERIVATION OF THE POLE CURVE EQUATIONS IN THE PROJECTIVE PLANE

Traditional four-position mechanism synthesis focuses on the poles corresponding to four displacements of a rigid body. From these poles equations for the center-point and circle-point curves are developed. However, if a pole is infinite (the associated displacement is a pure translation), the established derivations for the pole curve equation break down. One can eliminate this problem by expressing the pole curve equation in the projective plane, because all points, including points at infinity, have finite homogeneous coordinates. In this paper, the pole curve equation is derived in the projective plane. This projection derivation is an analytic expression of Alt's graphical construction of the pole curve. The pole curve is the intersection of a pair of projective pencils of circles (one pencil for opposite sides of the opposite-pole quadrilateral) which are defined by the homogeneous coordinates of the poles. The resulting equations are applied to several problems in the synthesis of four-bar linkages. In addition the rotation curves, the locus of displacement poles for a four-bar linkage, are computed for linkages with sliding joints.