# What Is the Shape of Geographical Time-Space? A Three-Dimensional Model Made of Curves and Cones

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## Abstract

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## 1. Introduction

## 2. The Properties of Geographical Time-Space

## 3. The Speed of Air and Road Transport Systems

## 4. Cartographic Representations of Geographical Time-Space

- Cities, considered as nodes of the transport network, remain at their conventional geographical location.
- Edges are drawn in the third dimension, proportionally to the travel time needed between nodes.
- The geographic surface is attached to the slowest network, i.e., the road network.

## 5. The Principles of A Representation of Geographic Time-Space Based on Three-Dimensional Cones

## 6. Theoretical Validation

## 7. Dataset

## 8. Implementation of the Geographical Time-Space of China

- On this representation, no geographical projection is used: Cities are located on the terrestrial sphere which explains that the fastest arcs, in red, are not straight lines but curves as they follow the geodesic.
- An angle of 45 degrees between the camera axis and the tangent to the surface of the earth, has been chosen as a compromise between the readability of the final image, and the preservation of proportionality of north-south compared to east-west lengths.
- Lights have been introduced in order to maximise the readability of the three-dimensional structure: A directional light is completed by punctual lights between the cones in order to improve contrasts.
- Cones are set in white colour with shadows to express their three-dimensional nature.
- Aerial edges are coloured in two colours according to the categories of long and short haul. Long-haul speed, i.e., the fastest observed speed on this geographical space, determines the parameters of the geometry of all representations: Cones slope, and the geometry of the other edges; given their importance we attributed them a highly visible colour, red. Slower edges, on short haul routes, are coloured in green to highlight their relatively weaker performance in geographical time-space.

- The need to consider the phenomenon of occultation of three dimension shapes.
- The conflict between the desire to rotate the structure in all directions—a common practice in three-dimensional modelling–and the convention of cartography that invite to restrict it to rotate exclusively on an east-west axis.
- The combination of geographical projection, from the geoid to a two-dimensional flat surface, with other types of projection used to transform a three-dimensional scene into a computer screen image.
- The need for shadings to reveal three-dimensional shapes that may conflict with the greying conventions of cartography used to express terrestrial relief, and conventional colour choices in cartography [50].
- The need for a background, which is not generally dealt with in cartography.
- The need to adjust the graphical parameters of networks–width, colour, shape, and transparency–to allow for reading the three-dimensional surface situated below.
- The treatment of terrestrial and maritime borders.

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Cones Parameters in Projected Geometry

## Appendix B. Cones and Straight Edges in Spherical Geometry

## Appendix C. Drawing Smoothed Edges in Projected Geometry

**Figure A3.**Several graphical solutions to the constraint of drawing an edge with a given length between cities a and b: Two straight segments (dotted) and different Bézier curves (red with one control point $Qr$; blue with two control points $Qb$ and $Rb$; yellow with two control points $Qy$ and $Ry$; and green with four control points $Qg$, $Rg$, $Sg$, and $Tg$). The position of control points was adjusted to constrain the curve length.

## Geolocation Information

- People’s Republic of China.
- Taiwan.

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**Figure 1.**Space inversion between three places located in geographical space (kilometres) and in geographical time-space (duration).

**Figure 2.**The commercial speed of aircraft services on a sample of origin destination pairs (data from www.flightglobal.com in 2016) and a linear approximation.

**Figure 4.**A representation of the Chinese geographical time-space in 2006. View of a model generated by the Shriveling world software, unprojected Chinese cities, flight information from openflights.org, UN WUP cities data.

## Short Biography of Authors

Alain L’Hostis Alain L’Hostis is senior researcher at the City, Mobility, Transport laboratory (LVMT) from the Université Gustave Eiffel. He holds a PhD and an habilitation thesis in spatial planning. His research topic is about geographical distances: he works on time-space cartography, on inter-urban distances through the measurement of accessibility, and on intra-urban distances in relation with the urban model of TOD. | |

Farouk Abdou Farouk Abdou works in Ministère de la Défense as project leader in software development. |

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**MDPI and ACS Style**

L’Hostis, A.; Abdou, F.
What Is the Shape of Geographical Time-Space? A Three-Dimensional Model Made of Curves and Cones. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 340.
https://doi.org/10.3390/ijgi10050340

**AMA Style**

L’Hostis A, Abdou F.
What Is the Shape of Geographical Time-Space? A Three-Dimensional Model Made of Curves and Cones. *ISPRS International Journal of Geo-Information*. 2021; 10(5):340.
https://doi.org/10.3390/ijgi10050340

**Chicago/Turabian Style**

L’Hostis, Alain, and Farouk Abdou.
2021. "What Is the Shape of Geographical Time-Space? A Three-Dimensional Model Made of Curves and Cones" *ISPRS International Journal of Geo-Information* 10, no. 5: 340.
https://doi.org/10.3390/ijgi10050340