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A continuation method follows a solution branch of a parameter-dependent equation from a known solution, using nearby solutions to guide the next solve. This is useful when one wants to understand how solutions change, rather than obtain just one isolated answer. A continuation predictor-corrector step follows a tangent vector and then uses a Newton method correction on a transverse section. The same geometry supports both pseudo-arclength continuation through a continuation fold and transversal surface intersection tracing.

Consider the original scalar polynomial exampleIts solution branch bends back twice in $\lambda$. Increasing $\lambda$ alone eventually stops following the same nearby solutions; following the curve in the $(x,\lambda )$ plane can continue around both bends. The continuation folds areFor $|\lambda |<2/(3\sqrt{3})$ there are three distinct real zeros of a function at that fixed $\lambda$, all lying on this one connected curve. A continuation method keeps their geometric relationship visible.

Every continuation method needs a seed, supplied by an independent solve, an explicitly known solution, or another search. Following one solution branch does not guarantee finding every connected component of the solution set. A large step can miss detail or make the correction converge to a distant solution. At branch intersections or a loss of rank of the full Jacobian matrix, the regular-curve argument needs separate analysis. The seed and component limitations also matter in transversal surface intersection tracing.

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.

Work first in the parameter rectangle of the parametric surface. Set $g(u,v)=f(F(u,v))$. With $F_u,F_v$ linearly independent, the chain rule givesThe intersection is transversal precisely when the implicit normal vector $\nabla f(F)$ and the parametric normal vector $F_u\times F_v$ are not parallel. Consequently $(g_u,g_v)\ne (0,0)$, and the implicit function theorem makes $g=0$ a locally regular parameter curve. This is the setting for transversal surface intersection tracing by a predictor-corrector continuation method.

Find one seed on the required component. Subdivide parameter patches, enclosing their images by bounding volumes and testing whether a certified range of $f$ excludes zero. For a positive-weight NURBS, the active control-point convex hull supplies a convenient enclosure; interval arithmetic can tighten the range of the composed function. Search the surviving patches for a bracketed root or a local corrected seed. Merely checking signs at four corners is inadequate: a small closed intersection can lie wholly inside a patch with the same corner signs. An adaptive or certified patch search, or a supplied seed, avoids this failure. For a sphere centered at $C$, use $f(P)=\Vert P-C{\Vert }^2-r^2$ and $\nabla f=2(P-C)$; for a torus, use its implicit polynomial and its differentiated polynomial in the same algorithm.

At a corrected seed ${\eta }_k=(u_k,v_k)$, an unnormalized parameter tangent vector and a unit spatial tangent vector areChoose the sign consistently with the previous tangent vector. Normalize $q=q_0/\Vert DF{q}_0\Vert$ and predict ${\eta }_p={\eta }_k+hq$, so $h$ is approximately a physical arc length step. Correct the predictor by Newton method applied toThe Jacobian has rows $(g_u,g_v)$ and $(T_k\cdot F_u,T_k\cdot F_v)$. At the starting point its first row vanishes along $q$, whereas its second row applied to $q$ is one. The two rows are therefore independent. The transverse-plane condition prevents the corrector from sliding arbitrarily along the intersection. Restrict correction to a neighborhood of the predictor, use damping, and reduce $h$ when correction fails or its displacement becomes excessive; this prevents jumps to another nearby branch.

To obtain a sufficiently dense sequence, impose a physical point-spacing bound, a bound on tangent turning, and a chord-error estimate. For local curvature $\kappa$, the small-step sagitta is approximately $\kappa h^2/8$; estimate curvature from successive tangent vectors and reduce $h$ until this and the desired spacing are acceptable. The correction residual should be small relative to the drawing tolerance. Uniform steps in $u$ or $v$ alone do not ensure spatial density because the surface metric can vary considerably.

For an open component, trace in both tangent directions from the seed. On reaching a parameter boundary, locate the endpoint by solving $g=0$ together with that boundary's parameter equation, rather than accepting an overshoot. For a closed component, stop only after nontrivial travel and return to the initial parameter location with compatible tangent orientation; near-coincidence in three-dimensional space alone can mistake a self-intersection or neighboring branch for closure. Cross periodic seams using the appropriate chart identification. Adaptive physical stepping, local correction, and component-aware termination give the required point sequence. A vanishing parameter gradient signals a tangency or singularity outside the transversal hypothesis and must be reported rather than passed through by this regular algorithm.

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.

A plane section of a subdivision surface is a zero set of the plane functional evaluated on the limit parametric surface. Certified patch bounding volumes reject sign-definite patches. On a regular transverse section, $\nabla g\ne 0$ and the parameter tangent is $(-g_v,g_u)$; predictor-corrector continuation method traces a branch after every component has been isolated. Closed loops, tangencies, coplanar patches and extraordinary-vertex neighborhoods require more than corner-sign tests. Degenerate sections can contain isolated points or two-dimensional portions rather than only regular curves.

At regular transverse intersections, the displayed map has a rank-three $3\times 4$ Jacobian matrix. Its one-dimensional kernel is the parameter tangent, while the physical tangent is parallel to the cross product of the surface normals. A continuation method predicts along that tangent and corrects $F=0$ together with a transverse section equation using Newton method. Patch bounds and component isolation supply seeds; one seed alone cannot discover every disconnected curve or closed loop. Tangency or coincident surfaces violate the regular-curve assumptions and require separate analysis.

For a parametric surface $F(u,v)$ and a scalar implicit equation $f(P)=0$, set $g=f\circ F$. At a regular transversal intersection, $\nabla g\ne 0$, and $(-g_v,g_u)$ is a parameter tangent vector. Normalize its image under $DF$ to choose physical arc length steps. A predictor followed by Newton method correction of $g=0$ and a transverse plane equation gives a local continuation method. Certified patch bounds are useful for finding seeds; corner signs alone do not detect a closed intersection entirely inside a patch.