曲线曲面降阶方法综述

Overview of curves and surfaces degree reduction

曲线曲面的降阶变换处理是指对于一条(片)给定次数的几何曲线曲面,找到另外一条(片)或多条(片)更低次数的同类型曲线曲面来逼近它,并使误差保持在预先给定的范围内.在计算机辅助设计/制造(CAD/CAM)领域中,由于不同造型系统间经常需要进行几何描述信息的数据交换或数据集成,作为曲线曲面的基本运算,降阶运算具有非常重要的意义.因而降阶逼近算法的理论研究成了当前的热点之一.此综述结合作者在该领域的最新研究成果,从Bézier曲线降阶、有理Bézier曲线降阶、B样条曲线降阶、Bézier曲线降多阶以及曲面降阶等方面综述了近年来国内外专家学者开展曲线曲面的降阶逼近研究的方法、成果及工业应用情况.

Degree reduction means finding a lower degree approximate curve or surface within certain error tolerance for a given curve or surface of certain degree. In the field of computer aided design/manufacturing (CAD/CAM), data exchange and data integration of geometric description information are always needed among different modeling systems. Degree reduction, as the basic operation of curves and surfaces, plays an important role. Thus degree reduction becomes one of the hottest topics in researches of computer aided geometric design. This paper gives an overview about the methods, results and applications of curves and surfaces degree reduction research in recent years, from degree reduction of Bézier curves, rational Bézier curves, B-spline curves, multi-degree reduction of Bézier curves and degree reduction of surfaces, combined with the authors' work in this area too.