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.

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

Edit: More description of the principles and formulas and can be found in the annexes of the article: A L’Hostis, F Abdou, 2021, What Is the Shape of Geographical Time-Space? A Three-Dimensional Model Made of Curves and Cones, ISPRS Int. J. Geo-Inf

Measuring visual length

We want to build a coherent representation of geographical time-space. In such a representation the measurements of length, which I introduced as measurements of the visual length, must be proportional to time-distances. A simple way to check this property of the representation lies in the direct measurement by means of a ruler on the computer screen!

Measuring visual length with the ruler on the computer screen

And the measurement shown that the ratio of the length of long and short routes, here around 2.3 is not consistent with the ratio of speed which is 7.5 when confronting a terrestrial speed of 100 km/h with the mean long haul airline speed of 750 km/h. We still have to work!

Another Blender test

This is another Blender rendered test of the same enigmatic area as in the previous post. Billel Helali has introduced visual effects in a search for readability of the three dimensional structure.

Dark blue cones, white edges, with Blender’s bloom effect

This is three dimensional cartography, hence we are working at the intersection of cartography, with its strict rules, and three-dimensional representation, with a lot more freedom. We want to build the most intelligible image and this is a true challenge. Why not trying to exploit the domain of visual effects? This is the bloom effect of Blender that highlights the white edges. Nice, isn’t it? Probably more for fun than for geographical analysis. But definitely worth sharing it here.

Tests in Blender

We are testing the production of the image with Blender that give way more latitude to control all the three dimensional image parameters. These are first attempts they are far from what we want to achieve but they already give an idea of a final result.

Edges as white cylinders, cones with facets

We had issues to move from the app to Blender: lines without width did not show in the rendered image from Blender; they need to have a volume; we converted these lines into cylinders.

All the Blender work is due to Billel Helali, great work!

I do not resist to leaving it as an enigma: which part of the world time-space is represented here?

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