Study On The Dimensions Of Spline Spaces Over L-meshes

Traditional tensor-product B-spline functions which are a basic tool in the design of free-form surfaces, are defined over special L-meshes, where no T-junctions and L-junctions appear. B-spline surfaces have the drawback that arises from the mathematical properties of the tensor-product B-spline basis functions. For example, it do not allow local modification of the domain partition. If we want to construct a surface which is flat in the most part of the domain, but sharp in a small region, to maintain the tensor-product mesh structure, we have to add many superfluous control points which are a big burden to modeling systems. So to overcome this limitation we need meshes which allow local refinement. So the study on the dimensions of spline spaces over L-meshes is very necessary.A L-mesh (i.e. Δ_L )is basically a rectangular grid that allows T-junctions and L-junctions. A T-mesh (i.e. Δ_T)is basically a rectangular grid that allows T-junctions. Deng et al. proposed a method based on B-nets to calculate the dimensions of spline spaces over T-mesh, when the smoothness is less than half of the degree of the spline functions (m≥2a + 1,n≥20 +1). Chong-Jun Li , Ren-Hong Wang ,Feng Zhang obtained the dimensions of spline spaces over T-meshes s(m,n,a, β,Δ_T) by using the Smoothing Cofactor-Conformality method. The result of Deng etal is a special case of their result. In this paper, we study on the dimensions of spline spaces S(m, n,a, β,Δ_L) over L-meshes by using the similar method. We also study on the Applicable Conditions of the dimension formula of S(m,n,a, β,Δ_L) over some special L-meshes, and expand the area in which the dimension formula works.