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Surfaces define distances between the objects on them. Surfaces can contain
spatial information far more complex than that which can be shown on any flat
plane. It is this property of surfaces, the geometry they create, that holds most
promise for advanced visualization and that has been least exploited to date. 12
A Euclidean plane (one with 'normal' geometry) has to obey the triangle
inequality, which states that the distance from one place to another must be less
than or equal to the distance of a route via another location. Euclidean planes
are flat; the shortest routes in one are found by following straight lines. On a
nonflat surface, however, the straight line distance between two points may well
not be the shortest. It is often advisable to travel via another route, such as round
mountains, avoiding gorges.
Given a set of distances defined between points, to visualize the space those
distances create a surface is formed on which the shortest routes between points
are given from a matrix of distances. This matrix has to be symmetrical (the dis-
tance is equal irrespective of the direction travelled) and only the shortest possible
routes are successfully depicted. Nevertheless, in this surface we have a valuable
visual image, which is not a mere elaboration of some simpler information. 13
Such a surface creates a two-dimensional space in three dimensions, which
cannot be arbitrarily stretched and remain valid, although it can be rotated and
internally reflected. This property could be used to indicate if real distance were
greater in one direction than another, by deciding which way to make uphill and
hence which downhill. However, it is uncertain whether this could always be
truly depicted and if the ratio of the differences in direction could be shown in
a very reliable way.
One further detail of this approach is that the surface could be built upon any
two-dimensional, flat spatial distribution. Therefore, when viewed from directly
overhead, a familiar geographical picture would be seen, while bringing the ori-
entation of the camera down would show discrepancies from the simpler metric.
The most useful possible employment of the technique here is in the depiction
of travel time.
12 This means least exploited by 1991 when these words were first written and still by 2011
when they are being edited. Computers are now easier to use and it is easier to print and save files,
but much of the early enthusiasm for developing new types of visualization went when the software
packages came in.
13 Tobler saw surface geometry as being of paramount importance in geography: 'The geometry
with which we must deal is rarely Euclidean, and it is, in general, not possible to obtain completely
isometric transformations.
...
The maps at first may appear strange, but this is only because we
have a strong bias towards more traditional diagrams of our surroundings and we tend to regard
conventional maps as being realistic or correct' (Tobler, 1961, p. 164).
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