Note this is possible only if u
∗ r
, u
∗ s
, and u
∗
t do not bridge all the gaps between w c 0
, wc rs, wc st and w c 1
. In what follows, we refer to this as the *gap condition*.

The gap condition is fulfilled by imposing the following constraints:

$$x\,O\,v_{c}(x\in X_{c}),\tag{34}$$ $$v_{c}\,E\!:\!SE\!:\!S\,w_{0}^{c},v_{c}\,S\!:\!SW\!:\!W\,w_{1}^{c}$$ (35) $$v_{c}\,E\!:\!SE\!:\!S\!:\!SW\!:\!W\,x$$ (36) $$(x\in\{u_{r}^{*},w_{rs}^{c},u_{s}^{*},w_{st}^{c},u_{t}^{*}\}).$$

Note that the constraint of vc to a spatial variable x in Xc does not contain tile name O. This means that the interior of vc is disjoint from x. From constraints w c 0 O vc and vc E : SE : S wc0
, we know M(vc) has the same upper left corner point as M(w c 0
) does.

Similarly, M(vc) has the same upper right corner point as M(w c 1
) does.

Assume the gap condition is violated. This means, there is no gap between any two consecutive regions in Xc (cf. Figure 9). In this case, the union of these regions contains the rectangle Ix(vc) × Iy(w c 0
), which should be excluded from the interior of vc. This contradicts the requirement that M(vc) shares the same upper left corner point with w c 0
.

Therefore, the constraints are not satisfiable.

![0_image_0.png](0_image_0.png)

On the other hand, suppose there is a gap between two consecutive regions in Xc (cf.

![0_image_1.png](0_image_1.png)

Figure 10). It is straightforward to check that all the constraints are satisfied if we let vc be the region obtained from (after necessary regularization) subtracting regions in Xc from the rectangle M(vc).