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The centered subdivision mask $[1,2,1]/2$ retains old control points and inserts edge midpoints. Its control polygon is unchanged as a geometric curve by refinement. Its basic limit is the hat $\varphi (t)=\max (1-|t|,0)$, with support $[-1,1]$. This linear spline representation reproduces constants and linear functions, and is $C^0$ but generally not $C^1$ at its spline knots.

If a triangular-lattice subdivision mask has the displayed residue-sum equality, refining data $p_{i,j}=q_j$ gives $p_{I,J}^{\prime }={\sum }_k{b}_{J-3k}{q}_k$, independent of $I$. If affine coordinates reproduce their parameter positions, a geometrically straight extrusion is preserved. Rotationally symmetric masks give the corresponding property in all three lattice-edge directions. A boundary curve compatible with this extrusion uses the collapsed mask $b$, not just the central row of the two-dimensional mask.

Lateral artifacts of a subdivision surface are unwanted variations along an intended straight extrusion: input cross-sections identical along a lattice direction acquire ripples in that direction under refinement. For a stationary arity-$m$ lattice rule $a_{i,j}$, extrusion along the first lattice coordinate is preserved exactly when ${\sum }_{i\equiv r(\mathrm{mod}m)}{a}_{i,j}$ is independent of the residue $r$ for every transverse offset $j$. Then summing along each row gives a single associated univariate subdivision mask. This direction-specific criterion does not promise invariance under every possible oblique extrusion.

Fix an explicit stationary-grid convention. For subdivision arity $m>1$, write the univariate rule asA column of the subdivision matrix is one translated subdivision mask; the next column is translated by $m$ fine-grid rows. Thus the column displacement is the arity. If $a_k$ is active from $k=\alpha$ to $k=\beta$, the subdivision mask width is $w=\beta -\alpha +1$ fine-grid positions, after removing zero padding. Equivalently the mask span is $\beta -\alpha$ fine intervals. This is different from counting old control points in a row stencil: that count measures how many controls influence one newly created point. Stating the convention avoids confusing these two widths.

For a convergent subdivision curve scheme, its support is the influence region of one original control point, equivalently the support of a function of the basic limit $\varphi$ obtained from data $p_j={\delta }_{j0}$. Translation gives the representation $P(t)={\sum }_j{p}_j\varphi (t-j)$. Descendant indices of index zero have parameter positionsThus the limiting support enclosure is $[\alpha /(m-1),\beta /(m-1)]$, as in the support of a stationary subdivision scheme. Endpoint activity gives equality for the ordinary positive convergent masks under discussion.

The functional precision set of a subdivision scheme consists of functions exactly recovered from their uniformly sampled data using the scheme's consistent parameter placement. Polynomial precision records the exactly reproduced polynomial space. Sampling-grid spacing and origin must be specified; the usual arbitrary-grid precision requirement uses every spacing and shift, rather than merely membership in the fixed-grid reconstruction space.

Center the given mask at indices $-1,0,1$. Its binary linear interpolatory subdivision rule isEach new control point is either an old point or an edge midpoint. Hence every refined control polygon is the same curve with extra vertices. Starting from scalar impulse data, the limit is explicitlyThe mask width is three positions, its mask span is two, its largest row stencil uses two old control points, and its subdivision arity is two. Translating the mask indexing to $0,1,2$ translates the basic support to $[0,2]$; the support width remains two original grid intervals.

On $[j,j+1]$, reconstruction is $(j+1-t)p_j+(t-j)p_{j+1}$. Constant and affine samples are therefore recovered exactly. A non-affine polynomial cannot agree with a linear expression on a whole open interval. Thus its polynomial functional precision set of a subdivision scheme isFor continuous functions reproduced at every translated sampling grid, the same conclusion holds: agreement with all chord interpolants forces the affine interpolation identity, hence affinity. At one fixed grid, the larger exactly representable class is the space of linear splines with those spline knots; this is not a higher polynomial precision claim.

For arbitrary data, the one-sided derivatives at $j$ are $p_j-p_{j-1}$ and $p_{j+1}-p_j$. They need not coincide. Therefore the generic limit is $C^0$, not $C^1$. Its first derivative is piecewise constant; exceptional affine data, or individual matching neighboring slopes, can be smoother. This conclusion follows directly from the limit formula, not just from a heuristic mask-width test.

Lateral artifacts of a subdivision surface are spurious changes along an intended straight extrusion. For example, if every row of a control net is the same profile translated along a straight direction, a refinement rule should not create periodic ripples along that direction. This is distinct from changing or smoothing the transverse profile itself.

Use triangular-lattice coordinates $(i,j)$ with basis vectors $e_1=(1,0)$ and $e_2=(1/2,\sqrt{3}/2)$. The displayed mask is supported on $|i|,|j|,|i+j|\le 3$, with the central entry at $(0,0)$. Write its coefficients as $a_{i,j}$, including their division by12. For data constant along horizontal rows, $p_{i,j}=q_j$, ternary refinement givesThus the three residue sums in every mask row must agree if arbitrary row-constant data are to remain row-constant. Equal whole-stencil sums alone only reproduce constants and do not guarantee straight-extrusion preservation.

For the original row $j=1$, the numerator coefficients at $i=-3,-2,-1,0,1,2$ are $2,4,5,5,4,2$. Their three residue sums are $7,8,7$, which do not agree. For $j=2$ the sums are $5,5,4$. In particular input $q_j={\delta }_{j0}$ yields heights $7/12,8/12,7/12$ along the refined row $J=1$. The original mask has lateral artifacts, since it introduces a three-phase ripple into a straight extrusion. The residue-sum criterion is also the usual symbol-factor test: the mask polynomial must contain the ternary averaging factor in the extrusion direction. The original does not. This connects the concrete calculation with the directional artifact analysis in Deriving Box-Spline Subdivision Schemes, while the coefficients and correction here are computed directly from the exam PDF.

A symmetric correction with the same denominator and support is to replace the six nearest-center coefficients5 by6, and the six second-ring corner coefficients3 by2. Leave every other coefficient unchanged, including the central6. The added and subtracted weights balance. For a complete computational specification, the corrected numerator rows areEvery entry in this new table is divided by12. The corresponding horizontal residue sums, before division, areConsequently the corrected rule has extrusion invariance of a ternary triangular scheme. Sixfold symmetry gives the same residue-sum property in the other two lattice-edge directions. Every two-dimensional residue class still sums to12 before normalization, so each refinement stencil sums to one. The weights are nonnegative. Symmetry about the stencil's refined parameter location also reproduces affine coordinates: the class first moments vanish, so an initial straight geometric translation direction remains straight.

The associated univariate boundary subdivision mask is the common sum in each transverse row, rather than the central row of the bivariate mask. Centering it at indices $-3,\dots ,3$ givesIts ternary stencils can be written explicitly asEach stencil sums to one and has the correct affine parameter moment. Applying this rule to a boundary profile agrees with the bivariate rule on a straight extrusion. Endpoint or corner rules must still be chosen for a finite open boundary. The artifact-free conclusion established here concerns the three lattice-edge extrusion directions; it does not promise exact extrusion preservation for every arbitrary off-grid direction. The directional limitation is intrinsic to the mask test, not a reason to substitute the central mask row as a boundary rule.

A norm estimate controls the worst possible refined differences for every choice of the initial controls. Submultiplicativity turns a length-$p$ contraction into a bound for arbitrary long products, so it proves a certain number of continuous derivatives. Its absolute values can lose cancellations; failure of the estimate is inconclusive. It therefore gives a lower bound on smoothness.

Conversely, an observable eigenmode of a repeated local product exhibits a particular sequence of shrinking neighborhoods with inadequate decay. General-position controls have a nonzero coefficient in that mode, so greater derivative continuity is impossible. Special symmetric or polynomial data may suppress the bad mode; this is why the general-position qualification matters. The obstruction gives an upper bound on smoothness, while failure to find an obstruction proves nothing. A lower bound on spectral growth becomes an upper bound on smoothness because large surviving differences correspond to rough behavior.

For example, the binary Cardinal cubic B-spline subdivision mask has symbol $M(z)=(1+z{)}^4/8$, up to an irrelevant index shift. At derivative order $r=2$, the first-difference subdivision mask is $B_2=(1+z)/2$, whose maximum parity sum is $1/2$: the limit is at least $C^2$. At $r=3$, $B_3=1$, and the surviving difference mode does not decay. For generic controls the third derivative is piecewise constant with nonzero jumps, so the subdivision curve is not $C^3$. Here the lower and upper tests determine the exact integer continuity, but they need not coincide for an arbitrary subdivision mask. Stability, sum rules and borderline modes must be handled explicitly before interpreting any numerical estimate as a continuity level.

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.

Fix the indexing conventionwhere the subdivision arity $a>1$ is an integer. A change of the initial control $P_0^0$ can influence only indices $i=j_1$ after one step, where $M_{j_1}\ne 0$. After $\ell$ steps the possible descendant indices areLet $j_{\min },j_{\max }$ be the smallest and largest active subdivision mask indices. Dividing by $a^{\ell }$ gives the enclosureTherefore the support of a stationary subdivision scheme, in initial control-grid units, is enclosed byFor the stated symmetric subdivision mask with active extremes $-w,w$, the width is $2w/(a-1)$. For a nonzero compactly supported basic limit function $\varphi$, this is an exact endpoint calculation: its refinement equation is $\varphi (x)={\sum }_j{M}_j\varphi (ax-j)$. If $\beta$ is its right support endpoint, only the largest active subdivision mask index can contribute sufficiently close to the rightmost point of this sum; all smaller indices have argument beyond $\beta$. Hence $\beta =(\beta +j_{\max })/a$, giving $\beta =j_{\max }/(a-1)$. The left endpoint follows in the same way. This argument also excludes cancellation at an extreme for signed subdivision masks. The support for initial control $P_k$ is translated by $k$. If the nominal subdivision mask was padded by zero coefficients, use its actual extremes; and a limiting subdivision curve has to exist before its support is meaningful. This calculation describes the entire possible influence region, rather than the width of a single refinement stencil on its finer grid.

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.