  Although the average population density of a city is an extremely simple
indicator, it is often used as a determinant factor for describing various
aspects of urban phenomena. On the other hand, a plethora of different measures
that aim at characterizing the urban form have been introduced in the
literature, often with the risk of redundancy. Here, we argue that two measures
are enough to capture a wealth of different forms of the population density.
First, fluctuations of the local density can be very important and we should
distinguish almost homogeneous cities from highly heterogeneous ones. This is
easily characterized by an indicator such as the Gini coefficient $G$, or
equivalently by the relative standard deviation or the entropy. The second
important dimension is the spatial organization of the heterogeneities in
population density and we propose a dispersion index $\eta$ that characterizes
the degree of localization of highly populated areas. We argue that these two
dimensions are enough to characterize the spatial organization of cities, and
we discuss this approach using a dataset of about $4,500$ cities belonging to
the $10$ largest urban areas in France, for which we have high resolution data.
Representing cities in the plane $(G,\eta)$ allows us to construct families of
cities. On average, compactness increases with heterogeneity, and we find four
large categories of cities (with population $>10,000$ inhabitants): (i) first,
homogeneous and dispersed cities with small density fluctuations, (ii) very
heterogeneous cities with a compact organization of large densities areas. The
last two groups comprise heterogeneous cities with (iii) a monocentric
organization or (iv) a more delocalized, polycentric structure. Integrating
these two parameters in econometric analysis could improve our understanding of
the impact of urban form on various socio-economical aspects.
