  The spatial distributions of cities fall into two groups: one is the simple
distribution with characteristic scale (e.g. exponential distribution), and the
other is the complex distribution without characteristic scale (e.g. power-law
distribution). The latter belongs to scale-free distributions, which can be
modeled with fractal geometry. However, fractal dimension is not suitable for
the former distribution. In contrast, spatial entropy can be used to measure
any types of urban distributions. This paper is devoted to generalizing
multifractal parameters by means of dual relation between Euclidean and fractal
geometries. The main method is mathematical derivation and empirical analysis,
and the theoretical foundation is the discovery that the normalized fractal
dimension is equal to the normalized entropy. Based on this finding, a set of
useful spatial indexes termed dummy multifractal parameters are defined for
geographical analysis. These indexes can be employed to describe both the
simple distributions and complex distributions. The dummy multifractal indexes
are applied to the population density distribution of Hangzhou city, China. The
calculation results reveal the feature of spatio-temporal evolution of
Hangzhou's urban morphology. This study indicates that fractal dimension and
spatial entropy can be combined to produce a new methodology for spatial
analysis of city development.
