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A subdivision curve interrogation interface provides its control/topology data, a parameter interval or address, and a refinement operation returning restricted child pieces with their parameter correspondence. It also provides a certified enclosing bounding volume, endpoints or limit-point enclosures, and an error estimate for replacing a sufficiently small piece by its chord. Positional and derivative limit queries may be implemented by refinement or by conversion to the appropriate spline. Nonnegative masks often imply a control-point convex hull bound; a generic scheme with negative weights needs a separate bound.

These enquiries can be applied to a parametric curve by using interval restriction as the subdivision operation and constructing its enclosure from its representation. For example, a Bézier curve is split by De Casteljau's algorithm, and each child's control-point convex hull encloses exactly that child arc. More generally, if a restricted curve has a certified bound $\Vert C^{\prime }(t)\Vert \le L_I$, the midpoint $m$ suppliesThis ball gives a signed plane-range enclosure. Such an upper derivative bound must be supplied or proved; taking the largest derivative seen in a few samples is not a certificate.

The recursive intersection algorithm is as follows. Start with the whole domain. For each piece with enclosure $B_I$, computeIf ${\alpha }_I>0$ or ${\beta }_I<0$, discard that piece. Otherwise split it and process the children, unless a root has already been uniquely isolated and can be polished as in Question 3. For a convex hull bound, the extrema of this affine plane function are attained among its control points, so only dot products are required. For an axis-aligned bounding box, choose the low or high coordinate in each term according to the sign of $n_i$.

At a small flat piece, the chord-plane parameter iswhen the denominator is nonzero. It supplies an initial estimate, which is corrected on the actual curve. A chord of a piece with geometric error $\epsilon$ gives plane-function error at most $\Vert n\Vert \epsilon$; a transverse root additionally needs a positive derivative lower bound to convert that into a parameter error. A chord hit is not by itself a proof of the exact crossing count. Keep unresolved tangent contacts, merge shared-endpoint hits, and return any certified coplanar parameter intervals separately.

Good speed comes from a bounding volume hierarchy, conservative cheap rejection before accurate evaluation, caching shared subdivision work, local rather than uniform refinement, and fast root polishing only after isolation. Convex-hull or box extrema cost far less than solving every potential intersection. Tight bounds shrink rapidly with subdivision and reduce the active search to neighborhoods of actual crossings. Derivative information can prove monotonicity, accelerate with the Newton root-finding iteration, or tighten range bounds using interval arithmetic. There is no universal fixed speed guarantee when the curve is nearly coplanar or has arbitrarily many intersections: those are genuinely difficult inputs. The representation must provide meaningful bounds and restriction operations; point/derivative enquiries alone do not imply such capabilities.

A useful subdivision curve interrogation supplies the following connected capabilities. It describes the parameter interval, open or closed topology, endpoints and break points; it evaluates the limit point $C(t)$ and, where the scheme guarantees them, $C^{\prime }(t)$ and $C^{\prime \prime }(t)$; and it refines or restricts an interval while retaining the mapping to the original parameter. It also provides enclosing bounds and a certified curve-to-chord error on each restricted piece. These are enough to build clipping, intersections, closest-point searches, tangents, curvature and adaptive sampling.

The enquiries concern the subdivision limit, not just a finite refined control polygon. For a stationary local refinement matrix $A$, a limit vertex can be obtained from a normalized left eigenvector $w^{T}A=w^T$, $w^{T}1=1$, as $w^{T}P$. Derivatives use the appropriately scaled difference schemes or equivalent spline evaluation on regular intervals. The convergence and derivative hypotheses must actually hold for the scheme; no second derivative can be requested at an intrinsic crease.

If the subdivision mask is nonnegative and preserves constants, descendants and limit points lie in the relevant control-point convex hull, giving a useful enclosing volume. For subdivision masks with negative weights, that hull need not be an enclosure: the interface must supply another valid bound, for example from a proven refinement-error estimate. This distinction is important for safe spatial rejection in an algorithm.

A subdivision curve should support subdivision curve interrogation rather than merely supply successively larger control polygons. Its interface should provide the parameter interval, orientation and open/closed status; endpoint and limit-point evaluation to a prescribed accuracy; and tangent vector or derivative evaluation where defined. It should be possible to split a parameter interval, commonly at its midpoint, into equivalent restricted curve definitions while retaining all neighboring controls needed by the refinement rule.

For geometric algorithms, request a certified bounding volume for each restricted curve, such as a convex hull or an axis-aligned bounding box, and the minimum and maximum of a plane's linear function over that bound. Request also a certified flatness bound: the maximum distance of the restricted limit curve from its endpoint chord, together with its spatial extent. These allow safe rejection, adaptive drawing and accuracy control. A positive, constant-reproducing subdivision mask often makes a control-point convex hull a valid bound. With negative mask coefficients this must be proved separately or replaced by a padded bound; it cannot simply be assumed.

Further useful enquiries include roots of a scalar function along the curve, closest-point searches, arc length, and inversion of an arc-length parametrization. They can be implemented through evaluation, differentiation, certified bounds and recursive subdivision. Evaluation, restriction, enclosure and error bounds are the essential primitives needed for reliable clipping and rendering; an unqualified control-polygon approximation is insufficient.