Subdivision and Spline Spaces

Schenck, Hal; Sorokina, Tatyana

doi:10.1007/s00365-017-9367-5

Cited by 11 publications

(10 citation statements)

References 8 publications

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“…It is remarked in [10,Remark 4.3] that for r = 1, the dimension formula (9) also holds even without the collinearity condition, from which we conclude that for an arbitrary n−1,n split:…”

Section: Theoremsupporting

confidence: 59%

“…It can be constructed by first making an Alfeld split A,n (= n,n ) of an n-simplex T using some interior point v. We then choose an interior point of each boundary face F (an (n−1)-simplex) of T and use it to split F into n subsimplices and then connect them to v. Let us say that n−1,n is aligned if, for every face F , the splitting point chosen for F is the unique point in F that is collinear with v and the vertex of T opposite F . This is what Schenck and Sorokina [10] called a facet split.…”

Section: The N−1n Splitmentioning

confidence: 83%

“…Proof The dimensions of the polynomial spline spaces for an aligned split n−1,n were derived by Schenck and Sorokina [10]. For 0 ≤ r ≤ d:…”

Section: Theoremmentioning

confidence: 99%

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A characterization of supersmoothness of multivariate splines

Floater

2020

Adv Comput Math

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We consider spline functions over simplicial meshes in $\mathbb {R}^{n}$ ℝ n . We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method. In this paper, we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.

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“…It is remarked in [10,Remark 4.3] that for r = 1, the dimension formula (9) also holds even without the collinearity condition, from which we conclude that for an arbitrary n−1,n split:…”

Section: Theoremsupporting

confidence: 59%

Section: The N−1n Splitmentioning

confidence: 83%

See 1 more Smart Citation

A characterization of supersmoothness of multivariate splines

Floater

2020

Adv Comput Math

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“…As we saw in §5, for k ≥ 3 this is nontrivial. More generally, find formulas for special configurations, as in [36] and [37]. 6.2.…”

Section: Open Questionsmentioning

confidence: 99%

Algebraic methods in approximation theory

Schenck

2016

Computer Aided Geometric Design

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Abstract. 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 R k , 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.

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“…Several conceptually similar approaches have been recently formulated, inspired by applications of splines in numerical analysis and geometric modelling. For instance, this approach was adopted to study splines on locally subdivided meshes in [15]; to study splines with local polynomial-degree adaptivity in [21,22]; and to study mixedsmoothness splines on T-meshes in [24].…”

Section: Introductionmentioning

confidence: 99%

Counting the dimension of splines of mixed smoothness: A general recipe, and its application to meshes of arbitrary topologies

Toshniwal,

DiPasquale

2020

Preprint

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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.

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Subdivision and Spline Spaces

Cited by 11 publications

References 8 publications

A characterization of supersmoothness of multivariate splines

A characterization of supersmoothness of multivariate splines

Algebraic methods in approximation theory

Counting the dimension of splines of mixed smoothness: A general recipe, and its application to meshes of arbitrary topologies

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