Partitioning of Data - Spatial Mathematics: Theory and Practice through Mapping - page 136

Figure 6.10 A curve that crosses itself is an example of a curve that is not a Jordan
curve.

starting point. This path is traced out in Figure 6.11a. Suppose that you have
been directed to color the area to your left as a light red. The result appears in
Figure 6.11a and it is probably not what was intended—instead the intention
was to color what intuitively appeared to be the interior. The solution is simple.
Break this single curve that crosses itself into two separate simple closed curves
at the starting node. Then color the inside of each rectangle individually.

Another sort of unintended consequence that can arise from having polygons
whose edges cross appears in Figure 6.11b. Now, the path is the same here, for
the first four edges as in Figure 6.11a. But this time, when the starting node is
returned to, turn south on Woodlawn and go around the block the other way.
This path is indicated by the arrows in Figure 6.11b. In this area of Chicago, the
even-numbered addresses are on the west side of the north-south streets and the

(a) (b)

E 57th St E 57th St

N N

E 58th St E 58th St

Figure 6.11 (a and b) Unintended consequences coming from Jordan Curve Theorem
failure. The red arrows trace out a curve similar to that in Figure 6.10.