A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries

2026-03-30

AI Summary

This work proposes the first unified subdivision framework capable of operating across constant-curvature spaces—Euclidean, spherical, and hyperbolic—addressing the limitations of traditional interpolation-based subdivision schemes that rely on a single global tension parameter and struggle to handle diverse geometries cohesively. The method employs a lightweight neural network (140K parameters) to learn local intrinsic features and a learnable geometric embedding, enabling per-edge prediction of structurally safe insertion angles. It introduces a constrained Sigmoid output head and a Riemannian manifold-aware subdivision operator, supported by theoretical convergence guarantees and an adaptive tension heuristic. Experiments demonstrate consistent superiority over baselines across 240 validation curves, achieving lower bending energy and angular roughness; notably, on out-of-distribution orbital trajectories, it reduces bending energy by 41% and angular roughness by 68%.

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.

Problem

• subdivision
• constant curvature geometry
• tension parameter
• interpolatory curves
• Riemannian manifolds

Innovation

• neural tension operator
• interpolatory subdivision
• constant curvature geometry
• learned geometric predictor
• Riemannian manifold