曲率单调的NURBS曲线插值容许G2 Hermite数据的方法

Two points, together with their tangents and curvatures are named as G2 Hermite data. Admissible G2 Hermite data is the Hermite data that there exists a monotone curvature curve interpolating it. This project presents a framework to solve the problem of interpolating admissible G2 Hermite data by NURBS curves with monotone curvature. The four steps are as follows: (1) determining the sufficient conditions for a NURBS curve to be a monotone curvature curve; (2) determining the sufficient conditions for the admissible G2 Hermite data which can be interpolated by NURBS curves with monotone curvature, we call such data as fair-interpolated G2 Hermite data; (3) splitting the admissible G2 Hermite data into a group of fair-interpolated G2 Hermite data by inserting points and their tangents and curvatures; (4) interpolating the fair-interpolated G2 Hermite data by a group of NURBS curves with monotone curvature and get the interpolation curve. Most of the existed methods for interpolating admissible G2 Hermite data find the curves with monotone curvature according to the admissible G2 Hermite data, while we find the fair-interpolated G2 Hermite data according to the interpolation curves with monotone curvature. Compared with the NURBS curves, G2 Hermite data is simple, so we avoid the complexity and difficulty of controlling the curvature of NURBS curve and the problem of interpolating admissible G2 Hermite data is easier than before.

存在曲率单调的曲线插值的G2 Hermite 数据(两点及其切向、曲率)称为容许G2 Hermite 数据。本项目研究用曲率单调的NURBS 曲线插值容许G2 Hermite 数据的方法。该方法分为四个步骤：（1）确定NURBS 曲线曲率单调的充分条件；（2）确定能用曲率单调的NURBS 曲线插值的容许G2 Hermite 数据，称之为可光顺插值G2 Hermite 数据；（3）插入若干点及其切向、曲率把待插值容许G2 Hermite 数据分解为若干组可光顺插值的G2 Hermite 数据；（4）用曲率单调的NURBS 曲线插值每组可光顺插值的G2 Hermite 数据即得插值曲线。本项目的特色是根据曲率单调的插值曲线找可光顺插值的G2 Hermite 数据。与NURBS曲线相比，可光顺插值的G2 Hermite 数据简单直观，所以化解了NURBS 曲线难以控制曲率的难点。

给定容许G2 Hermie 数据和指定的NURBS 曲线类型，本项目旨在求出插值该容许G2 Hermite 数据且曲率单调的NURBS 曲线。项目执行3年以来，项目组成员齐心协力，刻苦攻关，在用曲率单调的NURBS曲线插值容许G2 Hermite数据方面取得了一定的进展，为拓展其在计算机辅助几何设计领域的进一步应用打下了基础。在计算机图形学和计算机辅助几何设计领域国际重要期刊Computer Aided Geometric Design发表论文1篇, 国内期刊《计算机辅助设计与图形学学报》发表论文2篇。另在其它期刊发表论文7篇，其中SCI收录论文4篇，协助培养硕士研究生6名。项目执行期间，协助主办国际会议1次，项目组成员赴国(境)外学术交流1次，参加国内会议4人次，邀请国(境)外学者来访6人次，国内学者来访4人次。主要成果如下：1. 提出了基于双圆弧插值的G2 Hermite数据容许分割方法，并根据双圆弧连接弧的不同情形，提供了3种不同的算法，发表于《计算机辅助设计与图形学学报》。2. 证明了非均匀三次B 样条曲线插值的GS-PIA(Gauss-Seidel Progressive Interpolation and Approximation，简称GS-PIA)算法的收敛性，发表于 《计算机辅助设计与图形学学报》。3. 发现了平面双线性映射的特征圆锥曲线，发表于Journal of Computational and Applied Mathematics。4. 基于局部算子实现2N点插值细分的方法，提高了计算新点的效率且无需存储系数，发表于Computer Aided Geometric Design。5. 等距节点上有理Berrut插值的Lebesgue常数的一个紧上界，极限情况下与最佳上界一致。发表于Applied Mathematics Letters。6. 细分曲线与其控制多边形的距离估计，优于以往的距离估计公式，发表于Applied Mathematics and Computation。7. 对偶2n的细分方法的局部算子及其插值性质，发表于国际期刊Journal of Computational and Applied Mathematics。8. 基于双参数的几何细分法，其极限曲线G1连续，且易于调整，发表于国内期刊《图学学报》。