Adaptive Interpolatory Curve Subdivision with Learned Local Angles

Curve subdivision is pivotal in computer graphics for generating smooth geometric objects from control polygons. Interpolatory subdivision is especially attractive because the refined curve is guaranteed to pass through the designer's control points. Classical four-point and six-point schemes preserve this property, but their behaviour is governed by a single global tension parameter, limiting their ability to adapt across flat regions, sharp turns and varying local geometries. We introduce an adaptive local-angle formulation that keeps the interpolatory structure intact while learning how each new vertex should be inserted. A compact edge-wise predictor assigns one insertion angle per edge, while the original vertices are copied exactly at every refinement level. Interpolation is therefore a structural property of the operator and does not depend on the trained weights. The same predictor is used with geometry-specific geodesic primitives on the Euclidean plane, the two-sphere and the Poincar'e disk. Under a matched-density evaluation protocol, the method reduces nearest-neighbour error by factors of five to seventeen over the best validation-tuned fixed-tension baseline, and by about 1.8 over centripetal Catmull-Rom in the Euclidean case. It also substantially reduces bending energy and tangent roughness, while remaining competitive with separately trained per-geometry models.