The Cayley-Bacharach Theorem for Continuous Piecewise Algebraic Curves over Cross-cut Triangulations  

作　　者：Renhong WANG Shaofan WANG 

机构地区：[1]School of Mathematical Sciences, Dalian University of Technology, Dalian 116085, P. R. China and National Engineering Research Center of Digital Life, Guangzhou 510006, P. R. China [2]Beijing Key Laboratory of Multimedia and Intelligent Software Technology, College of Computer Science and Technology, Beijing University of Technology, Beijing 100124, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2011年第9期1717-1724,共8页数学学报（英文版）

基　　金：The first author is supported by National Natural Science Foundation of China （Grant Nos. U0935004, 11071031, 11001037, 10801024） and the Fundamental Research Funds for the Central Universities （Grant Nos. DUT10ZDll2, DUT10JS02）; the second author is supported by the 973 Program （2011CB302703）, National Natural Science Foundation of China （Grant Nos. U0935004, 60825203, 61033004, 60973056, 60973057, 61003182）, and Beijing Natural Science Foundation （4102009） We thank the referees for valuable suggestions which improve the presentation of this paper.

摘　　要：A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.

关 键 词：Bivariate spline function piecewise algebraic curve Cayley-Bacharach theorem 

分 类 号：O187.1[理学—数学] O157.5[理学—基础数学]