S-\lambda 曲线曲面造型方法和技术的研究

The theories and algorithms of curve and surface modelling are widely used in industrial manufacturing, medical, information technology, computer graphics and other areas. The demands of the practical problems and the introductions of the approaches of Mathematics and some inter-disciplines continually promoted the development of curve and surface modelling methods and technologies. Our project focuses on the S-\lambda curves and surfaces modelling methods, studies the approximation error, computer implemented technology, the stitching and cutting technologies of S-\lambda curves and surfaces by the theories and technologies of Approximation Theory and Computer-Aided Geometric Design, to meet the demand of the modelling method from theory to practical application. The research contents are showed as follows: 1) The estimation of approximation error on the S-\lambda curve (surface) and the aimed curve (surface) and the solving of the minimum error curve (surface); 2) The blossoming technology of S-\lambda modelling method; 3) The stitching and cutting technologies of S-\lambda curves and surfaces. The research results of our project will improve S-\lambda curves and surfaces modelling methods and technologies, play an important role in promoting the practical application of the modelling technology.

曲线曲面造型的理论与算法在工业制造、医学、信息技术、图形学等领域中有广泛的应用。实际问题的需要以及数学与其他一些交叉学科方法的引入促进了曲线曲面造型方法和技术的不断发展。本项目针对 S-\lambda 曲线曲面造型方法，根据逼近论与计算机辅助几何设计的理论和技术，研究了S-\lambda 曲线曲面的逼近误差、计算机实现技术、拼接和裁剪，满足该造型方法从理论到实际应用的需求。研究内容包括: 1) 研究S-\lambda 曲线(曲面)与目标曲线(曲面)或控制多边形(网格)之间误差估计与最小误差曲线(曲面)的确定；2) 研究S-\lambda 造型方法对应的开花技术；3) 研究 S-\lambda 曲线曲面的拼接与裁剪技术。本项目的研究成果将完善S-\lambda 曲线曲面造型方法和技术，对促进 S-\lambda 曲线曲面造型技术的实际应用具有重要的推动作用。

连续曲线曲面造型方法的研究是 CAGD 热点问题之一. S-λ曲线曲面作用一类包含传统 Bezier 曲线曲面造型方法的一般化造型框架, 对其进一步深入地研究有重要的意义. S-λ曲线曲面的造型理论是从概率论和算子逼近论发展而来. 因此, 进一步挖掘算子逼近论的理论在几何造型的应用具有一定的理论价值...本项目在论文《Bivariate tensor product (p,q)-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators》中定义了一类张量积 (p,q) 型 Kantorovich 类 Bernstein-Stancu-Schurer 算子及其基函数, 并研究了它们的收敛性质, 并通过数值例子验证了收敛性; 在论文《张量积 Baskakov-Szász-Mirakjan 曲面的构造及性质》中构造了新的曲面张量积 Baskakov-Szász-Mirakjan 曲面, 并研究了其基函数的非负性, 单位分解性, 线性无关性, 插值性; 推导出了基函数的导函数, 积分和升降阶公式...本项目所研究内容定义了一种能够用于造型的算子以及构造了一种新的张量积曲面, 拓展了计算机辅助几何设计能够表示的曲面范围, 具有重要的理论意义和应用背景.