Morgan-Scott型剖分上的样条空间的奇异性
On Singularity of Spline Space Over Morgan-Scott's Type Partition

DOI：10.3770/j.issn:1000-341X.2010.01.001

 作者 单位 罗钟铉 大连理工大学应用数学系, 辽宁 大连 116024 刘凤山 特拉华州立大学应用数学研究中心, 美国 多佛 19901 施锡泉 特拉华州立大学应用数学研究中心, 美国 多佛 19901

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines (or ponits), and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space $S_{\mu 1}^\mu(\Delta_{MS}^\mu)$ are geometrically given for any smoothness $\mu$ by recursion. Moreover, the famous Pascal's theorem is generalized to algebraic plane curves of degree $n\geq 3$.