On Explicit Equations of the Burmester Curves for the PPP-P Case in Spherical Motion

In this paper, an analytic treatment to investigate the Burmester curves of the PPP-P case in spherical motion is presented. With a rotation angle of a coupler about a pole and an instant centre associated with an inflection point specified, a geometric construction is established to determine the corresponding spherical centre-points and circle-points. According to the geometric construction, the explicit equations of spherical Burmester curves for the PPP-P case are directly derived by using spherical trigonometry, without the use of differential technique. The explicit equations obtained give a clear insight into the geometric properties and special cases of the spherical Burmester curves, which include degenerated curves and symmetric curves. A design application is presented.