Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 59 / 2 / Solution

An efficient parametric curve interrogation interface should provide the parameter domain and open/closed status, point evaluation, one-sided endpoint enquiries, first and second derivatives wherever defined, and information about knots or singular parameters. It should also restrict a curve to an interval and subdivide it without changing its limit, return a certified bounding volume for that restriction, and supply derivative or chord error bounds when supported. These enquiries allow cheap geometric rejection before expensive exact evaluation. For a subdivision curve, limit evaluation is distinct from returning a vertex of a finite refined control polygon; the latter is usually an approximation.

For a query point $Q$, minimize $d(t)=\frac{1}{2}\Vert P(t)-Q{\Vert }^2$. At a differentiable interior minimizer,At a regular curve point this says that the query displacement is perpendicular to the tangent vector. If a second derivative exists, the local second-order test isa strictly positive value is sufficient for a strict local minimum. The stationarity equation alone also includes local maxima and is automatic at $P^{\prime }=0$, so it cannot be used as the only global search criterion.

A reliable efficient closest-point search on a subdivision curve uses branch and bound. Cover the compact curve by restricted pieces with certified enclosures $B_I$. For each piece set $L_I=\mathrm{dist}(Q,B_I)$, a lower bound on its true distance. Distances to actually evaluated limit points give upper bounds; keep the best point $P(t_{\ast })$ and distance $U$. For an axis-aligned bounding box ${\prod }_k[l_k,h_k]$, the squared lower bound is inexpensive:For positive partition-of-unity refinement, a convex hull of the active control points is an enclosure. A negative-weight scheme requires a separate certified enclosure; its control hull need not contain the limit.

The algorithm is:

Because ${\min }_I{L}_I\le d_{\min }\le U$, the reported point is at most $ϵ$ farther from $Q$ than a true nearest point. Include endpoints, nondifferentiable joins, and singular parameters in the candidate bookkeeping, or split them into separate pieces. With enclosures converging in diameter under subdivision on a compact continuous curve, the bounds converge to arbitrary prescribed positive distance tolerance. Interval bounds or outward-rounded arithmetic are needed if the guarantee is intended to be certified in floating-point computation. A single Newton root-finding iteration from a single seed supplies no such global guarantee.