Geometrically Continuous Interpolation in Spheres

Abstract:

      In this paper, a new method for geometrically continuous interpolation in spheres is proposed. The method is entirely based on the spherical B\'ezier curves defined by the generalized de Casteljau algorithm. Firstly we compute the tangent directions and curvature vectors at the endpoints of a spherical B\'ezier curve. Then, based on the above results, we design a piecewise spherical B\'ezier curve with $G^1$ and $G^2$ continuity. In order to get the optimal piecewise curve according to two different criteria, we also give a constructive method to determine the shape parameters of the curve. According to the method, any given spherical points can be directly interpolated in the sphere. Experimental results also demonstrate that the method performs well both in uniform speed and magnitude of covariant acceleration.