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PH2T-splines, Part I: A Reasonable Mesh Assumption

Bingru Huang, Yue Xi

arXiv:2609.11443v1math.NAcs.NA

TL;DR

Constructing a mesh-compatible spline basis requires dimensional stability, but hierarchical refinement conditions do not always provide it. The paper examines two refinement settings and introduces template translations; neither setting guarantees stability, motivating a (d-1) × (d-1) template condition as a reasonable assumption for subsequent basis construction.

  • Problem

    A mesh-compatible basis construction requires a dimensionally stable spline space, yet spline-space dimension can depend on T-mesh geometry as well as topology.

    Source: §1 · p. 2

  • Method

    The paper studies four examples of unrestricted and no-vanishable-edge hierarchical refinement, then describes selected regions using translated templates in the cell-index grid.

    Source: §1 · p. 2, §4.3 · pp. 15–16

  • Results

    Neither unrestricted refinement for d = 3, 4 nor refinement without vanishable T l-edges for d = 5, 6 guarantees dimensional stability; the latter increases dimension at every refinement level.

    Source: §1 · p. 3, §4.2.2 · p. 14

  • Takeaways & Limitations

    The (d-1) × (d-1) template condition is a sufficient, reasonable assumption for dimensional stability, but is not claimed necessary for every stable mesh or optimal in every degree.

    Source: §5 · p. 17

    Dimensional stability is necessary but not sufficient for construction-suitability; linear independence and completeness must still be established for a proposed basis.

    Source: §3 · p. 8

Abstract

from arXiv · show

This paper is the first in a three-part series on the construction of polynomial splines with the highest order of smoothness over hierarchical T-meshes, referred to as $\PHtwoT$-splines. For splines of bi-degree $(d,d)$, we study suitable refinement conditions for the subsequent basis construction, which requires dimensional stability of the underlying spline space. We present two groups of examples, considering unrestricted hierarchical refinement and refinement without vanishable T $l$-edges, respectively. The first setting permits new edges without additional degrees of freedom. In the second group, every refinement level excludes vanishable T $l$-edges and increases the dimension. Nevertheless, the dimension is unstable in both groups. We then introduce template translations to describe each refined region as a union of translates of a fixed template in the cell-index grid. Together with the known stability result under $(d-1)\times(d-1)$ template refinement, these examples justify this condition as a reasonable mesh assumption for the subsequent $\PHtwoT$-spline construction.

1 Introduction

The introduction motivates mesh-based basis construction for maximally smooth splines and explains why refinement conditions must ensure dimensional stability. Its examples show that two natural refinement restrictions fail, motivating a translated-template assumption for the later construction.

1 Introduction

  • Tensor-product NURBS refinement can add degrees of freedom outside the target region, motivating locally refinable spline constructions.The introduction situates T-splines, hierarchical B-splines, locally refined splines, and polynomial splines over hierarchical T-meshes among these approaches.Source: §1 · p. 1
  • For high-smoothness spline spaces, dimension may depend on mesh geometry as well as topology.Earlier work observed this instability and studied meshes containing T-cycles and nested T-cycles.Source: §1 · p. 2
  • Dimensional stability is necessary for a basis construction compatible with the mesh structure.The paper introduces construction-suitable T-meshes in this setting.Source: §1 · p. 2
  • Neither unrestricted refinement for d = 3, 4 nor no-vanishable refinement for d = 5, 6 guarantees dimensional stability.In the latter group, every refinement level strictly increases the dimension.Source: §1 · p. 3
  • Template translations describe refined regions, but the (d −2) × (d −2) condition remains insufficient for stability when d = 5, 6.The examples motivate the (d −1) × (d −1) template condition, for which a stability result is known [27].Source: §1 · pp. 2–3

2 Preliminaries

The preliminaries define hierarchical T-meshes and the maximally smooth spline spaces studied in the paper, then establish notation and concepts used to discuss dimension stability. In particular, refinements proceed by local cross subdivision, while spline smoothness is encoded through conformality equations on T l-edges.

2.1 T-meshes and hierarchical T-meshes

  • A T-mesh is a finite collection of axis-aligned rectangular cells with disjoint interiors and connected union.Interior vertices where three edges meet are T-nodes; those where four meet are cross-vertices.Source: §2.1 · p. 3
  • A hierarchical T-mesh is formed by successive local subdivisions; cross subdivision splits selected cells into 2 × 2 subcells.The refined region at each level is the collection of cells subdivided at that level.Source: §2.1 · p. 4

2.2 Spline spaces and smoothing cofactors

  • The spline space consists of bi-degree (d, d) polynomials with C^(d−1) continuity in both coordinate directions, without imposed boundary conditions.The smoothing cofactor method expresses smoothness through edge and vertex cofactors; on a T l-edge, the cofactor system has rank min{d + 1, r}.Source: §2.2 · pp. 4–5
  • A T l-edge is vanishable when its conformality equations have only the zero solution, equivalently when it has at most d + 1 vertices.For d = 5, isolated 2 × 2 refinement yields vanishable edges, whereas isolated 3 × 3 refinement yields non-vanishable edges; this edge-level distinction does not determine full-space dimension.Source: §2.2 · p. 5

2.3 Structurally isomorphic T-meshes and dimensional stability

  • Structurally isomorphic meshes preserve l-edge types, intersections, and relative positions of parallel edges, even when knot intervals differ.Dimensional stability is defined relative to a specified collection of meshes; the paper notes that invariance fails for all hierarchical T-meshes when d = 3, 4, 5, 6.Source: §2.3 · pp. 6–7

3 Construction-suitable T-meshes

The section defines construction-suitable meshes through a well-defined basis construction over a structurally isomorphic class, and proves dimensional stability is necessary for such a construction.

3 Construction-suitable T-meshes

  • A well-defined construction must span the full spline space on each mesh and give corresponding families under structurally isomorphic maps.For linearly independent constructed families, the spanning condition makes them bases; correspondence is independent of the chosen mesh within the class.Source: §3 · pp. 7–8
  • A T-mesh is construction-suitable when a well-defined basis construction exists on its structurally isomorphic class.The construction is defined across the class rather than as a single family on one fixed domain.Source: §3 · p. 8
  • Construction-suitability implies stable spline-space dimension across the class; unstable dimensions still allow separate meshwise bases, but not one common construction.Linear independence and completeness must still be established for a proposed construction.Source: §3 · p. 8
  • For the bilinear HB example, the 17-function family is a basis throughout [Ts]I and defines a well-defined construction βHB.Its function indices, levels, supporting cells, and corresponding B-spline formulas match under structurally isomorphic maps.Source: §3 · p. 9

4 Instability examples and a mesh assumption

Four examples show that neither unrestricted refinement nor refinement excluding vanishable T l-edges guarantees dimensional stability. Template translations lead to a (d −1) × (d −1) refinement assumption, under which the dimension is stable on the specified admissible class.

4.1 Unrestricted hierarchical refinement

  • Unrestricted hierarchical refinement can add edges without increasing spline-space dimension, and the dimension may depend on mesh geometry.Vanishable T l-edges force their vertex cofactors to zero, so a refinement can introduce edges without adding free cofactors.Source: §4.1 · p. 10

4.1.1 A bicubic example

  • For the bicubic example, the dimension is unstable: the first two refinements add edges without increasing it, while the third-level increase is two for τ = 1 and one for τ = 2.The meshes are structurally isomorphic as τ varies, but their dimension depends on τ.Source: §4.1.1 · p. 10, Table 1 · p. 11

4.1.2 A biquartic example

  • For the biquartic example, the dimension is unstable: from an initial tensor-product dimension of 100, refinement adds one degree of freedom at τ = 1 and none otherwise.The four dashed T l-edges shown in the example are vanishable for S4.Source: §4.1.2 · pp. 11–12, Figure 8 · p. 12

4.2 Refinement without vanishable T l-edges

  • No-vanishable refinement requires every newly generated T l-edge to contain at least d + 2 vertices, yet that restriction does not secure dimension stability.The degree-five and degree-six examples increase dimension at each refinement level despite ending with different dimensions.Source: §4.2 · p. 12, §4.2.2 · p. 14

4.2.1 An example of degree five

  • For d = 5, both refinement levels add degrees of freedom, yet dimension is unstable; every level is no-vanishable, with each newly generated T l-edge containing at least seven vertices.Source: §4.2.1 · p. 13

4.2.2 An example of degree six

  • For d = 6, each refinement level increases dimension, yet the final dimensions differ between the two meshes, although every new T l-edge is non-vanishable.Each new T l-edge has at least nine vertices.Source: §4.2.2 · p. 14, Figure 10 · p. 14

4.2 Refinement without vanishable T l-edges

  • For d = 5, 6, excluding vanishable T l-edges and increasing dimension at every level still does not ensure dimensional stability.The examples also show that replacing (d −1) × (d −1) templates with (d −2) × (d −2) templates remains insufficient.Source: §4.2 · p. 12, §4.2.2 · p. 14, §4.3 · p. 16

4.3 Template translations and the mesh assumption

  • A refined region has channel length at least m exactly when generated by translated m × m templates; the smallest template side universally preventing vanishable T_l-edges for S_d is q_d.For d = 5, 6, translated (d − 2) × (d − 2) templates do not ensure dimensional stability. Under Assumption 4.9, the dimension of S_d(T) is stable on its admissible structurally isomorphic class.Source: §4.3 · pp. 15–16

5 Conclusion

The paper examines refinement conditions for constructing maximally smooth PH2T-splines over hierarchical T-meshes, where dimensional stability is necessary for a mesh-compatible basis construction. Its examples motivate the (d −1) × (d −1) template condition as a reasonable sufficient assumption, without claiming it is necessary or optimal.

5 Conclusion

  • Unrestricted refinement can add edges without increasing dimension, while excluding vanishable T l-edges still fails to ensure stability even when each level increases dimension.The bicubic and biquartic examples illustrate the first behavior; the degree-five and degree-six examples illustrate the second.Source: §5 · p. 17
  • Template translations and channel length express refinement conditions through the arrangement of selected cells.The degree-five and degree-six examples satisfy the (d −2) × (d −2) template condition but have unstable dimensions.Source: §5 · p. 17
  • Together with the known stability result in [27], the examples justify the (d −1) × (d −1) template condition as a reasonable mesh assumption for later basis construction.The condition is sufficient for dimensional stability, but is not claimed necessary for every stable hierarchical T-mesh or optimal in every degree.Source: §5 · p. 17

A Conformality matrices and dimension calculations

This appendix calculates spline-space dimensions using the smoothing cofactor method and T-connected-component decomposition. It describes conformality matrices, reductions of their rank, and exact certificates.

A.1 The conformality matrix

  • The conformality matrix collects equations for T l-edges, and its kernel is the conformality vector space associated with L(T).For the regular example meshes, the stated dimension formula applies when there are no rays or homogeneous boundary conditions.Source: §A.1 · p. 17

A.2 Vanishable edges and the CNDC

  • A T-connected component with a reasonable edge ordering is diagonalizable and has a full-row-rank conformality matrix; the residual component is the CNDC.The extraction procedure removes edges with at least d + 1 relative mono-vertices, leaving a completely non-diagonalizable component when the process terminates with one.Source: §A.2 · p. 18

A.3 Elimination of mono-vertices

  • Eliminating mono-vertex variables contributes pivots and reduces the remaining system to equations on multi-vertices.The reduction uses a polynomial basis including Lagrange polynomials at mono-vertices; distinct edges have disjoint mono-columns, so their reductions can be performed independently.Source: §A.3 · p. 18

A.4 Exact rank certificates

  • The appendix establishes rational-matrix ranks by combining a nonsingular minor with block elimination, using exact rational arithmetic rather than a numerical tolerance.The selected index sets specify the minor and complementary blocks used in the certificate.Source: §A.4 · p. 19

B The bicubic example

The bicubic example uses conformality-matrix reductions to calculate dimensions across refinement levels and to establish the rank behavior of the final reduced system.

B The bicubic example

  • The final mesh has 19 T l-edges and 86 interior vertices, of which 70 lie on T l-edges.The mesh also has eight cross-cuts.Source: B The bicubic example · p. 19
  • The final reduction first removes 25 forced-zero variables, leaving 11 edges, 20 mono-vertices, and 25 multi-vertices.The same elimination applies for every τ > 0.Source: B The bicubic example · p. 19
  • The reduced matrix K3(τ) is specified through edge blocks and multi-vertex columns, with only row 13 depending on τ.The passage describes the matrix construction, while the supplied table captions identify the edge-block and kernel-vector tables.Source: B The bicubic example · pp. 19–20, Table 3 · p. 20, Table 4 · p. 20
  • At levels one and two, forced-zero elimination gives full column ranks 21 and 38; at level three, the reduced-system ranks are 68 and 69.These are the stage counts reported for Table 1.Source: B The bicubic example · p. 21

C The biquartic example

The biquartic example establishes rank variation in a reduced matrix, while the degree-five example calculates spline dimensions at successive refinement levels and specifies a final reduced system.

C The biquartic example

  • For the biquartic example, rank K4(τ) is 3 at τ = 1 and 4 for 0 < τ < 2 with τ ≠ 1.At τ = 1, the vector (3, 5, 5, 3)T spans the kernel.Source: C The biquartic example · p. 22

D The degree-five example

  • The degree-five example maps the uniform-mesh coordinates to the prescribed midpoint-subdivision coordinates using a piecewise affine map.The same coordinate convention is used for the degree-six example.Source: D The degree-five example · p. 22

D.1 The first refinement level

  • At the first refinement level, both degree-five meshes have dimension 147.The initial dimension is 144.Source: §D.1 · p. 22

D.2 The final mesh and its reduced matrix

  • The final degree-five mesh has 20 T l-edges, and its reduced matrix is specified by edge data, mono-coordinate sets, and multi-vertex intersections.The passage also describes extracting two central edges before forming the remaining component.Source: §D.2 · pp. 22–23, Table 6 · p. 23

D.3 Rank certificates and dimensions

The rank calculations establish distinct dimensions for the compared degree-five spline spaces and give dimension 267 at the first refinement level of the degree-six example.

D.3 Rank certificates and dimensions

  • The final mesh has full ranks 116 and 117 and dimensions 150 and 149 for S5(T5,A) and S5(T5,B), respectively.The extra zero column accounts for a free variable of the extracted component.Source: §D.3 · p. 23

E The degree-six example

E.1 The first refinement level

  • At the first refinement level, the conformality matrix has two disjoint 3 × 3 crossing arrays of rank five, yielding dimension 267.The calculation extracts central horizontal and vertical edges, with relative mono-vertex counts eight and nine.Source: §E.1 · p. 24

E.2 The final mesh and its reduced matrix

The final degree-six mesh’s edge structure is reduced to a component whose edge blocks determine the two matrices used in the rank calculation.

E.2 The final mesh and its reduced matrix

  • The final mesh has 28 T l-edges and 214 vertices on their union; after extracting two central edges, the remaining 26 edges form the CNDC.The CNDC contains 124 mono-vertices and 73 multi-vertices.Source: §E.2 · p. 24
  • The extracted central horizontal and vertical edges retain relative mono-vertex counts of eight and nine.Source: §E.2 · p. 24
  • Table 7 gives the edge blocks for K6(0), and with (47) and (34) these specify both matrices exactly.Source: §E.2 · p. 24, Table 7 · p. 24

E.3 Rank certificates and dimensions

The rank certificates give distinct dimensions for the two final degree-six meshes, proving the stated dimension result despite their shared mesh counts.

E.3 Rank certificates and dimensions

  • The selected minors have orders 55 and 56, and the cited determinant is det K6(1)[I1, J1] = −380522718 · 11 · 132 · 19 · 232 · 292 · 592.The row universe is {0, . . . , 57}, and the column universe is {0, . . . , 72}; the listed index sets for u = 0 and u = 1 differ.Source: §E.3 · pp. 24–25
  • The final meshes have dimensions 277 and 276 for S6(T6,A) and S6(T6,B), respectively.Their full ranks are 193 and 194, with c = 18 and nv = 295.Source: §E.3 · p. 25
  • Together with the first-level calculation, these certificates prove (24), while three zero columns retain the extracted component’s free variables.Source: §E.3 · p. 25

F The template conditions

Template translations characterize selected regions and support mesh conditions that exclude vanishable edges and stabilize dimension; smaller templates can fail to ensure stability.

F.1 Channel length and template translations

  • A region has channel length at least m exactly when it is a union of translated m × m square templates.The equivalence follows by covering each selected cell with a contained m × m subtemplate.Source: §F.1 · p. 25

F.2 Exclusion of vanishable edges

  • Subdividing regions built from qd × qd templates produces no vanishable T l-edges.A horizontal selected run has r ≥ qd cells, so its new edge has at least 2r + 1 vertices; the vertical case is identical.Source: §F.2 · p. 25
  • The qd template size is minimal for this guarantee: an isolated m × m submesh with m < qd creates vanishable edges with 2m + 1 ≤ d + 1 vertices.Source: §F.2 · p. 25

F.3 The adopted mesh assumption

  • For d ≥ 3, the adopted (d−1) × (d−1) template condition excludes vanishable edges and yields stable dimension on the specified mesh class.The argument applies the dimension formula in [27, Theorem 4.2] and uses unchanged mesh counts and isolated subdivision-component counts.Source: §F.3 · p. 25
  • For d = 2, the 1 × 1 template condition adds no restriction, and dimension is stable because the relevant counts remain unchanged.The biquadratic dimension formula depends on vertex and edge counts and isolated subdivisions at successive levels.Source: §F.3 · p. 25
  • For d = 5 and d = 6, (d−2) × (d−2) templates exclude vanishable edges and each level increases dimension, yet the final dimensions differ on structurally isomorphic meshes.Thus this smaller template condition does not ensure stability.Source: §F.3 · p. 26
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