= Solution
In a regular region, moving one <control point> adds its displacement multiplied by a translate of the <twice-smoothed four-direction box spline>. Measure parameter distances in original grid-edge units, with the affected <control point> at the origin. The convex hull of the binary mask offsets is
$$
Z=[-1,1](1,0)+[-1,1](0,1)+[-1,1](1,1)+[-1,1](1,-1).
$$
After successive binary refinements the physical displacements are scaled by $2^{-1},2^{-2},\ldots$. Their possible accumulated offsets therefore form $\sum_{j\geq1}2^{-j}Z=Z$. The <box spline> description confirms that this <zonotope> is the actual support, not merely a loose enclosure. Equivalently,
$$
\boxed{Z=\{(u,v): |u|\leq3,\ |v|\leq3,\ |u|+|v|\leq4\}}.
$$
It is an octagon with vertices $(3,1),(1,3),(-1,3),(-3,1),(-3,-1),(-1,-3),(1,-3),(3,-1)$. Its enclosing square is six original grid intervals wide in each direction. Cutting four right triangles of area two from this $6\times6$ square gives area $36-8=28$ parameter-square units. Outside this octagon the control-point displacement has no effect; its closure is the support, although the basis function can vanish on its boundary.
\b[The regular influence footprint is this width-six, area-28 parameter octagon.] For a nonplanar control mesh, the affected part of the spatial surface is the image of the octagon, not necessarily a planar octagon of the same metric size. Near an <extraordinary subdivision vertex>, use the actual refinement neighborhood instead of assuming a globally regular lattice footprint.
Back to article page
