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A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries

Hassan Ugail, Newton Howard

Abstract

Interpolatory subdivision schemes generate smooth curves from piecewise-linear control polygons by repeatedly inserting new vertices. Classical schemes rely on a single global tension parameter and typically require separate formulations in Euclidean, spherical, and hyperbolic geometries. We introduce a shared learned tension predictor that replaces the global parameter with per-edge insertion angles predicted by a single 140K-parameter network. The network takes local intrinsic features and a trainable geometry embedding as input, and the predicted angles drive geometry-specific insertion operators across all three spaces without architectural modification. A constrained sigmoid output head enforces a structural safety bound, guaranteeing that every inserted vertex lies within a valid angular range for any finite weight configuration. Three theoretical results accompany the method: a structural guarantee of tangent-safe insertions; a heuristic motivation for per-edge adaptivity; and a conditional convergence certificate for continuously differentiable limit curves, subject to an explicit Lipschitz constraint verified post hoc. On 240 held-out validation curves, the learned predictor occupies a distinct position on the fidelity--smoothness Pareto frontier, achieving markedly lower bending energy and angular roughness than all fixed-tension and manifold-lift baselines. Riemannian manifold lifts retain a pointwise-fidelity advantage, which this study quantifies directly. On the out-of-distribution ISS orbital ground-track example, bending energy falls by 41% and angular roughness by 68% with only a modest increase in Hausdorff distance, suggesting that the predictor generalises beyond its synthetic training distribution.

A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries

Abstract

Interpolatory subdivision schemes generate smooth curves from piecewise-linear control polygons by repeatedly inserting new vertices. Classical schemes rely on a single global tension parameter and typically require separate formulations in Euclidean, spherical, and hyperbolic geometries. We introduce a shared learned tension predictor that replaces the global parameter with per-edge insertion angles predicted by a single 140K-parameter network. The network takes local intrinsic features and a trainable geometry embedding as input, and the predicted angles drive geometry-specific insertion operators across all three spaces without architectural modification. A constrained sigmoid output head enforces a structural safety bound, guaranteeing that every inserted vertex lies within a valid angular range for any finite weight configuration. Three theoretical results accompany the method: a structural guarantee of tangent-safe insertions; a heuristic motivation for per-edge adaptivity; and a conditional convergence certificate for continuously differentiable limit curves, subject to an explicit Lipschitz constraint verified post hoc. On 240 held-out validation curves, the learned predictor occupies a distinct position on the fidelity--smoothness Pareto frontier, achieving markedly lower bending energy and angular roughness than all fixed-tension and manifold-lift baselines. Riemannian manifold lifts retain a pointwise-fidelity advantage, which this study quantifies directly. On the out-of-distribution ISS orbital ground-track example, bending energy falls by 41% and angular roughness by 68% with only a modest increase in Hausdorff distance, suggesting that the predictor generalises beyond its synthetic training distribution.

Paper Structure

This paper contains 38 sections, 5 theorems, 36 equations, 9 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Mathematical Foundations
  4. Notation and Setting
  5. Exterior Angles as Universal Intrinsic Descriptors
  6. The Angle-Based Subdivision Operator
  7. The Classical Tension Assignment
  8. Evaluation Metrics
  9. Theoretical Analysis
  10. Structural G1-Safety Bound
  11. Necessity of Adaptive Tension
  12. Conditional C1 Convergence via Proximity
  13. Tension Predictor Architecture and Loss
  14. Problem Formulation
  15. Input Feature Vector
  16. ...and 23 more sections

Key Result

Theorem 1

Let $f_\theta : \mathbb{R}^7 \to \mathbb{R}$ denote the scalar output of the tension predictor before the final rescaling. Define, with $(\alpha_{\min}, \alpha_{\max}) = (-\tfrac{\pi}{4}+0.02,\; \tfrac{\pi}{4}-0.02)$ and $\sigma$ the standard logistic function. Then for any $\theta \in \mathbb{R}^n$ and any input $\mathbf{x}_j \in \mathbb{R}^7$,

Figures (9)

  • Figure 1: Qualitative comparison on Euclidean $\mathbb{E}^2$ (near-circular polygon). Four panels: control polygon; four-point ($\mu{=}0$); six-point ($\mu{=}{-}0.25$); learned predictor ($k{=}4$ levels). Ground truth dashed (grey).
  • Figure 2: Qualitative comparison on Euclidean $\mathbb{E}^2$ (elongated ellipse). Four panels: control polygon; four-point ($\mu{=}0$); six-point ($\mu{=}{-}0.25$); learned predictor ($k{=}4$ levels). Ground truth dashed (grey). High-curvature tips visible at either end of the major axis.
  • Figure 3: Qualitative comparison on Spherical $\mathbb{S}^2$ (four-lobe curve). Four panels as in Fig. \ref{['fig:qual_eucl_1']}. Angular oscillations are visible at each lobe in the classical outputs.
  • Figure 4: Qualitative comparison on Spherical $\mathbb{S}^2$ (irregular open-style curve). Four panels as in Fig. \ref{['fig:qual_eucl_1']}. Severe jagged artefacts are visible in the classical outputs.
  • Figure 5: Metric comparison for the three primary baselines across 240 validation curves at the epoch-200 checkpoint. Rows: $\mathbb{E}^2$, $\mathbb{S}^2$, $\mathbb{H}^2$. Columns: symmetric Hausdorff distance ($d_H$), bending energy and $G^1$ proxy (all lower is better). Error bars show $\pm1$ standard deviation. Annotated percentages give Hausdorff change relative to the four-point baseline. Log-exp lifts and the oracle are omitted for clarity; see Table \ref{['tab:aggregate']} for the full comparison.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2: Heuristic motivation for adaptivity
  • Remark 3: Empirical ceiling of fixed-tension
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Remark 4
  • ...and 3 more