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Geometry-dependent rank defect in $C^1$ cubic spline space

Xinyu Wu, Jiansong Deng

arXiv:2609.02424v1math.NAcs.NAmath.CO

TL;DR

The dimension of C1 cubic splines on arbitrary nondegenerate planar triangulations remained unresolved, and it was unknown whether Schumaker’s local correction captured every dimension increase. The paper studies a one-parameter family of realizations of a fixed triangulation. It finds an extra dimension at a nonsingular parameter, arising from a dependence across all seven interior vertex cycles.

  • Problem

    It was unresolved whether Schumaker’s lower bound, with its correction for singular four-stars, always gives the C1 cubic spline dimension on arbitrary nondegenerate triangulations.

    Source: §1 · pp. 1–2

  • Method

    The authors construct a one-parameter family of nondegenerate realizations of a fixed triangulation and determine its rank profile.

    Source: §1 · p. 2

  • Results

    At t = 1/5, dim S_3^1(T(1/5)) = 34 > P_T(1, 3) = 33 although all interior vertices are nonsingular, and the dependence couples all seven vertex cycles.

    Source: §1 · p. 2, §5 · pp. 8–9

  • Takeaways & Limitations

    The singular-four-star correction does not capture every geometry-dependent dimension contribution; the nonsingular counterexample has a global dependence across vertex cycles.

    Source: §5 · pp. 8–9

    The reported rank profile concerns the specified one-parameter family on its admissible interval; rescaling edge-line forms does not alter rank statements.

    Source: §1 · p. 2, §2.2 · p. 4, §3 · p. 5

Abstract

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Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $\sigma$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $\sigma=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $\sigma=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein--B\'ezier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.

1 Introduction

A one-parameter family of realizations of a fixed triangulation disproves the conjecture that Schumaker’s lower bound always gives the cubic C1 spline dimension. At one nonsingular parameter, a global rank defect adds a dimension not captured by the local correction.

1 Introduction

  • The dimension of C1 cubic splines on arbitrary nondegenerate planar triangulations remained unresolved since the 1970s, including whether a realization with σ = 0 could have an additional dimension.Schumaker’s bound incorporates σ for singular interior four-stars, but its attainment for every realization was conjectured.Source: §1 · pp. 1–2
  • Earlier degree-two work showed that geometry can change spline dimension without changing the combinatorial triangulation, while the arbitrary-geometry cubic case remained open.The seven-triangle example changes dimension when three distinguished lines become concurrent.Source: §1 · p. 2
  • The authors construct an 18-triangle complex and vary only its central vertex along a parameter interval whose realizations are nondegenerate.The exact rank profile separates the family into three cases.Source: §1 · p. 2
  • Except at t = 1/5 and t = 3/83, the family has dimension 33, equal to Schumaker’s lower bound; at t = 3/83, the singular central four-star raises the dimension to 34, as the local correction predicts.The generic conformality-matrix rank is 49; at 3/83, a single local vertex block loses rank.Source: §1 · p. 2
  • At t = 1/5, all interior vertices are nonsingular, yet dim S_3^1(T(1/5)) = 34 > P_T(1, 3) = 33 because the rank dependence couples all seven interior vertex cycles.Every individual vertex block has full row rank, making this a global rather than a local defect.Source: §1 · p. 2

2 Preliminaries

The preliminaries present two matrix approaches to computing cubic spline dimensions: direct Bernstein–Bézier edge matching and smoothing cofactors organized around interior-vertex cycles. The cofactor formulation gives a smaller matrix whose block structure exposes whether rank loss is local or global.

2.2 Smoothing cofactors and vertex cycles

  • The smoothing-cofactor method eliminates common C0 traces, encoding each interior-edge polynomial jump as a multiple of the squared edge-line form.The resulting compatibility equations are imposed around cycles in the dual graph.Source: §2 · p. 3, §2.2 · p. 4

2.1 Bernstein–Bézier edge matching

  • The B-net method represents cubic patches by Bernstein ordinates and imposes C1 joining conditions across each interior edge.The four shared-edge equations match traces, while three further equations match transversal derivatives; the kernel of the assembled system is the space of globally C1 B-nets.Source: §2.1 · p. 3

2.2 Smoothing cofactors and vertex cycles

  • For a triangulated disk, the spline dimension is 10 + dim ker C(T) = 10 + 3E0 − rank C(T).Interior-vertex cycles form a basis of the dual cycle space, and their compatibility equations make the polynomial pieces path independent.Source: §2.2 · pp. 4–5
  • Schumaker’s bound states δ(T) ≥ 0, and the cited example demonstrates that the discrepancy can be δ = 1 even when σ = 0.Changing the scale of an edge-line form merely rescales its three cofactor variables and does not alter any rank statement.Source: §2.2 · pp. 4–5

3 A one-parameter triangulation

The section constructs a moving interior fan in a fixed triangulation and identifies its admissible geometry and singular configuration. It frames the rank question for realizations with no singular vertices.

3 A one-parameter triangulation

  • The complex has 13 vertices and 18 triangles; its seven interior vertices supply the cycles, while only the central fan point moves.Boundary vertices close the disk but do not contribute vertex-cycle equations.Source: §3 · pp. 5–6, Figure 1 · p. 6
  • The moving fan point lies inside the convex quadrilateral exactly for t in (-3/4, 24/55), so its four triangles form a nonoverlapping fan.The remaining fixed triangles are nondegenerate, and the edge set has 24 interior edges.Source: §3 · p. 6
  • Only the central vertex can be a singular four-star; its opposite edge pairs become collinear simultaneously exactly at t = 3/83.At every other admissible parameter its four incident edges have distinct projective directions.Source: §3 · pp. 6–7
  • The central rank question is whether the cofactor matrix has full row rank 49 whenever σ = 0.The section sets up the family’s lower-bound expression before stating this test.Source: §3 · p. 7

4 The exact rank profile

The exact rank profile has two isolated exceptional parameters, with dimension 34 there and 33 elsewhere. The paper distinguishes a local singular-star dependence from a global dependence across full-rank vertex blocks.

4 The exact rank profile

  • Dimension is 34 at t = 1/5 or 3/83 and 33 elsewhere in I; the conformality-matrix rank is respectively 48 and 49.The complementary Bernstein–Bézier calculation reports the same dimension profile.Source: §4 · pp. 7–8
  • A maximal-minor factorization and exact rank certificates establish full row rank except at the two admissible roots, where the rank drops by exactly one.The endpoint roots correspond to a collapsing fan, while 3t − 5 lies outside I.Source: §4 · pp. 7–8
  • At t = 1/5, σ = 0 despite dimension 34; changing t to 101/500 raises rank from 48 to 49 and lowers dimension from 34 to 33.Both realizations are nondegenerate and have the same combinatorial complex.Source: §4 · p. 8
  • Each exceptional point is isolated along the parameter line, and the additional spline mode disappears under an arbitrarily small generic displacement.The selected maximal minor vanishes to first order at each root.Source: §4 · p. 8
  • At t = 3/83, the dependence is confined to the central block of rank 6; at t = 1/5, all seven blocks have rank 7 but the assembled matrix has a global dependence.The latter dependence has nonzero components in all seven vertex-cycle blocks and remains beyond Schumaker’s local correction.Source: §4 · p. 8
  • The Bernstein–Bézier matching matrix independently gives rank 146 at both exceptional parameters and 147 elsewhere, yielding dimensions 34 and 33 respectively.This agrees with the smoothing-cofactor calculation.Source: §4 · p. 8

5 Conclusion

The paper disproves the conjecture that Schumaker’s lower bound always gives the dimension of the C^1 cubic spline space. In a one-parameter family of realizations, a nonsingular configuration exceeds the bound through a global dependence among interior vertex cycles.

5 Conclusion

  • The family fixes an 18-triangle complex and moves only its central vertex over the admissible interval I = (−3/4, 24/55).The lower bound is attained for every other parameter in I.Source: §5 · p. 8
  • At t = 3/83, the central four-star is singular and the local correction σ = 1 accounts for dimension 34.For t in I excluding 1/5 and 3/83, the dimension is 33 and the lower bound is attained.Source: §5 · p. 8
  • At t = 1/5, dim S_3^1(T(1/5)) = 34 exceeds P_T(1/5)(1, 3) = 33 despite all interior vertices being nonsingular.This provides the counterexample to the conjecture that Schumaker’s lower bound is always attained.Source: §5 · p. 8
  • At t = 1/5, dependence couples all seven interior vertex cycles although every individual block has full row rank; the Bernstein–Bézier calculation gives the same dimension profile.The exceptional dependence at t = 3/83 is instead confined to the central vertex block.Source: §5 · p. 9
  • The singular-four-star correction does not capture every geometry-dependent dimension contribution; global compatibility also matters.Source: §5 · p. 9
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