Page 189 - Geometric Modeling and Algebraic Geometry

10 Cube Decompositions 191

Fig. 10.5. The eigencomponents of a Z 4-circulant matrix (top) and a Z 2 -circulant matrix
2
bottom.

direction, each one corresponding to an eigencomponent with σ(g)=1, cf. Fig. 10.6
(left). If we want the corresponding eigencomponent to be zero, the middles of the
segments joining the middles of opposite edges should be the same. That means that
for each subset, the middles of the four edges should form a parallelogram.
The fourth condition says that the barycenter of the points with σ(g)=0, 2
should be the same with the barycenter of the points with σ(g)=1, 3, cf. Fig. 10.6
(right). This is similar to the n =2 case, where a quadrilateral is a parallelogram if
a only if the points with σ(g)=0, 2 and the points with σ(g)=1 have the same
barycenter.

Fig. 10.6. Left: Four edges with the same direction are shown in bold. Right: The vertices
with σ(g)=0, 2 (filled circles) and σ(g)=1, 3 (empty circles).

From the above discussion it is intuitively clear that the existence of large compo-
nents with σ(g)=2, 3 leads to hexahedra with convoluted shapes. Fig. 10.7 shows
the effect of adding a single non-zero component with σ(g)=2, 3 to the unit cube.

10.4.1 Application: the multivariate quadratic spline

n
Up to now we studied general Z -circulant matrices without any reference to their
2 n
elements. In this section we study the Z -circulant matrix corresponding to the evo-
2
lution of one cell of an n-dimensional grid under the quadratic B-spline subdivision