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.

形状变形是两个形状（曲线，曲面等）之间随时间的连续变形。对于平面曲线，在两个闭合曲线之间进行混合的最有效方法是基于变形曲线的构造，该变形曲线包含其有符号的曲率函数。后者是通过对源曲线和目标曲线的有符号曲率函数进行线性插值获得的（Sederberg等（1993），Saba等（2014）以及Surazhsky和Elber（2002））。

当关键曲线（源曲线和目标曲线）关闭时，中间曲线不一定是闭合的，但是在闭合过程中，我们要寻找与打开的中间曲线足够接近的闭合曲线。在本文中，我们提出了两种在两条球形闭合曲线之间混合的算法，以使变形曲线保持闭合和球形。我们的两种方法首先基于测地多边形对平滑曲线的逼近，其次基于离散测地曲率和多边形球面边长的内插。我们通过将变形测地线多边形的封闭条件强加于球体上并最大程度地减小离散测地线曲率的差异来解决封闭变形测地线多边形的问题。