= 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.
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