Research On The Dimensions Of Several Multivariate Spline Spaces On Special Partitions

It's well known that the dimension of a spline function space is very important for studying approximation theory of spline functions. However some dimensions of spline spaces over particular partitions not only depend on the topological quantities,but also depend on geometrical quantities.In 2003 Sederberg etal~[11] invented T-spline,which is a spline space over T-mesh.Using a method based on B-nets, Jiansong Deng etal~[10] gave the dimension of a spline function space over a T-mesh in 2004.When the smoothness is less than half of the degree of the spline functions, the formula derived involves only the topological quantities of the T-mesh.Because of the constrain the formula doesn't exist when the smoothness is close to the degree of the spline functions.In this paper we take advantage of the smoothing co-factor method to calculate the dimensions of spline function over some particular T-meshes and derive the more general dimension formula. Furthermore We explain the singularity of a spline space from geometric point of view,rnaking it more easier to understand.