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Dimension of Bi-degree $(d,d)$ Spline Spaces with the Highest Order of Smoothness over Hierarchical T-Meshes

Bingru Huang, Falai Chen

arXiv:2507.11047v2math.NAcs.NA

TL;DR

General dimension formulas for T-mesh spline spaces can contain undetermined terms, and some dimensions depend on geometric as well as topological information. The paper uses the smoothing-cofactor conformality method, a tensor product T-connected component, and recursive level-by-level dimension calculations under tensor product subdivision. The dimension formula is dim Sd(T ) = v + db + (d −1)2 + γ0 −t(d + 1), (6).

  • Problem

    General dimension formulas for T-mesh spline spaces can contain undetermined terms, and some dimensions depend on geometric as well as topological information.

    Source: §1 · p. 2

  • Method

    The paper uses the smoothing-cofactor conformality method, a tensor product T-connected component, and recursive level-by-level dimension calculations under tensor product subdivision.

    Source: §1 · p. 3, §7 · p. 26

  • Results

    The dimension formula is dim Sd(T ) = v + db + (d −1)2 + γ0 −t(d + 1), (6).

    Source: §4 · p. 19

  • Takeaways & Limitations

    Tensor product subdivision generalizes previous subdivision modes, which the paper treats as special cases of its results.

    Source: §7 · p. 26

    Level-by-level calculation is not established when tensor product subdivision fails the condition in Theorem 3.2, and the m ≠ n case remains open.

    Source: §7 · p. 26

Abstract

from arXiv · show

In this article, we study the dimension of the spline space of di-degree $(d,d)$ with the highest order of smoothness over a hierarchical T-mesh $\mathscr T$ using the smoothing cofactor-conformality method. Firstly, we obtain a dimensional formula for the conformality vector space over a tensor product T-connected component. Then, we prove that the dimension of the conformality vector space over a T-connected component of a hierarchical T-mesh under the tensor product subdivision can be calculated in a recursive manner. Combining these two aspects, we obtain a dimensional formula for the bi-degree $(d,d)$ spline space with the highest order of smoothness over a hierarchical T-mesh $\mathscr T$ with mild assumption. Additionally, we provide a strategy to modify an arbitrary hierarchical T-mesh such that the dimension of the bi-degree $(d,d)$ spline space is stable over the modified hierarchical T-mesh. Finally, we prove that the dimension of the spline space over such a hierarchical T-mesh is the same as that of a lower-degree spline space over its CVR graph. Thus, the proposed solution can pave the way for the subsequent construction of basis functions for spline space over such a hierarchical T-mesh.

1 Introduction

Locally refinable splines have promising applications, but understanding spline spaces over T-meshes—especially their dimensions—remains a fundamental challenge. General dimension formulas can retain undetermined terms, and some spline-space dimensions depend on geometric as well as topological information.

Prior work

  • Earlier dimension methods include B-net, smoothing-cofactor, minimal-determining-set, and homology approaches, but general formulas can retain terms that cannot be determined for general T-meshes.Prior work also addressed three-dimensional cases and particular hierarchical meshes; highest-order smoothness is a popular setting, while earlier hierarchical results covered specific cases such as S2(T) or restricted mesh types.Source: §1 · p. 2

Motivation and problem

  • Some T-mesh spline dimensions depend on geometric as well as topological information, making general formulas based only on topology challenging or impossible.Hierarchical T-meshes are identified as the most practical among the particular mesh types studied.Source: §1 · p. 2

Contributions

  • The paper studies highest-order-smoothness spline spaces over hierarchical T-meshes for arbitrary degree, introducing tensor product T-connected components and a conformality-vector-space dimension formula.The formula is developed using the smoothing cofactor method.Source: §1 · pp. 2–3
  • For a hierarchical T-mesh under tensor product subdivision, the conformality-vector-space dimension over a T-connected component can be calculated level by level.The paper describes this calculation as recursive.Source: §1 · pp. 2–3
  • The paper proposes a special hierarchical T-mesh under cross-subdivision, gives a spline-dimension formula for it, and proves a conjectured relationship with its CVR graph.The study describes these results as generalizing prior results and containing results on hierarchical B-spline-space dimensions.Source: §1 · p. 3

2 Preliminaries

This section introduces the mesh and spline-space concepts used in the paper, then presents the smoothing-cofactor framework for dimension calculations. It also describes partitioning T-connected components to simplify those calculations.

2.1 T-meshes and hierarchical T-meshes

  • A T-mesh is a set of axis-aligned rectangles whose pairwise intersections are empty or consist of boundary points; hierarchical meshes add recursive levels of subdivision.The paper focuses mainly on regular T-meshes and cross subdivision, which divides a cell into four equal subcells.Source: §2.1 · pp. 3–4

2.2 Spline spaces over T-meshes and homogeneous boundary conditions

  • The highest-order smoothness space uses bi-degree (d,d) with continuity order d−1 in both directions, and its dimension equals that of a related space on the extended mesh.The extended mesh copies each boundary l-edge d times and extends l-edges ending at the boundary.Source: §2.2 · pp. 5–6

2.3 Smoothing cofactor method

  • Local conformality relates polynomial pieces around a vertex through a vertex cofactor, while global conformality imposes relations among cofactors along T l-edges.The global conditions form linear systems for all T l-edges; vanished edges are assumed not to occur.Source: §2.3 · pp. 6–7
  • The smoothing-cofactor method expresses spline dimension through the conformality vector space, alongside counts of cross-cuts and interior vertices.The stated formula is dim S_d(T) = (d + 1)^2 + c(d + 1) + n_v + dim CVS[L(T)].Source: §2.3 · p. 8

2.4 Bipartite partition and diagonalizable component

  • A bipartite partition separates a T-connected component into a sub-component and its complement, and a reasonable ordering can reduce dimension calculation to one part.The cited proposition gives this reduction when the edges in the other part have a reasonable order.Source: §2.4 · p. 8
  • The framework also distinguishes diagonalizable from completely non-diagonalizable components according to whether a suitable bipartition and reasonable ordering exist.A reasonable order requires each successive l-edge to contain at least d + 1 vertices not on earlier edges.Source: §2.4 · pp. 8–9

3 Tensor product T-connected component

This section develops dimension formulas for conformality vector spaces of tensor-product T-connected components, including a level-0 identity under stated mesh conditions and recursive calculations for hierarchical T-meshes under tensor-product subdivision. It also gives a related formula for homogeneous-boundary spline spaces.

3 Tensor product T-connected component

  • A tensor product T-connected component comprises l-edges forming an aligned horizontal-vertical grid, possibly excluding boundary edges, and excludes T-cycle structures.T-cycle structures can cause dimensional instability, while non-diagonalizable cases still complicate dimension calculations.Source: §3 · p. 9
  • For a tensor product T-connected component, dim CVS is m(p-d-1)+n(q-d-1) when diagonalizable and (p-d-1)(q-d-1) when non-diagonalizable.Here m and n count horizontal and vertical T l-edges, p and q count vertices on each, and d is the spline degree.Source: §3 · pp. 10, 11, 14
  • Theorem 3.1 gives the first systematic dimension formula; unlike prior work, it goes beyond upper and lower bounds, with the main complexity arising from non-diagonalizable components.The cited prior bounds enabled calculations in specific cases but did not provide a general formula.Source: §3 · p. 15
  • For hierarchical T-meshes with mutually disjoint (d-1) × (d-1) tensor product subdivisions at each level, the conformality-vector-space dimension can be computed recursively.The subdivision is assumed non-degenerate, meaning none of its l-edges vanish; the level-by-level calculation sums the dimensions associated with each level's T-connected components.Source: §3 · p. 15
  • For a completely non-diagonalizable tensor-product T-connected component with 4d − 3 vertices on each T-l-edge, dim CVS[L(T ′)] = (3d − 4)^2.This follows by subtracting d + 1 from the vertex count on each edge and multiplying the resulting factors.Source: §3 · p. 15
  • For a completely non-diagonalizable component with 2d − 1 vertices on each T-l-edge, dim CVS[T1] = (d − 2)^2.The dimension is obtained from [2d − 1 − (d + 1)]^2.Source: §3 · p. 15
  • If the tensor-product mesh T ′ has at least d + 1 horizontal and vertical l-edges, then dim S̄_d(T) = dim S̄_d(T ′) + dim CVS[L].Here T ′ is formed from the boundary l-edges and cross-cuts, and L comprises the mesh's rays and T-l-edges.Source: §3 · p. 16
  • For a hierarchical T-mesh under tensor-product subdivision, if level 1 has one tensor-product T-connected component and level 0 has at least d + 1 horizontal and d + 1 vertical l-edges, dim CVS[N0] = dim CVS[T1] + dim CVS[N1].Source: §3 · p. 16

4 Dimension of the spline space Sd(T ) over a hierarchical T-mesh

The section derives a dimension formula for highest-smoothness spline spaces on hierarchical T-meshes with partially overlapping tensor-product subdivisions. It also gives a recursive calculation and a strategy for modifying arbitrary meshes to obtain a stable dimension.

Dimension formula

  • For the specified hierarchical meshes, dim Sd(T) = v + db + (d − 1)^2 + γ0 − t(d + 1), with partially overlapping (d − 1) × (d − 1) subdivisions allowed.Here v, t, b, and γ0 count vertices, edges, boundary edges, and isolated components subdividing interior cells, respectively.Source: §4 · p. 19

Recursive calculation

  • The conformality-space dimension is calculated recursively, handling isolated tensor-product components and non-isolated components level by level.The proof uses a reasonable ordering of l-edges for non-isolated components; the formula also extends to spline spaces with homogeneous boundary conditions.Source: §4 · pp. 18–19

Dimension formula

  • The dimension formula is consistent with the formula for (m,n)-subdivision T-meshes when m = n = d, while allowing partial overlap.The section describes this as a more general case than the one in.Source: §4 · p. 21

Mesh modification

  • An arbitrary hierarchical T-mesh can be modified into a mesh whose spline-space dimension is stable by covering subdivision-region boundaries with tensor-product submeshes.The construction slides a (d − 1) × (d − 1) template along the boundary and subdivides the cells it covers.Source: §4 · p. 21
  • In the example, successive 3 × 3 submeshes extend coverage of the subdivided-region boundary to produce the modified mesh.The original S4(T) dimension is described as unstable, while the resulting S4(T′) dimension is stable.Source: §4 · pp. 21–22

5 Hierarchical T-mesh and its CVR graph

The section relates a hierarchical T-mesh spline space to the lower-degree spline space on its CVR graph. Under the stated subdivision conditions, their dimensions are equal, and the CVR conformality-space dimension can be calculated recursively.

CVR graph

  • The CVR graph retains cross vertices and edges whose endpoints are both cross vertices, removing other vertices and edges.The resulting graph supports a lower-degree spline space with a readily constructible basis, which the paper connects to basis construction for the original mesh.Source: §5 · pp. 22–23

Dimension relation

  • When all T l-edges in L(T) have a reasonable order, dim CVS1[L(T)] = dim CVS2[L(T)] = v − (d + 1)t, where v is the number of vertices and t the number of T l-edges.Source: §5 · p. 24

Recursive calculation

  • The conformality-vector-space dimension for a CVR-graph T-connected component can be calculated recursively across levels.The result applies when each level uses possibly overlapping (d − 1) × (d − 1) tensor-product subdivisions.Source: §5 · p. 24

Dimension relation

  • Theorem 5.1 considers a hierarchical T-mesh T and its CVR C, where T is formed at each level by subdividing a collection of (d−1)×(d−1) tensor-product submeshes that may overlap.Source: §5 · p. 25

7 Conclusion and Future Work

The paper derives a dimension formula for highest-smoothness spline spaces on hierarchical T-meshes under tensor product subdivisions, and relates their dimensions to those of CVR graphs. It also identifies conditions and degree settings where further work is needed.

7 Conclusion and Future Work

  • A dimensional formula is obtained for the highest-smoothness spline space under (d −1) × (d −1) tensor product subdivisions.The result applies to a hierarchical T-mesh.Source: §7 · p. 26
  • Tensor product subdivision generalizes previous subdivision modes, and conformality-vector-space dimensions can be computed recursively for qualifying T-connected components.The paper also derives a conformality-vector-space dimension formula using the smoothing cofactor method.Source: §7 · p. 26
  • The paper confirms a conjecture relating spline-space dimensions on a hierarchical T-mesh to dimensions over its CVR graph.Source: §7 · p. 26
  • Level-by-level calculation remains unresolved when tensor product subdivision fails Theorem 3.2; dimensions for highest-smoothness bi-degree (m, n), where m ≠ n, remain open.Source: §7 · p. 26
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