A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries

• URL: http://arxiv.org/abs/2603.28937v1
• Date: Mon, 30 Mar 2026 19:16:25 GMT
• Title: A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries
• Abstract summary: Interpolatory schemes generate smooth curves from piecewise-linear control polygons by repeatedly inserting new vertices.<n>We introduce a shared learned tension that replaces the global parameter with per-edge insertion angles predicted by a single 140K- parameter network.<n>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.
• Score: 0.8694591156258423
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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