On Singularity of Spline Space Over Morgan-Scott’s Type Partition

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 impossible to avoid 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. Using these concepts and the relevant results, a polished necessary and suffi-cient conditions to the singularity of spline space Sµµ+1(∆ µ MS) are geometrically given for any smoothness µ by recursion. Moreover, the famous Pascal’s theorem is generalized to algebraic plane curves of degree n ≥ 3