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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 tensor-product surface basis is formed from two independent univariate basis families ${\varphi }_i(u)$ and ${\psi }_j(v)$ on separate parameter intervals. Given a rectangular control net $p_{ij}$, its parametric surface isThe parameters and indices vary independently over a product domain. The representation can be evaluated first in either parameter, giving a convenient computational advantage. Assume a finite basis, or locally finite spline bases, so differentiation and summation below are justified locally.

For positivity, if each factor basis is nonnegative, then ${\varphi }_i(u){\psi }_j(v)\ge 0$. If both factors are strictly positive at a point, their product is strictly positive there. Nonnegative weights are the usual geometric meaning of positivity; zero boundary weights are allowed. With partition of unity, this gives the local convex hull enclosure of the active control points.

For continuity, suppose the first family is $C^r$ and the second $C^s$, including matching derivatives across their respective spline knots. On every polynomial piece, and by matching at the knot lines,Every factor on the right is continuous in its own variable, so its product is jointly continuous. Thus all these mixed derivatives are continuous, in particular the surface is jointly $C^{\min (r,s)}$. The separate coordinate smoothness can be better: $r$ orders in $u$ and $s$ in $v$. For piecewise polynomials of degrees $d_u,d_v$, an interior knot of multiplicity $k$ in the first factor normally gives $C^{d_u-k}$ matching in $u$, inherited along that knot line; similarly for $v$. The statement is an inherited lower bound. Special control points can cancel derivative jumps, so a particular surface may be smoother than a generic member of its basis.

For summation to unity, independent summation factors exactly:This also explains why translations of all control points translate the surface by the same vector, as in affine equivariance of a geometric basis. These three arguments establish tensor-product inheritance of geometric basis properties directly from the factor identities.

A triangular Bézier patch is a non-tensor-product example. For barycentric parameters $u,v,w\ge 0$ with $u+v+w=1$, a quadratic patch isThe six basis functions have total degree two and sum to $(u+v+w{)}^2=1$. Their index constraint and triangular parameter domain do not arise from two independently indexed univariate factors on a rectangle. An individual triangular polynomial could be embedded in another larger representation, but this triangular basis and control net are not a tensor-product surface definition.

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.

A triangular Bézier patch is a non-tensor-product example. On barycentric parameters $u,v,w\ge 0$ with $u+v+w=1$, defineThe indices satisfy a total-degree constraint, and $w=1-u-v$ couples the two independent parameters. The control net is triangular with $(n+1)(n+2)/2$ points, rather than a rectangular product index set. The multinomial theorem gives nonnegative basis functions summing to one, but the entire basis is not a product of two independent univariate bases.

A tensor-product surface basis combines two independent univariate bases:The coefficients $P_{ij}$ are three-dimensional control points. Fixing one parameter gives a curve in the other univariate basis; the coefficients of that curve are the corresponding weighted combinations of rows or columns of controls.

The control net must have a rectangular two-index topology, with row/column neighbors and quadrilateral mesh cells. “Rectangular” describes the index grid, not the geometry of the points: its faces need not be flat or rectangular in space. A surface requiring a triangular domain or arbitrary extraordinary mesh connectivity is not directly described by one such tensor-product patch.

A subdivision surface interrogation interface needs the initial control net, its face/edge/vertex connectivity, the subdivision rule, patch addressing and refinement into child regions that cover the same limit surface. It must supply limit-point evaluation, available tangent/normal data and smoothness information, including the treatment of extraordinary subdivision vertices. A finite refined mesh point is generally not the exact limit point.

For geometric search it must provide certified bounding volumes for each descendant region, bounds that shrink under refinement, and a mesh-to-limit or chord/patch error estimate. With positive convex subdivision masks the active-control convex hull can provide such an enclosure; schemes with negative coefficients need another valid bound. A closed surface also needs consistent orientation, component navigation and containment queries for the enclosed body. The $C^1$ hypothesis supplies first-order tangent information at regular points, not automatically second derivatives or curvature everywhere. These enquiries distinguish an accurate limit-surface algorithm from a calculation only on an approximate control polyhedron.

For nonnegative barycentric parameters, these total-degree Bernstein basis functions define a surface over a triangle. The triangular control net has $(n+1)(n+2)/2$ points. Its index constraint couples the three exponents, so it is not a tensor product of two independent univariate bases. The multinomial theorem gives nonnegativity and partition of unity.