On the surface - The Visualization of Spatial Social Structure - page 218

7.6 Travel time surface

Let us suppose that after an appropriate rotation two dimensions rep-
resent the classical longitude and latitude forming a 'basic' plane, and
the third dimension, the altitude above the plane thus defined, repre-
sents the 'inaccessibility' of a city. The higher above the basic plane,
the worse a city's linkages with the global network.
(Marchand, 1973, p. 519)

Geographers have attempted to depict travel time on maps for many years.
Because they have usually limited themselves to flat two-dimensional representa-
tions, this has proved to be impossible. Correct travel times from a single origin
can be drawn, and have been on many occasions. These linear cartograms are
created by showing isolines of equal time distance from a point and then trans-
forming those lines into circles around that point. Where the travel time space
is inverted, however, even depiction of a single point may not be possible in
Euclidean space. Imagine what happens as the isolines reach round the globe.
Statistical multidimensional scaling has often been used to try and find the
best fitting two-dimensional representation of a set of distances. Frequently all
this achieves, geographically, is the reconstruction of the original map with a bit
of distortion - only useful when you did not know the original. The essential
problem is that travel time, unless exactly equal to physical distance, cannot be
drawn on a flat plane, 14 just as, over large areas of the globe, conventional maps

distort shape.
The answer to how to create a time surface is to begin with the simple flat
geography, and raise or lower points in some third dimension until the correct
distances are achieved, creating a surface where distance is drawn in inverse
proportion to speed. If you can travel quickly between two points they are drawn
close together (at a similar height); if travel between them is only possible slowly
then they are drawn far apart by one being drawn much higher than the other.
Just as an infinite number of area cartograms can be created to any given
specification, so too can an infinite number of travel time surfaces. The actual
algorithm required must create the simplest such surface, containing the least
rucks or changes in vertical direction. Thus, for any given Euclidean space, a
unique travel time surface can be projected above and below it.
For Britain making a time surface would create a landscape dominated by
mountainous inner cities, with London supreme, as by road it takes the longest
time to travel into. The major motorways would cut great gorges through the hills

14 A time surface can be defined as: 'Given a velocity field on the Euclidean plane, we define
a transformation of the plane into a two-dimensional curved surface lying in three-dimensional
Euclidean space. The surface characterized by the transformation has the property that travel time
on any path in the original Euclidean plane is equal to the length of the image of that path on the
transformed surface. In particular, the image of the minimum-time path between two points on the
plane is the geodesic curve joining their image points on the surface. This surface has therefore been
referred to as the time surface' (Angel and Hyman, 1976, p. 38).