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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 isAfter successive binary refinements the physical displacements are scaled by $2^{-1},2^{-2},\dots$. Their possible accumulated offsets therefore form ${\sum }_{j\ge 1}{2}^{-j}Z=Z$. The box spline description confirms that this zonotope is the actual support, not merely a loose enclosure. Equivalently,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 $6\times 6$ 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.

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.

The factorization of the binary subdivision mask identifies eight box spline directions. Apart from the monomial that centers the support, it isThe four normalized factors correspond respectively to averaging in the directions $(1,0),(0,1),(1,1),(1,-1)$, each with multiplicity two. The factor four is the determinant of the binary dilation in two dimensions. The centering monomial translates the resulting box spline and changes neither its polynomial degree nor its smoothness. Thus this is the twice-smoothed four-direction box spline, rather than a tensor-product bicubic B-spline.