Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 66 / 5 / c

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.