 Solution

A subdivision surface interrogation interface needs the initial control net, its face/edge/vertex connectivity, the subdivision rule, patch addressing and refinement into child regions that cover the same limit surface. It must supply limit-point evaluation, available tangent/normal data and smoothness information, including the treatment of extraordinary subdivision vertices. A finite refined mesh point is generally not the exact limit point.

For geometric search it must provide certified bounding volumes for each descendant region, bounds that shrink under refinement, and a mesh-to-limit or chord/patch error estimate. With positive convex subdivision masks the active-control convex hull can provide such an enclosure; schemes with negative coefficients need another valid bound. A closed surface also needs consistent orientation, component navigation and containment queries for the enclosed body. The $C^1$ hypothesis supplies first-order tangent information at regular points, not automatically second derivatives or curvature everywhere. These enquiries distinguish an accurate limit-surface algorithm from a calculation only on an approximate control polyhedron.