Adaptive Interpolatory Curve Subdivision with Learned Local Angles

Technical Summary: Adaptive Interpolatory Curve Subdivision with Learned Local Angles

Problem Statement
Interpolatory curve subdivision is a fundamental technique in computer graphics for generating smooth geometric objects from control polygons while ensuring the limit curve passes through the original control points. Classical schemes, such as the four-point and six-point rules, rely on a single global tension parameter ($\mu$) to govern the insertion of new vertices. While these methods preserve the interpolatory property, the fixed global parameter limits their adaptability; a single tension value cannot simultaneously optimize performance across flat regions, sharp turns, and varying local geometries. Furthermore, existing adaptive interpolatory schemes (e.g., $\kappa$-curves, centripetal Catmull–Rom) are largely restricted to the Euclidean plane, while curves on non-Euclidean manifolds (the two-sphere and the Poincaré disk) typically rely on manifold extensions of fixed-tension rules. The authors investigate whether a compact, learned predictor can improve subdivision fidelity and smoothness across Euclidean, spherical, and hyperbolic geometries while strictly preserving the structural property of interpolation.

Methodology
The proposed approach reformulates interpolatory subdivision by replacing the deterministic mapping from local exterior angles to insertion angles with a learned function.

• Structural Interpolation: The method maintains a strict structural invariant: original control vertices are copied exactly at every refinement level ($q_{2j} = p_j$). New vertices ($q_{2j+1}$) are inserted along geodesics between existing vertices. This ensures that interpolation is a property of the operator's architecture, not a learned behavior, guaranteeing zero positional error for retained vertices regardless of the predicted angles.
• Learned Local-Angle Predictor: A compact, edge-wise multilayer perceptron (MLP) predicts a bounded insertion angle $\alpha_j$ for each edge. The input feature vector includes intrinsic local exterior angles, normalized edge lengths, and a discrete curvature code ($\kappa \in \{-1, 0, +1\}$) indicating the model space (Euclidean, Spherical, or Hyperbolic). A learned embedding maps this curvature code to condition the network.
• Geometry-Specific Primitives: The same learned predictor is applied across all three constant-curvature manifolds ($E^2, S^2, H^2$) by combining the shared weights with geometry-specific geodesic primitives (log-exp maps) for vertex insertion.
• Training Objective: The predictor is trained on synthetic control polygons with known dense ground-truth sampling. The loss function combines a symmetric Chamfer fidelity term (using geodesic distance), a smoothness penalty on predicted exterior angles, and a bending energy penalty. The training targets are dense reference curves intended as sparse-recovery references rather than unique optimal interpolants.
• Evaluation Protocol: To ensure fair comparison, all output curves are resampled to a matched density using geometry-aware arc-length interpolation before computing metrics. This prevents artifacts caused by differing native sampling densities (e.g., vertex clustering in parametric baselines).

Key Contributions

1. Learned Local-Angle Selector: Introduction of a compact, edge-wise predictor that maps intrinsic local features to bounded insertion angles, enabling local adaptivity without sacrificing the interpolatory structure.
2. Structurally Interpolatory Implementation: A formulation where interpolation is guaranteed by the operator's design (copying original vertices exactly), independent of the trained weights or predicted angles.
3. Unified Cross-Geometry Framework: A single shared predictor conditioned on geometry that operates effectively across the Euclidean plane, the two-sphere, and the Poincaré disk, utilizing geometry-specific geodesic primitives.
4. Corrected Evaluation Protocol: A rigorous comparison framework including a validation-set fixed-$\mu$ oracle, arc-length-matched smoothness metrics, and structural invariant tests to ensure fair benchmarking against fixed-tension and adaptive baselines.

Results
Under a matched-density evaluation protocol, the learned predictor demonstrates significant improvements over fixed-tension baselines:

• Fidelity: The method reduces nearest-neighbour error by factors of 6.1 (Euclidean), 4.7 (Spherical), and 17.0 (Hyperbolic) compared to the best validation-tuned fixed-$\mu$ baseline. In the Euclidean case, it also outperforms centripetal Catmull–Rom by a factor of 1.8 in mean nearest-neighbour error.
• Smoothness: Bending energy and tangent roughness ($G^1$ proxy) are reduced by one to three orders of magnitude compared to fixed-tension schemes.
• Generalization: On an out-of-distribution real-world dataset (the International Space Station ground track), the learned predictor produced a substantially smoother refined curve (reducing bending energy by factors of 66 to 184) while maintaining comparable fidelity to baselines.
• Efficiency: A single shared geometry-conditioned model (28,737 parameters) performs competitively against three independently trained per-geometry models (86,211 parameters total), recovering most of the performance at one-third the parameter budget.

Significance and Claims
The authors position this work as a specific study within the angle-based subdivision framework rather than a replacement for the broader class of classical interpolatory curve-design methods.

• Scope: The method is presented as an adaptive subdivision operator that improves the fidelity-smoothness trade-off over fixed global-tension rules. It does not claim to replace spline-based methods that offer strong analytical guarantees on curvature behavior (e.g., $\kappa$-curves, $C^2$ interpolating splines).
• Limitations: The paper explicitly acknowledges that the method is currently limited to closed curves with supervised training from dense reference samples. It does not provide formal convergence guarantees for the trained operator, nor does it offer a pathway for unsupervised refinement from control polygons alone.
• Conclusion: The primary contribution is demonstrating that a learned local-angle formulation can significantly enhance subdivision performance across multiple constant-curvature geometries while preserving the critical structural property of interpolation, offering a practical, compact solution for cross-geometry curve design.