Kinematic Geometry of Spherical Evolutes

Introduction In the last few decades, the study of spherical kinematics has been extensive. Veldkamp (1967) examined the analogies of spherical kinematics to plane kinematics. He derived the classic theorems of Kennedy, Euler-Savory, and Bobillier in spherical kinematics. Both Kamphius (1969) and Veldkamp (1969) studied spherical oscillating slider-crank mechanisms. They determined the locus of inflection-points and Ball-points for the coupler link of the mechanism. Kamphius (1969) used instantaneous invariants that would be used later by Let us consider a point P in a rigid body rotating about a fixed point O. The point P generates a spherical curve. Since the scale may be arbitrary, we set the length of the ray OP to unity. The spherical curve, lying on a unit sphere of center O, is denoted by C s . The curve C s represents the trajectory of point P. Corresponding to a position of P on C s , there is a ray emanating from O passing through the center of curvature P e of C s . As P moves along the trajectory C s , the ray generates a cone. The intersection of the cone and the unit sphere yields a unique spherical curve. We define this spherical curve, denoted by E s , as the spherical evolute of the spherical trajectory C s