Derivation of the functional relations between fractal dimension of and shape indices of urban form

The relationships between the size, scale, shape, and dimension of urban settlements are basic problems remained to be further resolved, and this paper provides an available perspective for understanding these problems. Based on the standard circle, the relations between the fractal dimension of urban boundary and the compactness ratios of urban shape were derived from a geometric measure relation in a simple way. The compactness ratios proved to be the exponential functions of the reciprocal of the boundary dimension. The results can be generalized and applied to the common indices of shape including circularity ratio, ellipticity index, and form ratio, which are defined by urban area, perimeter, or Feret’s diameter. The mathematical models are empirically verified by the remote sensing data of China’s 31 mega-cities in 1990 and 2000 and lend support to the assumption that urban boundaries are pre-fractals rather than real fractals. A conclusion can be drawn that there exist certain functional relations between the shape indices and the boundary dimension, and within certain range of scales, the fractal parameters can be indirectly estimated by the ratios of size measurements to reflect the features of urban shapes.

► The functional relation between boundary dimension and shape indices is derived.
► Fractal geometry and Euclidean geometry can be linked by the functional relation.
► The function implies the relation between of size, scale, and shape of cities.
► The scaling range is a key of fractal research of cities.
► The function relation can be employed to make urban spatial analysis in practice.