On Singularity of Spline Space Over Morgan-Scott's Type Partition
Received:June 17, 2008  Revised:October 08, 2008
Key Words: singularity of spline space   Morgan-Scott's partition   planar algebraic curve   characteristic ratio   characteristic mapping   characteristic number.  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.10771028; 60533060) and the program of New Century Excellent Fellowship of NECC, and is partially funded by a DoD fund (Grant No.DAAD19-03-1-0375).
Author NameAffiliation
Zhong Xuan LUO School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Feng Shan LIU Applied Mathematics Research Center, Delaware State University, Dover 19901, U. S. A 
Xi Quan SHI Applied Mathematics Research Center, Delaware State University, Dover 19901, U. S. A 
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Abstract:
      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 $n\geq 3$.
Citation:
DOI:10.3770/j.issn:1000-341X.2010.01.001
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