Research On The Spherical Beta Splines

With the maturity of computer technology and the increasing development of modern industry,the spline curves fitting technology on high-dimensional sphere has been widely used in key frame animation and aerospace vehicle trajectory planning.In practice,it is often necessary to use spherical splines with well continuity and high shape controllability to interpolate points on high-dimensional spheres.So far,the existing construction methods have some limitations: some of the parameters are too cumbersome,so the applicability of the proposed construction method in practical application is not high.some methods have lost their fundamental properties after extending the splines with good control performance in Euclidean space to the high-dimensional sphere.some methods can't be extended to high-dimensional space and are limited to low-dimensional space.Therefore,based on the knowledge of spherical Bézier curves and beta spline curves,the thesis provides a new method for constructing geometrically continuous spherical interpolation splines on any dimensional spheres.Firstly,The Beta splines in Euclidean space are extended to the high-dimensional spheres.The spherical Beta spline curves are defined,and its continuity is discussed.Secondly,the relationship between the auxiliary control vertex and the interpolation point is clarified.The algorithm of using known interpolation points to calculate auxiliary control vertices is given.Then,a geometrically continuous spherical beta interpolation spline curves are constructed by using the above algorithm,and a new method for constructing spherical interpolation spline curve is obtained.Finally,the construction of spherical beta interpolation spline curve is realized on two-dimensional sphere,which proves the feasibility of this method.This method is applicable to the construction of arbitrary dimensional spline curves.The parameters that can control the shape of spline curves are introduced,so that the constructed spline curves have flexible controllability.Numerical experiments show the specific influence of the change of the two shape parameters on the shape of the spline curve,and verify that the properties of the shape parameters of beta spline curves in Euclidean space are still applicable on the sphere.