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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.

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.

At $X=P(u,v)$, orient the unit normal vector $n=(P_u\times P_v)/\Vert P_u\times P_v\Vert$ towards the eye. Define the unit directions from the reflecting point to the source and eye byThe incident propagation direction is $-a$. The law of specular reflection is $b=-a+2(a\cdot n)n$, so $a+b$ is normal to the surface. Conversely, if the tangential components of $a+b$ vanish and $a\cdot n,b\cdot n>0$, its unit-length summands have equal positive normal components and obey the reflection law. Therefore solve the two scalar equationsThe stated orientation condition gives $b\cdot n>0$. It removes the back-facing ambiguity, but not multiple images or caustics. Exclude $Q(t)=P(u,v)$ and $E=P(u,v)$, where these directions are undefined.

At a root where the $2\times 3$ Jacobian matrix $DF$ has rank two, the implicit function theorem makes the solution set a local curve in parameter space. Compute a unit tangent $w$ in its nullspace; equivalently take the normalized cross product of the two rows of $DF$. A continuation method predicts $x_p=x+hw$, where $x=(u,v,t)$, then corrects with the Newton method applied to the three equationsThe last equation fixes a transverse section and allows tracing through a turning point of $t$, where solving only for $(u,v)$ at successive fixed source parameters would fail. The corresponding space point is $R=P(u,v)$, with derivative $R^{\prime }=P_u{u}^{\prime }+P_v{v}^{\prime }$ along the traced parameter-space curve.

Find initial roots by subdividing the parameter cube, using interval arithmetic bounds on $F$ to discard impossible boxes and validated local solves in the remaining boxes. Trace in both directions, stop at the boundary or a recognized closed loop, and retain a record of traced boxes to avoid duplicates. Search the remaining boxes as well: starting from just one seed would miss disconnected closed reflection curves. Where $DF$ loses rank, reduce the step and subdivide locally to find the outgoing branches instead of continuing a single guessed tangent. If physical visibility is required, discard roots for which either open segment $XE$ or $XQ(t)$ is occluded; test these segments against the surface or any supplied scene geometry.

Control the geometric approximation rather than just the parameter step. For any regular traced arc $R(s)$ with a bound $\Vert R^{\prime \prime }(s)\Vert \le M$, the chord error bound gives an arc-to-chord Hausdorff distance at most $M(\Delta s{)}^2/8$. Budget, for example, half of $ϵ$ for corrected-vertex error and half for chord error; accept a segment only whenBound the vertex error through the inverse transverse Jacobian matrix and a bound on $DP$, or use validated root enclosures. If analytic second-derivative bounds are unavailable, recursively subdivide and enclose each arc in a geometric tube of radius at most the remaining tolerance around its chord. A midpoint test alone is a useful heuristic but cannot certify the requested error for an arbitrary oscillatory curve.

The returned vertices $P(u_k,v_k)$ and their ordered connecting segments give the reflection curve to the prescribed tolerance on every resolved branch. Finite certified termination requires regular evaluable patches, derivative or enclosure bounds, and finitely resolvable branches; eye-facing normals alone do not guarantee these properties. Singular components or an identically vanishing reflection constraint must be handled as such rather than silently reported as an ordinary curve.

Use the limit subdivision surface and limit subdivision curve, rather than treating their initial control meshes as the reflecting geometry. Build adaptive refinement hierarchies for both. Each surface cell supplies a limit-position evaluator, local tangent vectors and a normal vector; each curve interval supplies a limit-position evaluator and tangent. Regular regions may be evaluated as local splines, while extraordinary neighborhoods require the evaluation machinery of the specified subdivision rules. A piecewise smooth surface is handled patch by patch, with explicit boundary or crease joins.

A useful initial polygonal construction refines the surface to small triangles and the source curve to small line segments. For a candidate mirror triangle in a plane $n\cdot (X-X_0)=0$, reflect the eye across that plane:For a source edge $Y(t)=(1-t)Y_0+t{Y}_1$, a physical reflection point is the plane intersection of the line $E^{\ast }Y(t)$:Keep only $0\le t\le 1$, $0<\lambda <1$, points inside the mirror triangle, and eye/source points on the appropriate reflecting side. Unfolding the reflected path across the mirror proves the specular reflection condition. As $Y$ moves along a straight edge, $X$ lies on the intersection of the mirror plane with the plane through $E^{\ast },Y_0,Y_1$, so the retained locus is a straight segment, a point or the empty set. Plane and triangle clipping computes its endpoints. If these defining points are collinear, or a denominator vanishes, use the corresponding limiting geometric case rather than a generic plane formula. A bounding volume hierarchy accelerates candidate search where geometric bounds exclude pairs; refined facet segments provide seeds and approximate connectivity.

To obtain the reflection curve of the actual limits, correct these seeds. In a local surface chart write $S(u,v)$, and on a source interval write $C(t)$. SetThe exact limit reflection locus is $F=0$ together with the front-side conditions. For a two-sided mirror, orient $n$ locally so that $b\cdot n>0$ and require $a\cdot n>0$; for a one-sided mirror use its prescribed orientation and discard the wrong side. At grazing incidence this orientation condition ceases to be strict and the case needs separate handling. Unlike the eye-facing assumption in the parametric question, no global orientation condition is supplied here.

Apply a continuation method: at a rank-two root of $DF$, predict in a nullspace tangent direction and correct by the Newton method on a transverse section. Map corrected parameters to $S(u,v)$, crossing patch and curve-interval boundaries through their hierarchy adjacency. Trace all branches, recognizing closed components and endpoints. Supplement facet seeds by recursive searches of the surface-cell/source-interval product, with interval arithmetic or other certified enclosures of $F$; otherwise a small loop absent from the initial polygonal approximation could be missed. Split singular neighborhoods where the Jacobian matrix loses rank, and test the open source-to-mirror and mirror-to-eye segments for occlusion when selecting actually visible components.

Allocate a geometric error budget to limit evaluation, root correction and polygonal chords. For example, require combined position-evaluation and corrected-root error at most $ϵ/2$ and, on every regular arc, require the chord error bound $M(\Delta s{)}^2/8\le ϵ/2$. Use subdivision position and normal bounds for evaluations: a tiny position error alone is insufficient, since a wrong normal vector changes the reflected ray. Away from singularities an inverse transverse Jacobian matrix bound converts reflection residual into parameter error and a bound on $DS$ converts this to space error. Refine until these bounds hold, or use validated enclosures of entire arcs and their distances to the chords.

Near a caustic the inverse bound can become large, so the hierarchy must refine more strongly or isolate the singular branch geometry. Mesh flatness alone does not certify a reflection tolerance. With regular convergent evaluators and finite resolvable components, the resulting ordered vertices and chords satisfyThe problem supplies no particular subdivision rules or smoothness guarantees, so this construction specifies what their evaluators and bounds must provide; an arbitrary nonsmooth or degenerate limit need not have an ordinary finite reflection curve. Refine for seeds, enforce reflection on the actual limit geometry, then refine the traced arcs to the required spatial tolerance.