黎曼流形上参数样条曲线的研究与应用

With the development of the modern science and technology and the improvement of the industry standard, the description, analysis and processing of the curve information in Riemannian manifold are widespread problems in the field of modern industrial design, and have very broad application requirements. This project takes the spline in Riemannian manifold as the research object. The main researches include the construction and shape optimization of the spline and the implementation of the efficient algorithm. Firstly in virtue of Riemannian geometry, this project constructs generalized rational Bézier curves in Riemannian manifold by the generalization of the classical rational de Casteljau algorithm, then give a new spline method in Riemannian manifold. Secondly On the basis of computational geometry, the geometric meaning of weights of generalized rational Bézier curves in Riemannian manifold is researched further, the effects of free parameters on the shape of spline are analyzed quantitatively and a new algorithm for choosing parameters optimally is designed. Finally, this project designs efficient and robust algorithms for applying the new spline method in Riemannian manifold to the field of aerospace, machinery manufacturing, CNC machining, 3D printing, panorama stitching and computer animation. The studies of this project are expected to obtain meaningful results in the above field.

随着现代科学技术的不断进步以及工业标准的不断提高，流形上曲线信息的描述、分析和处理是现代工业设计领域广泛存在的问题，具有非常广阔的应用前景。本项目拟以黎曼流形上的样条曲线为研究对象，围绕其构造表示方法、形状的优化控制、高效算法的设计与实现展开理论与应用研究。借助黎曼几何为理论支撑，通过推广有理de Casteljau算法，构建黎曼流形上广义有理Bézier曲线，提出流形上新的样条表示方法；借鉴计算几何的思想和方法，深入研究黎曼流形上广义有理Bézier曲线权因子几何意义，定量分析自由参数对样条曲线形状的影响，提出自由参数的优化选取算法；以上述研究为理论基础，设计高效鲁棒的交互式优化设计算法，实现广义有理Bézier样条曲线在路径规划与设计、曲面剪裁与分割以及3D刚性物体运动控制中的应用，获得一系列有意义的应用成果。

本项目主要以黎曼流形上的样条曲线为研究对象，围绕其构造表示方法、形状的优化控制、高效算法的设计与实现展开研究。以黎曼几何为理论支撑，通过推广有理德卡斯特里奥算法，我们构建了黎曼流形上广义有理Bézier曲线，提出了流形上新的样条表示方法。基于球面Bézier曲线，我们提出了一种仅利用插值点位置信息便可在任意维空间中构造球面C2 Hermite插值样条的新方法。数值实验表明，新方法适用范围更广，局部性质更好，灵活度更高。..为了使DP曲线具有形状调节的功能，我们提出了一类带有形状参数的三次DP曲线，并对曲线的性质进行了深入的研究。进一步地，基于三次DP曲线，从平面PH曲线的定义出发，我们构造了一类新的曲线——三次PH-DP曲线，并进一步分析了DP曲线和DP-PH曲线的误差。..由于最佳码本具有众多良好的性质，所以其在信号处理、图论以及组合设计中有着广泛的应用。我们对近似最佳码本问题进行了研究，给了出两类构造近似最佳码本的新方法。