Lateral artifacts of a subdivision surface

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 (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.

Table of contents
- Extrusion invariance of a ternary triangular scheme Lateral artifacts of a subdivision surface

Extrusion invariance of a ternary triangular scheme ( $b_{j} = \sum_{i \equiv 0 (mod 3)} a_{i, j} = \sum_{i \equiv 1} a_{i, j} = \sum_{i \equiv 2} a_{i, j}$ )

 0  0

If a triangular-lattice subdivision mask has the displayed residue-sum equality, refining data

p_{i, j} = q_{j}

gives

p_{I, J}^{'} = \sum_{k} b_{J - 3 k} 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