Geometrically Continuous Interpolation in Spheres

DOI：10.3770/j.issn:2095-2651.2012.04.001

 作者 单位 罗钟铉 大连理工大学数学科学学院, 辽宁 大连 116024 大连理工大学软件学院, 辽宁 大连 116600 王倩 大连理工大学数学科学学院, 辽宁 大连 116024

本文利用球面B\'ezier曲线,研究了球面上的几何连续插值问题,给出了一种在球面上构造几何连续的插值曲线的直接方法.首先计算出球面~B\'ezier~曲线在端点处的单位切矢及主法矢,然后构造出~\$G^1\$~和~\$G^2\$~连续的分段球面插值曲线,最后给出了确定插值曲线形状参数的方法.本文的方法克服了在构造~\$C^2\$~球面~B\'ezier~样条时对控制点选取的约束条件.数值实验表明, 用该方法构造出的球面插值曲线, 具有均匀变化的速度及加速度等特点.

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.