= Solution
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, $C'(t)$ and $C''(t)$; 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> $w^TA=w^T$, $w^T\mathbf1=1$, as $w^TP$. <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.
Back to article page
