Toric曲面单值性与NURBS曲面近似offset的研究

Injectivity of parametric surfaces means there exists no self-intersection. Injectivity plays an important role in reparameterization, surface morphing, Finite Element Analysis and iso-geometric Analysis. Computing the offsets of NURBS surfaces are the research hotspot in Computation Geometry, Computer Aided Design, Computer Aided Geometric Design etc.. Self-intersection is considerable in computing offset of NURBS surface, most of the existing research focus on computing the self-intersections by numerical methods, rather than the existence of the self-intersections. Then how to identify the injectivity of parametric surface and Bézier polytope, that is there exists no self-intersections in surfaces/polytope, has a strong theoretical and application value for geometric modelling field. This project will do some research on injectivity of Bézier polytope and n-dimension toric surface, together with computing non-self-intersecting approximate offset of NURBS surface. The main contents are as follows: Give the sufficient and necessary conditions and identify algorithms of injective Bézier simplex、Bézier simploid、Bézier volume; Give the sufficient and necessary condition and identify algorithm of injective n-dimension toric surface; On the basis of the completed research on injectivity of NURBS surfaces of the applicant, the project will propose a new algorithm of computing non-self-intersection offset of NURBS surface by considering the related exists algorithms.

参数曲面没有自交点称为具有单值性。单值性在曲面重新参数化、曲面变形、有限元分析等领域有重要的应用。NURBS曲面offset的计算，一直是计算几何、计算机辅助设计、计算机辅助几何设计等领域的研究热点。自交点问题在offset计算中是不可忽视的，目前的研究主要利用数值方法求解自交点，而非其存在性的判定。因此如何准确的判定单值性，对几何造型领域具有很强的理论和应用价值。本项目拟开展Bézier polytope、n维toric曲面的单值性与非自交NURBS曲面近似offset的计算的研究。包括给出Bézier simplex、Bézier simploid、Bézier volume单值性的充要条件及判定；n维toric曲面单值性的充要条件及判定；在申请人已完成NURBS曲面单值性结论的基础上，拟结合其已有的offset算法，提出计算非自交NURBS曲面近似offset的算法。

本项目对toric曲面单值性与NURBS曲面近似offset进行研究，主要包括三维Bézier体的单值性、toric曲面的理论和应用、NURBS曲线曲面非自交的近似offset等内容。通过本项目研究，我们取得了如下成果：给出了三维Bézier体单值的充分必要条件及其判定的数值算法；给出了toric曲面的升阶公式和de Casteljau算法；改进了Bézier曲线单值判定算法；给出了广义toric曲线曲面的定义和基本几何性质；改进了二维toric曲面单值判定的数值算法；给出了一类模型的渐进稳定分析；提出一种非自交NURBS曲线近似offset计算；本项目还对NURBS曲面非自交近似offset计算数值方法和toric曲线曲面非自交近似offset计算数值方法进行研究给出相应的结果。本项目共计发表与录用论文6篇，其中SCI检索5篇。本项目将计算几何、计算机辅助几何设计、代数几何等方向结合研究toric曲面的单值性及offset曲线曲面，所得结论不仅是对计算几何领域的理论上的补充，同时对计算机辅助几何设计的应用方面有一定的贡献。