Normalizing and classifying shape indexes of cities by ideas from fractals

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A standard scientific study comprises two processes: one is to describe how a system works, and the other is to understand why the system works in this way. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe the size and shape of cities. However, the comparability of a shape index is often influenced by the resolution of remote sensing images or digital maps because the calculated values depend on spatial measurement scales. This paper is devoted to exploring the scaling property in classical shape indexes with mathematical methods. Two typical regular fractals, Koch's island and Vicsek's figure, are employed to illustrate the property of scale dependence of many shape indexes. A set of formula are derived from the geometric measure relation to associate the fractal dimension for boundary lines with shape indexes. Based on the ideas of fractals, two main problems are solved in this work. First, several different correlated shape indexes such as circularity ratios and compactness ratios are normalized and unified by substituting Feret's diameter for the major axis of urban figure. Second, three levels of scaling in shape indexes are revealed, and shape indexes are classified into three groups. The significance of this study lies in two aspects. One is helpful for realizing the deep structure of urban form, and the other is useful for clarifying the spheres of application of different types of shape indexes.