The origin paper of the three-dimensional geographical time-space model

The original idea of the principle of three-dimensional geographical time-space must be credited to Pr Philippe Mathis in his 1990 paper for the Association de Science Régionale de Langue Française (ASRDLF) workshop on distance and spatial analysis in Chamonix.
The document is a word-processor printed file but with freehand  diagrams by Mathis and notably the key diagram of page 5 reproduced here.

Pr Philippe Mathis origin hand-drawing of a three-dimensional time-space coherent representation of a multimodal transport network, in a communication to the 1990 Chamonix ASRDLF workshop on distance and spatial analysis

On Mathis origin diagram fastest transport mode occupies the two dimensional usual plane, while slower edges of the transport graph are drawn above or under the surface as curves, proportionally longer than the Euclidean straight line. It is striking to observe that the most recent images of the model with cones and rounded edges are much more similar to this origin drawing than the first computer software implementation by L’Hostis in 1993 based on broken lines.

It is not surprising to know that Mathis has always conducted artistic activities beside and after his academic career with drawings, portraits, geometric drawings, and more recently sculptures.

New ideas for three dimensional structure of geographical time-space

We want to introduce new ways to draw the three dimensional structure of geographical time-space. Currently we have lines for high speed air links and cones for terrestrial transport modes, road and high-speed rail. Cities served by road only have a unique slope given by the ratio produced by comparing with the fastest speed, of long range flights.

In the current state of the code, cities served by even a unique high-speed rail link generate a cone with a different slope, given the different ratio with the fastest speed. This approach may easily be critiqued since it applies high speed rail speed to the whole area surrounding the city, id est, its cone.

For this reason we want to introduce new principles for generating the terrestrial time-space surface. High-speed lines between two cities A and B will be draw and used as the basis for a surface located between the cones A and B. In a first model the line will be drawn with two broken lines. This choice is coherent with the way cones are drawn, with broken segments for terrestrial path between cities, along the cones.

Sketch of the principle of drawing cones and edges with broken lines
Sketch of the principle of drawing cones and edges with broken lines

A variation of this model considers the high-speed line as a smooth curve instead of a broken line. This choice is coherent with the way aerial edges are currently drawn — as curved Bezier lines — a choice justified by the lack of geographical meaning of the broken segment despite a strong visual presence, as can be seen here.

Sketch of the principle of drawing cones and edges with smooth lines
Sketch of the principle of drawing cones and edges with smooth lines

While previously the terrestrial surface was an assemblage of cones, the new surface is generated from a more complex three dimensional geometry with cones and and complex shapes. Slope keeps its time-space meaning and several terrestrial speeds can be represented with a basis of cones.

We will see what the final visual result will be in hopefully a few days, thanks to Farouk. Thanks also to Jules for the drawing.

Data model for geographical time-space relief cartography

With Farouk, we have set up he data model for geographical time-space cartography. This consist in four tables. The two central table are for cities and networks. As we want to explore time-space through time, it is necessary to have time data on the population of cities. The idea is to introduce a threshold so that only cities with a given population will show at a given period in time.

Time-space relief cartography data model
Geographical time-space relief cartography data model

Transport networks are described as graphs with cities as edges. An edge in the network has a starting year, and also an end year. Transport modes are described through transport modes, characterized by their commercial speed. Start year and end year are provided in order to describe, for instance, the Concorde period. For the twentieth and twenty first centuries, the transport modes are road, motorway, rail, high-speed rail, planes, and supersonic planes.

To each transport mode is associated a table of speed varying in time, in order to consider the temporal acceleration of trains, of places in the historical time.