Artificial reality - The Visualization of Spatial Social Structure - page 80

The degree of autonomy that the final shape of the cartogram has, from the
influence of the areal division that was used to create it, was only achieved
by choosing a careful definition and measure of contiguity . In creating the car-
tograms shown here two places were said to be contiguous if they shared a
common border or were linked by a major tunnel, road or rail bridge. The measure
of contiguity was not absolute, but estimated as the proportion of the perimeter
of an area made up by the border in question or the length of an estuary coastline,
which a bridge, for instance, rendered traversable.
The algorithm for creating the cartograms began with each place at its original
(Euclidean) location, represented as a circle whose area was in proportion to
its population. Next, overlapping circles repelled each other while circles were
attracted to their neighbours in relation to the strength of their contiguity measure.
Places that bordered the sea expressed a degree of inertia because part of their
perimeter, being coastline, did not make up a common border, and this helped to
maintain prominent peninsulas. Thus, although the exact shape of the coastline
was sacrificed, many of its key locational features were retained.
The sustained combination of all these forces in parallel (Box 3.3) created the
new pictures of Britain used throughout this topic. 16 An algorithm was written
where the solution evolved towards the desired goal, releasing and tightening

Box 3.3 Deriving a constant

The following set of equations show how the damping factor K of 0.25 was
derived for the cartogram algorithm, where x is position, v is velocity and
A n is used as an ansatz :

v n + 1 = K(v n − x n )
x n + 1 = x n + v n + 1
x n + 1 = x n + K((x n − x n − 1 ) − x n )
x n + 1 − x n + Kx n -1 = 0

A n + 2 − A n + 1 + KA n = 0

by factors:

A n = 0 either

2

or A − A + K = 0

To avoid oscillation the solutions to A must be real; therefore the discriminant
must be positive: 1 − 4 K ≥ 0. Thus the largest nonoscillating K is 0.25.

16 These were also used in 'A New Social Atlas of Britain', published in 1995 and made available
as an open access 131 Mb pdf file on the web: http://www.dannydorling.org/?page_id = 81.