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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.

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.