New Proof of Dimension Formula of Spline Spaces over T-meshes via Smoothing Cofactors ∗1)

T-meshes are formed by a set of horizontal line segments and a set of vertical line segments, where T-junctions are allowed. See Figure 1 for examples. Traditional tensor-product B-spline functions, which are a basic tool in the design of freeform surfaces, are defined over special T-meshes, where no T-junctions appear. B-spline surfaces have the drawback that arises from the mathematical properties of the tensor-product B-spline basis functions. Two global knot vectors which are shared by all basis functions, do not allow local modification of the domain partition. Thus, if we want to construct a surface which is flat in the most part of the domain, but sharp in a small region, we have to use more control points not only in the sharp region, but also in the regions propagating from the sharp region along horizontal and vertical directions to maintain the tensor-product mesh structure. The superfluous control points are a big burden to modelling systems. In [5], Sederberg etal explained the troubles made by these superfluous control points in details. To overcome this limitation, we need the local refinement of B-spline surfaces, i.e. to insert a single control point without propagating an entire row or column of control points. In [4] hierarchical B-splines were introduced, and two concepts were defined: local refinement using an efficient representation and multi-resolution editing. In principle, Hierarchical B-splines are the accumulation of tensor-product surfaces with different resolutions and domains. Weller and Hagen [8] discussed tensor-product splines with knot segments. In fact, they defined a spline space over a more general T-mesh, where crossing, T-junctional, and L-junctional vertices are allowed. But its dimensions are estimated and its basis functions are given over the mesh induced by some semi-regular basis functions. In 2003, Sederberg etal [5] invented T-spline. It is a point-based spline, i.e., for every vertex, a blending function of the spline space is defined. Each of the blending functions comes from some tensor-product spline space. Though this type of splines supports many valuable operations within a consistent framework, but some of them, say, local refinement, are