Walk on City Maps - Mathematical and Physical phenomenology of the City, a Geometrical approach

We are interested in the phenomenology of cities by restricting them to the geometry of their street network. This study aims at being synthetic, functional and interdisciplinary. It follows the large work that has been performed from the early XXth century by townplanners, social scientists, geographers, statisticians, physicists. We try to demonstrate that the street - as a coherent alignment of street segments - can be considered as an essential elementary structure of the city. How much information is encoded in the street network? To what extent does it constraint the city use? How are the current urban layout and its evolution determined jointly by traffic axis and structuring elements? We present a mathematical and computational framework allowing to consider the map of a city as a geometric continuum associated to the topology of a planar graph. To this graph we juxtapose a hypergraph structure using the street geometry to obtain easily the notion of axis and a multi-scale representation of the city. In spite of an apparent shape diversity, we show that the street network of a city is subjected to general laws that leave hallmarks on a city map. We propose several morphogenesis models of the city, implementing the idea that the city's growth follows a structured extension / division of space logic and able to reproduce hallmarks observed on actual maps. The understanding of regulation mechanism of the city allows us to propose functional algorithms whose computational efficiencies are very interesting. We present an algorithm recovering streets from a collection of street segments; the notion of simplest centrality whose calculus on a map allows a hierarchical analysis of it, revealing for instance main traffic axis and ill-deserved area; a fast approximate algorithm to nd the shortest path between two random points; and a Spectral Clustering based algorithm to produce morphological segmentations of maps. We also work on the identi cation of random tessellation models to be substituted to an actual road network and to solve large optimization problems using statistical equivalence.