Study Of Splines Research Articles

Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.

Counting the dimension of splines of mixed smoothness

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.

Algebraic methods in approximation theory

This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis; their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in Rk, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree. The objects appearing here may be computed using the spline package of the Macaulay2 software system.

Spline space and its B-splines on an n+1 direction mesh in [formula omitted

It is well known that splines play an important role in many fields, especially, their close relationship with wavelets makes them have more widespread applications in numerous scientific and engineering domains. Univariate and bivariate splines have been well studied and lots of results have been obtained. Because of the intrinsic difficulty between bivariate case and higher-dimension (three or more dimensions) settings, the study of splines on higher dimensions are very limited. For example, the study of the bivariate splines on a three-direction mesh triangulation has obtained many important and excellent results, but almost all of those results have no analog generalization to higher dimensions. In this paper, we will study the higher-dimension splines defined on n+1 mesh simplical partitions which is the analog of bivariate splines on three-mesh triangulations. We have also pointed out many interesting differences between bivariate splines and higher-dimensional cases. Our main results are that, similar to bivariate and trivariate cases, a necessary and sufficient condition for S k r ( Δ) to contain a B-spline is k⩾ 1 2 (r+1)(n+1) for r⩾1 being odd and k⩾ 1 2 r(n+1)+1 for r⩾0 being even.