## Conceptual framework for geographical time-space representation

Extending the terminological orientation we expose here the conceptual framework for the representation of geographical time-space.

 Geography Mathematics Graphic representation Cities Graph (nodes and edges) Cones Transport networks Curves Geographical distance Minimum path distance Visual length ↓↑ Time-scale

In the graphic representation of geographical time-space we generate cones and edges. The length of these graphical elements can be converted into travel time by means of a “time scale” similarly to the the way the reader will measure kilometer distances using the kilometer scale found in classical cartography.

The graphic representation is designed to produce geographical distance measurements that will be consistent with observed measurement over transport networks between cities.

## Geographical time-space representation terminology based on concepts

In order to rationalize the code of the application, we introduce a new naming policy for variables and functions, based on the underlying concepts. We start from geographical objects, cities and transport networks that we model as graphs, and we finally produce an image made of cones and curves.

 Geographical objects Mathematical objects Graphical objects Cities Nodes of a graph Cones Transport networks Edges of a graph Curves

The geographical objects are modeled  in a graph. Producing the end image is a matter of graph drawing which is an identified scientific domain. In order to better distinguish the graph part and the graphical part of the code we emphasize the use of the words node and edges for the graph and the use of cones and curves for the graph drawing section. Depending on the choices made by the user, curves may be implemented as straight lines, broken segments or Bézier curves.

In the code, the use of the word line is restricted to the context of three.js objects.

## 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.

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.

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.

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.

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.

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.

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.

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.

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!

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.

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.

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?

## First coherent image with cones and edges

Today we finally produced this first image of a coherent geographical time-space representation with air and terrestrial transport modes.
Air links are following the geodesic curve when the distance is greater than 2000 kilometers. Below this threshold the speed of the air service is lower which is expressed by the curving of lines high above the earth surface where the fastest transport occurs.

Still un-projected, a lot to improve in terms of readability, but already the basis of what I had in mind since the start of this project is taking shape. A great day today.

## Setting the height or the radius of cones?

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.

## Sketch of the model

This is how it will look like when the model will be complete. Time-space is represented by cones. As explained here, the slope of the cone allows to produce longer itineraries proportionally to the maximum available speed on the space considered.

Then two issues must be resolved, one theoretical, and the other more linked to graphic and cartographic choices. Firstly this is a terrestrial time-space, and the surface should not extend in the maritime domain, where different transport conditions occur. Secondly, these representations have a major issue to overcome: the readability of the representation in reference to the conventions of classical maps. This second arguments implies to render shorelines and continents shapes in order to make the representation more easy to read.

For these two reasons the basic cone structure will be cut along shorelines, as in this example of an island carrying a city. It remain to be decided, judging again by the criterion of legibility, if the final structure will be a volume or a surface.

## Shriveling untill resembling a chestnut

“And yet it moves”, said Galileo. “And yet this is the earth as we experience it” could we now say, looking at this image.

Produced by a JavaScript code based on the library three.js, with decisive help from Farouk Abdou, this is the first accurately parametrized representation of the global time space, at the time of Concorde (1977-2003), when the maximum available transport speed, that indicates the standard for maximum speed, was 1500 km/h. The principle of construction of the image is detailed here.

This has been, so far, the most intense shriveling movement in transport history, when the maximum available speed raised from 700 km/h with jets to the supersonic speed of Concorde.

Considering, in a simplified model, terrestrial speed of 100 km/h, say on a motorway or by train, the slope of cones takes this geometry. During this Concorde period, terrestrial speed was 15 times slower than the maximum available  speed.

This image is very close to the idea of the chestnut that I announced in 2007. High quality image download link.

## The shriveling map construction principle

Unlike all previous time-space relief representations (1997, 2009) where the surface is based on the edges of a graph,  in the representations developed in 2016, we introduce conic shapes to represent time-space. The cone is the basic structure of this representation. Cones are characterized by a slope that follows the ratio between the speed of basic terrestrial networks, i.e. the road system, and the maximum available speed, which is attained, on the period we consider, through the air transport system.

On the figure we have two cities joined by a fast non-stop transport system, and also by a slower terrestrial transport mode. The fast transport link is represented through a red straight line, while the cones represent the surface on which slower terrestrial transport mode are drawn.  Assuming that travel speed are similar in all the geographic space considered, the slope of the cones centered on the two cities is a coherent representation of time-space. The two cones intersect midway between the two cities. The slope of the cones implies that the length of  the straight red line, and of the line drawn on the cone and joining the two cities, are proportional to the respective transport time. This forms a time space representation with different transport speed, and a time-space scale can be added, as in the USA map (p. 8).

## The first image of a shriveling world

Using the library Cesium, and with great help from Thomas Leysens and Farouk Abdou, this is the first image (in higher quality here) out of this new project, producing a representation of global time space.

Very excited when I first saw it on my computer screen. This is the first time this image, which I had in mind for years, finally materializes.

Unfortunately it has proven very difficult, for me at least, to control cones geometry. In addition in Cesium it is not possible to draw geometries under the surface of the globe. the problem is that, conceptually, the time-space relief map model, everything occurs under the surface.

An exchange with Cesium developers has indicated an alternative choice: the library three.js, which has less limitations for our purpose.

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