A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries

Hassan Ugail; Newton Howard

cs.LGMar 30, 2026

A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries

Authors: Hassan Ugail, Newton Howard

Organizations: Centre for Visual Computing and Intelligent Systems University of Bradford United Kingdom · School of Individualized Study Rochester Institute of Technology United States

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.

Figures & tables

Geometry	Method	Mean-NN ( $\downarrow$ )			Bending $E_{B}$ ( $\downarrow$ )	$G^{1}$ proxy ( $\downarrow$ )
Geometry	Method	Mean $\pm$ Std		vs. oracle	Mean	Mean
$\mathbb{E}^{2}$	Four-point ( $\mu{=}0$ )	0.03538 $\pm$	0.0089	—	1477.5	2.655
	Six-point ( $\mu{=}-0.25$ )	0.03623 $\pm$	0.0086	—	2355.3	2.824
	Log-Exp 4pt	0.00971 $\pm$	0.0057	—	2025.7	0.496
	Log-Exp 6pt	0.00884 $\pm$	0.0055	—	2788.4	0.359
	Best $\mu^{*}$ ( $+0.050$ )	0.03525 $\pm$	0.0089	—	1374.1	2.630

Table 1: Aggregate evaluation across all three geometries (240 validation curves) at the epoch-200 checkpoint (

k=5

subdivision levels, 12 control points

\to

384 output points). Bold indicates the best Neural Operator mean-NN value. “vs. oracle” gives the percentage reduction in mean-NN relative to the best fixed-

\mu

oracle; positive values indicate improvement.

\mu^{*}

is the oracle tension from the validation set grid search.

E_{B}

values are raw (not normalised);

G^{1}

values are in radians.

\mathbb{H}^{2}

raw

E_{B}

values are large for all methods owing to the Poincaré disk chord-length normalisation; the

G^{1}

proxy is the appropriate smoothness measure for

\mathbb{H}^{2}

(see Section 7.2 for full explanation).

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. 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. 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 $\pm 1$ 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 1 for the full comparison.

Method	$\sigma{=}0$	$\sigma{=}0.03$	$\sigma{=}0.06$	$\sigma{=}0.10$	$\sigma{=}0.15$	$\sigma{=}0.20$
Four-point ( $\mu{=}0$ )	2.655 (0.32)	2.70 (0.33)	2.76 (0.35)	2.84 (0.38)	2.95 (0.42)	3.08 (0.47)
Six-point ( $\mu{=}-0.25$ )	2.824 (0.35)	2.88 (0.36)	2.94 (0.38)	3.05 (0.42)	3.18 (0.47)	3.34 (0.54)
Lin. Adapt. Heur. (LAH)	1.480 (0.28)	1.52 (0.30)	1.57 (0.33)	1.65 (0.37)	1.76 (0.43)	1.89 (0.51)
Neural predictor	0.237 (0.08)	0.258 (0.09)	0.287 (0.11)	0.334 (0.14)	0.396 (0.17)	0.462 (0.21)
Improv. vs. 4pt	$91\%$	$90\%$	$90\%$	$88\%$	$87\%$	$85\%$

Table 2: Robustness to noisy control polygons (

\mathbb{E}^{2}

k{=}4

levels, 240 validation curves). Gaussian noise with standard deviation

\sigma

is added independently to each control-polygon vertex. Values show the mean

G^{1}

proxy (rad,

\downarrow

) with

\pm 1

SD in parentheses. The neural predictor degrades gracefully: its advantage over the four-point baseline narrows only from

91\%

\sigma{=}0

85\%

\sigma{=}0.20

. The linear adaptive heuristic (LAH) captures roughly half the neural predictor’s advantage, confirming that nonlinear learned adaptivity is required for the full smoothness benefit.

Condition	Params	Mean-NN ( $\downarrow$ )			$G^{1}$ proxy ( $\downarrow$ )
		$\mathbb{E}^{2}$	$\mathbb{S}^{2}$	$\mathbb{H}^{2}$	$\mathbb{E}^{2}$	$\mathbb{S}^{2}$	$\mathbb{H}^{2}$
Learned 8-dim	140,505	0.01673	0.05438	0.00434	0.219	0.256	0.167
One-hot 3-dim	139,841	0.01660	0.05498	0.00422	0.223	0.247	0.152
No-geom	139,457	0.01531	0.05116	0.00559	0.267	0.263	0.249

Table 3: Geometry-embedding ablation (100 epochs, early-stopping patience = 20). Mean-NN (

\downarrow

) and

G^{1}

proxy (

\downarrow

) per geometry at the best checkpoint. Cross-geometry NN is the arithmetic mean of the three per-geometry mean-NN values.

Condition	Mean-NN ( $\downarrow$ )			$G^{1}$ proxy ( $\downarrow$ )			Norm. Bending ( $\downarrow$ )
	$\mathbb{E}^{2}$	$\mathbb{S}^{2}$	$\mathbb{H}^{2}$	$\mathbb{E}^{2}$	$\mathbb{S}^{2}$	$\mathbb{H}^{2}$	$\mathbb{E}^{2}$	$\mathbb{S}^{2}$	$\mathbb{H}^{2}$
Full	0.01842	0.05436	0.00414	0.370	0.296	0.217	0.075	0.918	0.033
No equiv ( $\lambda_{E}=0$ )	0.01651	0.05195	0.00418	0.248	0.322	0.155	0.012	0.046	0.021
No bending ( $\lambda_{B}=0$ )	0.01681	0.05141	0.00450	0.248	0.254	0.160	0.018	0.096	0.023
No smooth ( $\lambda_{S}=0$ )	0.02211	0.05843	0.00751	0.442	1.121	0.420	0.029	0.599	0.043

Table 4: Loss-component ablation (100 epochs, early-stopping patience = 20). Mean-NN (

\downarrow

G^{1}

proxy (

\downarrow

) and normalised bending (NB, relative to four-point = 1.00,

\downarrow

) at the best checkpoint. “Full” reproduces the complete loss; each subsequent row zeros the indicated term.

Method	Hausdorff ( $\downarrow$ )	Bending $E_{B}$ ( $\downarrow$ )	$G^{1}$ proxy ( $\downarrow$ )
Four-point ( $\mu=0$ )	0.00414	129.6	2.235
Six-point ( $\mu=-0.25$ )	0.00529	253.6	2.906
Learned predictor	0.00699	76.0	0.721
Improv. vs. 4pt	$-69.0\%$	$+41.4\%$	$+67.8\%$

Table 5: ISS ground-track evaluation (

k=5

levels, 16 control points

\to

512 output points). Positive improvement favours the learned predictor; the negative Hausdorff value reflects the fidelity–smoothness trade-off (see Section 8.1 for interpretation).

Figure 6: Qualitative comparison on Hyperbolic $\mathbb{H}^{2}$ (near-circular curve). Four panels as in Fig. 1 . Boundary oscillations are visible in the classical outputs near the Poincaré disk edge.

Figure 7: Qualitative comparison on Hyperbolic $\mathbb{H}^{2}$ (second near-circular curve). Four panels as in Fig. 1 . Small oscillatory indentations are visible in the classical outputs.

Figure 8: ISS orbital ground track on $\mathbb{S}^{2}$ (16 control points $\to$ 512 output points, 5 subdivision levels). Top row: equirectangular map projections of the three methods against the 520-point ground truth (dashed grey). Bottom row: corresponding 3D globe views. Colour coding: four-point (blue), six-point (pink), learned predictor (orange).

Figure 9: ISS ground-track metric comparison ( $\mathbb{S}^{2}$ , $k{=}5$ levels, 16 control points $\to$ 512 output points). Normalised bar chart of Hausdorff distance, bending energy and $G^{1}$ proxy for all three methods; raw values annotated on each bar.

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Geodesics with Unified Tangent-constrained Priors and Curvature Regularization

Curvature-penalized geodesic models have proven their effectiveness in image segmentation by computing globally optimal curves. Unfortunately, these models remain susceptible to shortcuts when delineating objects with complex shapes and image intensity distributions, as they lack mechanisms to enforce shape-aware tangent constraints. To address this limitation, we propose a unified geodesic framework that integrates tangent-constrained priors with curvature penalization. The key idea is to formulate tangent admissibility directly within the orientation-lifted space, where path tangents are restricted to spatially varying angular sectors derived from intrinsic shape representatives (ISR) such as skeletons or interior landmarks. This formulation gives rise to a family of tangent-constrained Finslerian metrics, extending the classical curvature-penalized geodesic models while enforcing mandatory tangent constraints. The resulting Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) admit efficient numerical solutions via variants of the fast marching method, preserving the single-pass computational complexity. Experiments on synthetic, natural, and medical images demonstrate that the proposed geodesic framework indeed improves robustness against weak boundaries and topological shortcuts, yielding segmentation results with enhanced shape fidelity compared to existing geodesic models.

Chong Di, Li Liu, Jinglin Zhang +3

Shandong Artificial Intelligence Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan, Shandong, China · Yuanshen Rehabilitation Institute, Shanghai Jiao Tong University School of Medicine, Shanghai 200025, China · School of Control Science and Engineering, Shandong University, Jinan, Shandong, China +2

Finsler Geometry Medical Image Segmentation

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Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another

Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from

-1

0

, while the embedding remains at curvature

-1

. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.

Federico Larroca, Paola Bermolen, Marcelo Fiori +1

Facultad de Ingenier´ıa Universidad de la Rep´ublica Uruguay

Riemannian Optimization Hierarchical Representation Learning Graph Embedding Hyperbolic Representation Learning

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We introduce the directional mollification operator, which acts on polygonal chains to produce

C^{\infty}

curve approximants that are arbitrarily close to the original curve -- pointwise and uniformly on compact subsets -- while strictly interpolating its vertices. Unlike standard mollification, which fails to preserve vertices, this directional construction enables local, vertex-preserving smoothing; modifying a single segment alters the

C^{\infty}

output only within an explicitly controllable neighborhood. The operator admits closed-form curvature bounds and provides analytic control over curve geometry. Furthermore, we develop a parametric family of smoothing operators that unifies conventional and directional mollification into a single framework. Combining computational simplicity with strict waypoint fidelity, this method is directly applicable to geometric modeling, robotics, computer graphics, and CNC machining.

Alfredo González-Calvin, Juan F. Jiménez, Héctor García de Marina

Dept. of Computer Architecture and Automation, Faculty of Physics, Complutense University of Madrid, Plaza de las Ciencias, 1, Madrid, 28040, Madrid, Spain · Dept. of Computer Engineering, Automation, and Robotics& Institute of Mathematics, University of Granada, Calle Ventanilla, Granada, 18001, Granada, Spain

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