= Solution
The corresponding <subdivision surface interrogation> needs mesh connectivity and patch adjacency, boundary and crease flags, local coordinate charts with transition maps, and subdivision into smaller patches with their original-chart mappings. Its evaluation enquiry returns a limit point, tangent <derivatives> and a <normal vector>; <second derivatives> are supplied only where the limit is sufficiently smooth. Certified <bounding volumes> and patch approximation errors allow a hierarchy for intersection and proximity searches. Boundary-curve enquiries complete the interface for clipping.
On regular parts of a <subdivision surface>, these evaluations may use its equivalent tensor-product or <box spline> patch. Near an <extraordinary subdivision vertex>, the local <subdivision matrix> determines the limit position through its constant <left eigenvector>, and its tangent modes determine the <normal vector> under the scheme's <characteristic map of a subdivision surface> regularity conditions. Refinement alone does not establish those conditions. A <subdivision surface> may have a well-defined <tangent plane> there without possessing <second derivatives> or <Gaussian curvature>. An interrogation interface must distinguish these cases and can work on surrounding regular charts when a higher <derivative> is unavailable at the vertex itself.
Together, subdivision, bounds, topology and limit evaluation support ray–<subdivision surface> intersection: bounds eliminate irrelevant patches; surviving patches are refined, and regular candidate intersections are corrected using position and first <derivatives>. The returned intersection is a point of the limit <subdivision surface> with an error certificate, rather than automatically a point on the last control mesh.
Back to article page
