Morphing of spherical closed curves

Abstract

Shape morphing is a continuous deformation in time between two shapes (curves, surfaces, ...). For planar curves, most efficient methods for blending between two closed curves are based on the construction of the morph curve involving its signed curvature function. The latter is obtained by linear interpolation of the signed curvature functions of the source and target curves (Sederberg et al. (1993), Saba et al. (2014) and Surazhsky and Elber (2002)).When the key curves (source and target curves) are closed, the intermediate curve is not necessarily closed, but with a closing process, we look for a closed curve close enough to the open intermediate one. In this paper, we propose two algorithms for blending between two spherical closed curves such that the morph curves remain closed and spherical. Our two methods are based firstly on the approximation of smooth curves by geodesic polygons and secondly on the interpolation of the notion of discrete geodesic curvature and the spherical side lengths of polygons. We solve the problem of closing the morph geodesic polygon by imposing its closing conditions on the sphere and by minimizing the difference of discrete geodesic curvatures.