论文标题
计算混合光滑度的花纹的尺寸：一般配方及其应用于任意拓扑的网格
Counting the dimension of splines of mixed smoothness: A general recipe, and its application to meshes of arbitrary topologies
论文作者
论文摘要
在本文中，我们研究了多边形网格上混合平滑度的双变量多项式花纹的尺寸。在这里，“混合平滑度”是指在网格不同边缘上选择不同顺利的平滑度。为了研究这种花纹空间的尺寸，我们使用来自同源代数的工具。这些工具首先通过Billera（1988）应用于花样研究。使用它们，对样条空间维度的估计量相当于针对样条空间的广义Billera-Schenck-Stillman综合体的研究。特别是，当该络合物的一个位置中的同源性和零是微不足道的时，可以通过组合计算样条空间的尺寸。我们称这种样条空间为“下酰基”。在本文中，从较低隔离的样条空间开始，我们提出足够的条件，确保在放松跨网格边缘的子集的光滑度要求后获得的样条空间相同。该一般配方在特定的环境中应用：任意拓扑的网络。我们展示了如何使用我们的结果来计算带有孔的三角形，多边形网格和T网格的样条空间的尺寸。
In this paper we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, "mixed smoothness" refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from Homological Algebra. These tools were first applied to the study of splines by Billera (1988). Using them, estimation of the spline space dimension amounts to the study of the generalized Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions one and zero of this complex are trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces "lower-acyclic." In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes.
