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A subdivision surface enquiry acts on the limit surface, not just on the current control mesh. The basic subdivision surface interrogation operations are: enumerate initial patches and their topological adjacency; subdivide a patch into covering children with inherited boundary links; return conservative bounding volumes; estimate approximation error and, where available, normal variation; and evaluate limit positions and first derivatives or tangent data. Boundaries, creases and extraordinary subdivision vertices need explicit rules. Nonnegative convex subdivision weights can supply a local control-hull enclosure, but a scheme with negative weights needs another proved enclosure; an arbitrary control-mesh box is not automatically a bound on the limit.

The intersection of subdivision limit surfaces can be computed by a paired recursive search. Start with every relevant pair of root patches $(A,B)$, along with their domains and adjacency identifiers:

A termination criterion specifies geometric error and conditioning, not merely a fixed refinement depth. For isolated regular transverse components, shrinking limit enclosures and normal/error bounds permit reliable curve approximation. Candidate boxes cannot be discarded merely because no coarse edge has crossed another surface: a complete small closed component can lie in the interior. Tangency, coincident limit patches, creases and poorly conditioned extraordinary-point neighborhoods need special tests or must be reported as unresolved degenerate configurations. The intersection is that of the two subdivision limits; intersecting a finite refined mesh without error control only gives an approximation.

An efficient subdivision surface interrogation interface needs the mesh topology and patch adjacency, boundary and trimming information, patch-local parameter charts and their transition maps, and local restriction/refinement with the complete influencing control net. Numerical enquiries should supply limit positions $S$, first derivatives $S_u,S_v$, and second derivatives where defined; these determine normal vectors and curvatures at regular points. It must also return certified patch bounding volumes and convergent geometric/derivative error bounds. Near an extraordinary subdivision vertex, use the local subdivision matrix and limit/tangent evaluation appropriate to that scheme; second derivatives need not exist merely because a tangent plane does. Finite refined mesh vertices are not exact limit positions.

Represent the cutting plane as $n\cdot x=d$, with $n$ normalized. On each chart setThe desired plane section of a subdivision surface is the image of the zero set $g=0$. For a bounding volume $B$, compute a certified interval $[g_{\min },g_{\max }]$ for the plane functional over $B$. If that interval is strictly positive or strictly negative, reject the patch. With positive partition-of-unity weights, the minimum and maximum of $n\cdot p_i-d$ over all influencing control points already bound its limit patch. For masks without this convex hull property, the interface must provide a different valid enclosure.

An adaptive algorithm that isolates components before tracing is:

This is exhaustive to a chosen tolerance when certified subdivision isolates finitely many regular transverse branches and enclosures converge. If $\Vert \nabla g\Vert \ge \gamma >0$ locally and $\Vert DS\Vert \le M$, a residual tolerance $|g|\le \delta$ corresponds locally to a geometric correction bounded by $M\delta /\gamma$, using the gradient flow toward the level set. Bounds on second derivatives similarly control curve-segment chord error bound. As $\gamma$ approaches zero, a fixed residual tolerance no longer gives a fixed geometric error, so the step and stopping rules must reflect conditioning.

Seeding must not rely just on corner signs. For example $S(u,v)=(u,v,u^2+v^2-r^2)$ cut by $z=0$ has a closed circle wholly inside a patch whose corner values are all positive. Certified range tests plus subdivision retain it; a corner-only marching rule can miss it. Tangencies also change the nature of the output: $S=(u,v,u^2+v^2)$ meets $z=0$ in one isolated point, while a coplanar patch contributes an area rather than a curve. Thus a general plane section is not necessarily a union of regular curves; the algorithm should classify these cases instead of silently fabricating or deleting a branch.

For efficiency, cache subdivision stencils, patch extraction, limit-evaluation coefficients and bounds; reuse child data instead of reconstructing the entire refined mesh. Use local influencing neighborhoods, with the halo dictated by mask support, and keep adjacency/boundary topology separate from changing geometric coordinates. Plane-functional bounds can be computed by refining scalar signed-distance controls, which is cheaper than refining all three coordinates. Combine coarse inexpensive boxes with tighter local convex hull or interval arithmetic tests only where needed. Adapt steps to curvature and transversality, reuse branch seeds, and use regular spline evaluation away from extraordinary points. Stable edge ownership, outward-rounded bounds, scale-aware tolerances, and explicit degeneracy flags are correctness requirements as well as performance issues. The result is a limit-surface intersection computation; merely intersecting a finitely refined control polyhedron with the plane supplies only an uncontrolled approximation.

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.

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.