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related ideas: Thiessen polygons, Dirichlet regions, Delaunay triangulations,
and others (Coxeter, 1961a; Kopec, 1963; Rhynsburger, 1973; Thiessen and
Alter, 1911). Each zone contains exactly one Canadian city and all territory
within that zone is closer to that city than it is to any other city. There are
a number of subtleties (such as dissolving arcs within buffers or splitting
polygons) in building the map in Figure 3.6 and some of these are shown in
the animation associated with the QR code in Figure 3.6. Euclidean buffers
such as these are generally reserved for studies more local than this one. The
point here is to see the connection between the two-dimensional Euclidean
construction for perpendicular bisectors and the GIS creation of buffers as
concentric circles. The reader will have a chance to practice with buffers of
various kinds (and related ideas) later in this chapter.
3.2.4 Base maps: Know your data!
One of the themes running through this topic is the importance of under-
standing your data. It is particularly relevant to our discussion here. In this
example, note that even though the proximity zones extend beyond the
national boundary of Canada, points for which there are data exist only
within the boundaries of Canada (dotted white in Figure 3.6 ). Thus, within
most proximity zones, some points have associated data while others do not.
Hence any average (or other statistical) values across an entire zone should be
viewed with caution. Often, extrapolated data sets that lie far from established
data points cause problems. And the zones beyond the data points should be
viewed with the most caution of all. Know your base maps! Know your data!
3.3 Set theory
While this material is clearly based on a single ruler and compass construc-
tion of Euclid (that of perpendicular bisector), it is also based on elements of
set theory—a foundational branch of mathematics that systematically stud-
ies collections of objects (Hausdorff, 1914). The pair of intersecting circles
in Figure 3.5a partitions the space that contains them into sets of points:
All the points in one of the circles; all the points in the other circle; and the
points outside the two circles. The space containing this configuration is the
universe of discourse. Because the circles intersect, there are other ways to
look at this partition. Consider the circle on the left as A and the one on the
right as B . The points within both A and B are said to be within its intersec-
tion, denoted A B . The points in either A or B or both is called the union
of the two sets, denoted A B . The points outside are in the complement of
A B . The configuration in Figure 3.5a , described in set-theoretic terms, is
a Venn diagram on two circles (Venn, 1880). Try to tie some of these con-
cepts more closely with the examples given above. How does the idea of
complement relate to the idea of having concern about accuracy away from
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