球面Beta样条曲线相关问题研究

Research on the Spherical Beta Spline Curse

摘要: 计算机技术的成熟，使得高维球面上的样条曲线拟合技术在关键帧动画和航空航天飞行器运行轨迹规划等领域都得到了广泛的应用。实际中常需用连续性好，形状可控性高的球面样条曲线来插值高维球面中的数据点。因此，本文的研究目的是提供一种可在任意维球面上构造几何连续的球面插值样条曲线的新方法。首先，将欧氏空间中Beta样条曲线推广到高维球面上，定义了球面Beta样条曲线，并对其连续性进行了讨论；然后，明确了辅助控制顶点与插值点之间的关系；最后，利用已知的插值点来计算辅助控制顶点，从而构造出几何连续的球面Beta插值样条曲线。该方法属于整体构造方法，松弛了连续要求，引入了可以控制样条曲线形状的参数，从而使得构造的样条具有灵活控制性。

Abstract: The maturity of computer technology has made the spline curve fitting technology on the high-dimensional spheres widely used in the fields of 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. Therefore, the purpose of this paper is to provide a new method for constructing geometrically continuous spherical interpolation splines on any dimensional spheres. First, the Beta spline in Euclidean space is extended to the high-dimensional spheres. The spherical Beta spline curve is defined, and its continuity is discussed. Second, the relationship between the auxiliary control vertex and the interpolation point is clarified. Third, the known interpolation points are used to calculate the auxiliary control vertices, all then a geometrically continuous spherical Beta interpolation spline is constructed. This method has overall properties. It relaxes the continuous requirements and introduces parameters that can control the shape of the spline curve, which makes the spherical splines more flexible and controllable.