Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 66 / 6 / i

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.