Degree elevation operator and geometric construction of C-B-spline curves  

作　　者：ZHU Ping WANG GuoZhao YU JingJing 

机构地区：[1]Institute of Computer Graphics and Image Processing, Zhejiang University Hangzhou 310027, China [2]Laboratory of Scientific Computing, Southeast University, Nanjing 211189, China

出　　处：《Science China(Information Sciences)》2010年第9期1753-1764,共12页中国科学（信息科学）（英文版）

基　　金：supported by the the National Natural Science Foundation of China(Grant Nos.60773179,60970079,60933008);the National Youth Science Foundation of China(Grant No.60904070)

摘　　要：Unlike a Bezier curve, a spline curve is hard to be obtained through geometric corner cutting on control polygons because the degree elevation operator is difficult to be obtained and geometric convergence is hard to be achieved. In order to obtain geometric construction algorithm on C-B-splines, firstly we construct the degree elevation operator by using bi-order splines in this paper. Secondly we can obtain a control polygon sequence by degree elevation based on the degree elevation operator derived from a C-B-spline curve. Finally, we prove that this polygon sequence will converge to initial C-B-spline curve. This geometric construction algorithm possesses strong geometric intuition. It is also simple, stable and suitable for hardware to perform. This algorithm is important for CAD modeling systems, since many common engineering curves such as ellipse, helix, etc. can be represented explicitly by C-B-splines.Unlike a Bezier curve, a spline curve is hard to be obtained through geometric corner cutting on control polygons because the degree elevation operator is difficult to be obtained and geometric convergence is hard to be achieved. In order to obtain geometric construction algorithm on C-B-splines, firstly we construct the degree elevation operator by using bi-order splines in this paper. Secondly we can obtain a control polygon sequence by degree elevation based on the degree elevation operator derived from a C-B-spline curve. Finally, we prove that this polygon sequence will converge to initial C-B-spline curve. This geometric construction algorithm possesses strong geometric intuition. It is also simple, stable and suitable for hardware to perform. This algorithm is important for CAD modeling systems, since many common engineering curves such as ellipse, helix, etc. can be represented explicitly by C-B-splines.

关 键 词：C-B-spline degree elevation operator bi-order C-B-spline geometric convergence geometric construction 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术] O18[自动化与计算机技术—计算机科学与技术]