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There are no numerical continuity bounds determined by $w$ and $a$ alone: the subdivision mask coefficients matter. First check constant reproduction, ${\sum }_j{M}_{ϵ-aj}=1$ for every residue $ϵ$, and convergence. For the usual stationary scalar scheme, two useful calculations are a uniform contraction test and a spectral obstruction test. Higher smoothness is tested with derived difference subdivision masks.

Write $M(z)={\sum }_j{M}_j{z}^j$ and $s_a(z)=1+z+\cdots +z^{a-1}$. Using backward differences $\Delta P_i=P_i-P_{i-1}$, the generating-function identityproves the subdivision difference scheme relation $\Delta S_M=S_{M/s_a}\Delta$. Iterating gives ${\Delta }^r{S}_M=S_{M/s_a^r}{\Delta }^r$. To test a candidate $C^r$ limit, suppose the requisite sum rules hold, so $M/s_a^{r+1}$ is a finite Laurent polynomial. The scaled derivative controls $q_i^{\ell }=a^{r\ell }{\Delta }^r{P}_i^{\ell }$ are refined by the subdivision maskwhose residue sums are one. Their first differences are refined byThe powers of $a$ are essential because each refinement shrinks parameter intervals by $a$.

A lower continuity bound comes from proving uniform contraction. The infinity norm of one difference-refinement step is bounded byIf $q<1$, adjacent derivative controls contract geometrically at every location. More generally, form the $p$-step subdivision maskand test ${\max }_{ϵ\mathrm{mod}{a}^p}{\sum }_j|(B_r^{[p]}{)}_{ϵ-a^{p}j}|<1$. This proves convergence of the constant-preserving derivative scheme to a continuous limit. One justification is to compare consecutive piecewise-linear interpolants: their difference is bounded by a constant times the maximum adjacent-control difference, and the geometrically decreasing bound is summable. To identify the limit as $P^{(r)}$, use fixed-degree Cardinal B-spline interpolants of the original refined controls. Their $r$th derivatives are sums of the scaled finite differences; convergence of both functions and derivatives gives $P\in C^r$. Testing successive $r$ supplies a guaranteed lower bound.

An upper continuity bound comes from an observable noncontracting shape mode. Build finite local subdivision matrices for $B_r$ on a refinement-invariant control neighborhood. For a digit $ϵ$ their entries are $(A_{ϵ}{)}_{ij}=(B_r{)}_{ϵ+i-aj}$, since the new index is $ak+ϵ+i$. Restrict to the realizable difference space and discard modes that do not affect the limit. A word ${ϵ}_1,\dots ,{ϵ}_p$ corresponds to a nested parameter location. If its product has an observable eigenvalue $\lambda$ with $|\lambda |>1$, repetition of this word makes some scaled derivative differences grow rather than tend to zero. For a stable limit representation, $C^r$ regularity would force these differences to vanish; hence general-position data cannot have $C^r$ continuity. A modulus-one mode requires examination of whether it actually persists, rather than an automatic rounding rule.

The two calculations can be organized through the joint spectral radius $\hat{\rho }$ of these restricted matrices:These norm and spectral bounds for subdivision regularity make the test computable from the subdivision mask. The right bound certifies all-word contraction when it is less than one; the left bound detects a repeated-word obstruction when it exceeds one. Longer products improve the tests. A single matrix's spectrum is not a substitute for controlling arbitrary products. The product estimates also appear in Charina's analysis of subdivision regularity; the strict contraction and observable-mode arguments above explain their respective roles.

Suppose ${\varphi }_j\in C^r$ in $u$ and ${\psi }_k\in C^s$ in $v$, including their respective knot joins. The mixed derivatives of the tensor-product surface basis areThese are continuous because each factor is continuous. A finite or locally finite control sum has the same property. In the usual total-order notation the parametric surface is therefore at least $C^{\min (r,s)}$, and also has the stronger stated rectangular set of continuous mixed derivatives. Across a $u$-knot its guaranteed order is $r$, and across a $v$-knot its guaranteed order is $s$. No general improvement over a factor's smoothness is possible: setting $P_{jk}=A_j$ and using ${\sum }_k{\psi }_k=1$ recovers the univariate curve ${\sum }_j{A}_j{\varphi }_j(u)$ as a parametric surface constant in $v$.

The core parametric surface interrogation is evaluation of the position and its derivatives through second order:Position answers point queries; first derivatives give the differential, tangent directions and normal vector; second derivatives supply curvature. An evaluator should also report the parameter domain, patch adjacency, boundary curves and knot or crease locations, and whether these derivatives exist at the queried point. At a discontinuity boundary it should provide the relevant one-sided limits rather than silently average them.

For spatial searches, provide restriction or subdivision into equivalent smaller patches and certified bounding volumes, with error estimates when an approximate mesh is used. These allow intersection and closest-point algorithms to reject remote patches and refine the remaining candidates. A useful interface therefore combines differential evaluation with domain/topology and enclosure information. Inverse parameter lookup, ray intersection and nearest-point search can be built from these primitives using subdivision and local root or minimization solves; they need not be presumed exact basic evaluations of every representation.

Let the point light be at $L$. For a subdivision curve point $C(t)$, its projected shadow lies on the ray beyond the subdivision curve:The shadow is the first intersection of that ray with the receiving subdivision surface after the subdivision curve point. Thus solveand retain the smallest admissible $\lambda$. This is the defining equation for shadow tracing for a subdivision curve.

First organize the subdivision surface patches into a bounding volume hierarchy. For a restricted subdivision curve interval, rays through its enclosing volume define an enclosing ray cone from $L$; subdivision surface bounds that do not meet this cone can be rejected. Refine surviving subdivision curve intervals and subdivision surface patches. For a fixed $t$, isolate all possible ray intersections using those bounds, then correct each regular candidate with the Jacobian matrix for the Newton methodThe scalar triple product shows that $J$ is invertible exactly when $n\cdot (C(t)-L)\ne 0$. Enforce the patch domain and $\lambda >1$ throughout correction, and choose the smallest valid $\lambda$. Bounds or validated root isolation, rather than a fixed list of subdivision curve samples, are needed if every shadow component is to be found.

Once a regular intersection is known, follow it as $t$ changes. Differentiating $F=0$ gives a useful predictor:Take a small predictor step in $(u,v,\lambda )$ and apply a Newton method correction at the new subdivision curve parameter. Move through patch adjacency with its chart transition; shrink steps when derivatives or error estimates change rapidly. Subdivide further and test a geometric curve-to-chord bound to approximate the resulting shadow trace to a chosen tolerance.

Surface-boundary crossings terminate or clip a trace. If a ray has no admissible subdivision surface intersection, that subdivision curve parameter contributes no shadow on $S$. At grazing contact the Jacobian is singular, so use interval subdivision and root isolation to detect the event and seed any outgoing branches. If several intersections are possible, recheck which is first; facing the light locally is helpful but is not a proof of global uniqueness. Open-curve endpoints and closed-curve seams must also be accounted for. With a compact represented geometry, transversal intersections and validated subdivision bounds, this algorithm traces every visible component; a purely sampled implementation gives only a tolerance-dependent approximation and cannot certify arbitrary tiny missed components.

For a directional light, replace the ray by $C(t)+\lambda d$ for a fixed propagation vector $d$ and $\lambda >0$. The intersection equation becomes $S-C-\lambda d=0$, with Jacobian $[S_u,S_v,-d]$ and continuation right-hand side $C^{\prime }(t)$. The same enquiries and event handling apply.

The corresponding subdivision surface interrogation needs mesh connectivity and patch adjacency, boundary and crease flags, local coordinate charts with transition maps, and subdivision into smaller patches with their original-chart mappings. Its evaluation enquiry returns a limit point, tangent derivatives and a normal vector; second derivatives are supplied only where the limit is sufficiently smooth. Certified bounding volumes and patch approximation errors allow a hierarchy for intersection and proximity searches. Boundary-curve enquiries complete the interface for clipping.

On regular parts of a subdivision surface, these evaluations may use its equivalent tensor-product or box spline patch. Near an extraordinary subdivision vertex, the local subdivision matrix determines the limit position through its constant left eigenvector, and its tangent modes determine the normal vector under the scheme's characteristic map of a subdivision surface regularity conditions. Refinement alone does not establish those conditions. A subdivision surface may have a well-defined tangent plane there without possessing second derivatives or Gaussian curvature. An interrogation interface must distinguish these cases and can work on surrounding regular charts when a higher derivative is unavailable at the vertex itself.

Together, subdivision, bounds, topology and limit evaluation support ray–subdivision surface intersection: bounds eliminate irrelevant patches; surviving patches are refined, and regular candidate intersections are corrected using position and first derivatives. The returned intersection is a point of the limit subdivision surface with an error certificate, rather than automatically a point on the last control mesh.

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.

Put $h=\Delta x$, let $C_h$ denote the sum of the four axial shifts, and let $K_h$ denote the sum of the four diagonal shifts. Reading both sides of the PDF stencil gives $M_h\dot{U}=L_{h}U$, where $M_h=2I/3+C_h/12$ and $L_h=(-10I/3+2C_h/3+K_h/6)/h^2$. The left-hand stencil acts on the time derivative; treating it as a scalar multiplier would lose the compact correction.

Write $D={\partial }_x^2+{\partial }_y^2$. Taylor expansion of the symmetric shifts givesandFor the exact diffusion solution $u_t=Du$, subtracting givesThe leading coefficient is not identically zero for general smooth data. Thus the normalized local residual is fourth order, and in the question's $h^{p+1}$ convention $\boxed{p=3}$. This is the compact semidiscrete nine-point diffusion scheme; its usual spatial numerical consistency order is four, rather than three.

Use the Fourier amplification symbol for the semidiscrete generator. At a mode $e^{i(k\xi +l\eta )}$, the two stencil symbols areThe mass symbol satisfies $1/3\le m\le 1$. The numerator of $\ell$ is increasing in each cosine, since each partial derivative is at least two, and its maximum is zero at $(1,1)$. Hence $\ell \le 0$ and the generator ${\lambda }_h=\ell /m$ is real and nonpositive.

Every exact-time mode is multiplied by $e^{t{\lambda }_h}$, of modulus at most one. Parseval identity therefore provesThe method is stable, uniformly in the spatial mesh. This is a semidiscrete conclusion: it imposes no time-step condition until a particular time integrator is applied. For the stated merely square-integrable initial data, cell averages give a well-defined stable initialization, with $\Vert U(0){\Vert }_{{\ell }_h^2}\le \Vert v{\Vert }_{L^2}$ by Cauchy–Schwarz. Arbitrary point sampling of an L2 equivalence class is not required.

For a sufficiently smooth exact solution, the centered spatial residual isLet $e=U-R_{h}u$ be the nodal error. It satisfies $e^{\prime }=A_{h}e+{\tau }_h$, with $A_h=D_{xx}-\alpha D_x$ and $\Vert {\tau }_h{\Vert }_h\le C{h}^2$ on the fixed time interval. Part (a) gives $\Vert e^{t{A}_h}\Vert \le 1$, so Duhamel's formula yieldsThus the method is convergent, with second-order spatial error for smooth data and second-order compatible initialization. The homogeneous endpoint values introduce no boundary residual. For nonsmooth L2 data, stable approximating projections and a density argument extend numerical convergence, although the same order need not hold without the derivative bounds used above.