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.

Geographical time-space principles

Geographical time-space is a cartographic representation where the geographical surface is represented as cones, and the networks is represented as edges in three dimensions. The geographical time-space principles are based on a series of three equations that define the geometry of cones and of edges.

Equation 1: The equation of the Geometry of Cone
Figure 1: The Geometry of cones in the projected geometry

The geometry of cones depends on the ratio of speed. Terrestrial speed are slow while long distance flights are fast. The fastest speed is smax while the slower terrestrial speed samb is attached to the route amb. The height of the cone will follow the equation 1, where the length of the om’ depends on

Figure 2: Drawing edges with different speed in the spherical geometry

In spherical geometry, the fastest edge is not a straight euclidean line, but rather a geodesic, or great circle edge between two cities, following the curvature of the earth. In the simplest variant of the model, a slower edge is drawn as two chained rectilinear segments of equal length, in the plane formed by the two cities orthogonal to the surface of the earth. The general principle of the model states that the length of the edge amb, in figure 2, is proportional to the length of the fastest speed edge g, hence to the ratio of speed. The equation 2 give the height om’ in function of the maximum speed speed, of theta, and r the radius of earth.

Equation 2: Formula of the length of segment om' as a function of speed, or r and theta
Equation 2: Formula of the length of segment om’ as a function of speed, or r and theta

Finally, in the case of aircraft links with length below 2000 kilometers the formula of the length of the segment om′becomes as shown on equation 3.

Equation 3: Formula of the length of segment om’ in the case of flight links of less than 2000 kilometers

Setting the height or the radius of cones?

The Sahara as one of the largest terrestrial area devoid of cities, an issue for time-space global cartography
The Sahara desert as one of the largest terrestrial area devoid of cities, an issue for time-space global cartography (L’Hostis 2016 CC-BY)

In the testing of the model, the issue of the height  of cones came quite quickly to the fore. As, by construction according to the principle of the model, cones are entirely drawn under the surface of the earth, cones  where initially envisaged as having the height of the earth radius. Nevertheless, this choice rapidly became problematic.

The limit case is when the maximum speed equals the terrestrial speed. In this situation the principle of the model implies that the slope of cones is null. The cone turns into a disk. In this situation, if the height is fixed, the radius tends to an infinite value and this creates visual artifacts.

To deal with this issue, the way forward consists in fixing not the height of the cone but rather its radius. In consequence the next step is to set the default radius of cones. This radius should be set at the half of the longest terrestrial area devoid of cities. The Sahara desert is a candidate for this situation. On QGIS it was possible to measure 2600 kilometers between El Djefa in Algeria and Niamey in Niger. the proposed default radius of cones will then be set at 1500 kilometers.

To deal with eventual issues in the radius of cones, a new column of data is added. It may prove relevant to set a smaller value for the default radius, and to set the cities close to Sahara to 1500 kilometers radius. But this will be a next step, to be adjusted when the first images will come out.