# Identifying urban boundaries

MorphoLim is dedicated to the identification of the morphological boundary of urban agglomerations. The method used avoids the need for any pre-defined maximum distance threshold between buildings. It is based on the conceptual and methodological tools of fractal geometry.

MorphoLim is open source and distributed under GPL licence. The software is developped in Java and runs on any computer supporting Java 7 or later (PC under Linux, Windows, Mac...)

The conceptual bases of Morpholim have been set out by Cécile Tannier, Pierre Frankhauser and Gilles Vuidel. The computer application has been created by Gilles Vuidel and Cécile Tannier. All were members of the research laboratory ThéMA in Besançon (France).

## Description

### Step 1: Step-by-step dilation and counting of the number of built clusters

The data used are vector maps representing buildings in 2D (polygons). The spatial extent of a study area is necessarily large because it comprises an (either monopolar or multipolar) urban agglomeration and its hinterland (i.e., a suburban or rural area that is under the influence of its urban core).

A vector version of the Minkowski dilation is applied to each built polygon of a map. This consists in surrounding each built polygon with a buffer of increasing width. The width of the buffer increases according to a geometric logic, which corresponds to the fractal logic. The number of clusters is counted after each dilation step. The results are portrayed as a log-log plot, where the X-axis represents the width of the dilation buffer and the Y-axis represents the corresponding number of built clusters.

### Step 2: Identification of a significant threshold on the dilation curve

A distance threshold is located on the dilation curve corresponding to a major change in its behavior. We therefore measure the maximum curvature of the dilation curve.

To identify the point of main curvature, the dilation curve is approximated by a series of polynomials of increasing degrees. We then select the polynomial of the lowest degree that best represents the original curve. This choice is based on Schwartz’s Bayesian Information Criterion (BIC).

Once the estimated curve is obtained, points of maximum curvature are identified: they are characterized by a zero derivation value of the curvature function. The point of main curvature has the highest absolute value of curvature among the points of maximum curvature. Finally, the point of main curvature is located on the estimated curve, which gives the distance threshold for drawing an urban envelope.

### References

Tannier C., Thomas I. (2013). Defining and characterizing urban boundaries: A fractal analysis of theoretical cities and Belgian cities, Computers, Environment and Urban Systems 41, pp.234-248

Tannier C., Thomas I., Vuidel G., Frankhauser P. (2011). A Fractal Approach to Identifying Urban Boundaries, Geographical Analysis 43(2), pp 211-227.

### Source code

MorphoLim is licenced under GPL and the source code can be downloaded from the sourcesup git repository:

`git clone git://git.renater.fr/morpholim.git`

## Video tutorial

Video 4:38 - Excerpt from a course "Identifying urban boundaries and analyzing their shape. A fractal approach" by Cécile Tannier & Isabelle Thomas, Master Erasmus+ "Geographical Modelling", University of Luxembourg, February 2016.

## Contact

About the model and the results cecile.tannier@univ-fcomte.fr

About the software application and its use gilles.vuidel@univ-fcomte.fr