= Solution
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:
$$
E^*=E-2\big(n\cdot(E-X_0)\big)n.
$$
For a source edge $Y(t)=(1-t)Y_0+tY_1$, a physical reflection point is the plane intersection of the line $E^*Y(t)$:
$$
X(t)=E^*+\lambda(t)(Y(t)-E^*),\qquad \lambda(t)=\frac{n\cdot(X_0-E^*)}{n\cdot(Y(t)-E^*)}.
$$
Keep only $0\le t\le1$, $0<\lambda<1$, 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 $E^*,Y_0,Y_1$, 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 $C(t)$. Set
$$
a=\frac{C(t)-S(u,v)}{\lVert C(t)-S(u,v)\rVert},\qquad b=\frac{E-S(u,v)}{\lVert E-S(u,v)\rVert},\qquad F=\begin{pmatrix}(a+b)\cdot S_u\\(a+b)\cdot S_v\end{pmatrix}.
$$
The 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 $\epsilon/2$ and, on every regular arc, require the <chord error bound> $M(\Delta s)^2/8\le\epsilon/2$. 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 satisfy
$$
\boxed{d_H\big(\text{visible limit reflection locus},\text{returned polygonal locus}\big)\le\epsilon.}
$$
The 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. \b[Refine for seeds, enforce reflection on the actual limit geometry, then refine the traced arcs to the required spatial tolerance.]
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