Abstract
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 (Trans. Am. Math. Soc. 310(1), 325–340, 1988). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is 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.
Highlights
Piecewise-polynomial functions called splines are foundational pillars that support modern computer-aided geometric design [1], numerical analysis [2], etc
We study bivariate spline spaces—i.e., n = 2— of mixed smoothness—i.e., different orders of smoothness constraints are imposed across different edges of the partition
Piecewise-polynomial splines are extensively applied in the fields such as computeraided geometric design [1] and numerical analysis [31]
Summary
Piecewise-polynomial functions called splines are foundational pillars that support modern computer-aided geometric design [1], numerical analysis [2], etc. In order to examine the sufficient conditions in practice, we narrow our focus down to a specific application: dimension computation for spline spaces on meshes of arbitrary topologies; e.g., see Fig. 1. Such splines enable geometric modeling of and numerical analysis on arbitrary smooth surfaces [19], and are very useful in applications. – In Section 4.3, we derive simple geometric conditions that allow a combinatorial dimension formula for mixed smoothness splines with non-uniformly chosen bi-degrees on T-meshes, extending the results from [14, 18]. This section will introduce the relevant notation that we will use for working with polynomial splines on polygonal meshes
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