Dimensions Of Bivariate Spline Spaces And Bivariate Weak Spline Spaces

A bivariate spline space S dr( ) defined on a triangulation of a planarpolygonal region is a set consisting of piecewise polynomials with degree d andglobal Cr smoothness, which has been widely applied to many fields, such as theapproximation theory, the computer aided geometry design (CAGD), waveletsand finite elements. The most important problem about the spline spaces is todetermine their dimensions.Generally, is restricted to be a regular triangulation. we consider to de-termine the dimensions of bivariate spline spaces over a kind of special regulartriangulations. Specifically speaking, a kind of so-called non-strictly degeneratetriangulations with degree 2 is firstly defined, which includes many well-knownrefined triangulations as its special cases, such as the Clough-Tocher refined tri-angulation CT, the type-1 Powell-Sabin refined triangulation PS1, the type-2Powell-Sabin refined triangulation PS2, the Wang refined triangulation W andthe triangulated quadrangulation +?, then the dimension of bivariate spline spaceS 72( ) is given, which is an extension of Alfeld and Schumaker's dimensionalresult over non-degenerate triangulations.Besides the general spline spaces, bivariate weak spline spaces are also use-ful in the CAGD, Hermit interpolations and finite elements. In the second partin this thesis, we turn to study the dimension of these kind of weak spline spacesdefined on appointed point triangulations associated to regular triangulations.First, by using the Be′zier-net method and the technique of minimal determin-ing sets, the dimension of bivariate weak spline space Wkμ(I1 ) with k = 2μisdetermined, where the triangulation is an union of a series of star regions, i.e.,, and I1 means the appointed point triangulation with one pointin the interior of each edge. This result improves Xu and Wang's work from thecase k > 2μto the case k≥2μ. Second, the dimension of bivariate weak splinespace W21(I1 ) is further determined, where the triangulation can be an arbitrary regular triangulation and I1 also means the appointed point triangulation withone point in the interior of each edge. As a cost of the improvement in triangu-lation from an union of a series of star regions to a general regular triangulation,positions of those appointed points have to be restricted to satisfy a geometricalcondition.