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

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