# Geospatial Assessment of the Territorial Road Network by Fractal Method
## 2 Statement of the Problem
### Calculation of the Transport Provision of Territories Based on the Fractals Theory  
It is known that the Hausdorff-Besicovitch dimension is a natural way to determine the dimension of a subset in metric space. In three-dimensional Euclidean space, the Hausdorff-Besicovitch dimension of a finite set is zero, the dimension of a smooth curve is one, the dimension of a smooth surface is two, and the dimension of a set of nonzero volume is three. For more complex (fractal) sets, the Hausdorff-Besicovitch dimension may not be an integer [26]. The determination of the Hausdorff-Besicovitch dimension can be considered by measuring the dimension of a curve (Figure 1), which is covered by a fixed grid of squares with a side \(\varepsilon>0\). Each point of a linear object belongs to one of the squares. Squares in which there are no points are not taken into account. The sum over all the squares covering the object (Hausdorff measure) has the following form:  
\[m_{p}=\sum\epsilon^{p} \tag{4}\]  
where \(p\) is an arbitrary parameter. There is a critical value of \(\mathrm{p}_{0}\) such that \(\lim\limits_{\xi\to\infty}m_{p}=\infty\) for all \(\mathrm{p}<\mathrm{p}_{0}\) and \(\lim\limits_{\xi\to\infty}m_{p}=0\) for all \(\mathrm{p}>\mathrm{p}_{0}\). This value \(\mathrm{p}_{0}=\mathrm{D}_{\mathrm{H}}\) is the value of the Hausdorff-Besicovitch dimension. For example, for a square \(\mathrm{Q}\) having a unit size with \(\varepsilon=1/10\), the number of square boxes covering \(\mathrm{Q}\) equals \(\mathrm{N}\left(\varepsilon\right)=\varepsilon^{-2}=(1/10)^{-2}=100\). The Hausdorff measure is \(\mathrm{m}_{\mathrm{p}}\) (\(\mathrm{Q}\)) = \(\mathrm{N}(\varepsilon)\epsilon^{\mathrm{p}}=\epsilon^{\mathrm{p}-2}\). Let us say \(\varepsilon\to 0\). Then, \(\mathrm{m}_{\mathrm{p}}\) (\(\mathrm{Q}\)) \(\to\infty\) for all \(\mathrm{p}<2\), and \(\mathrm{m}_{\mathrm{p}}\) (\(\mathrm{Q}\)) \(\to 0\) for all \(\mathrm{p}>2\). Thus, \(\mathrm{D}_{\mathrm{H}}\) (\(\mathrm{Q}\)) = DimQ = 2. The dimension in general is determined by the law of similarity:  
\[N(\varepsilon)\approx\frac{1}{\varepsilon^{D}} \tag{5}\]  
By taking the logarithm of the right and left sides of Equation (5), we obtain  
\[lnN(\varepsilon)=-Dln\varepsilon \tag{6}\]  
\[D=\lim\limits_{\varepsilon\to\infty}\frac{lnN(\varepsilon)}{ln(1/\varepsilon)} \tag{7}\]  
where \(N(\varepsilon)\) is minimal number of the square boxes covering the object and \(\varepsilon\) is the square box size. Let us give a definition of the transport provision of the territory based on the fractals theory [20]. We will consider the geospace (territory) as a two-dimensional space, and the maximum transport provision of the territory is the possibility of getting from each point of this territory to any other point in the shortest distance. By destination points, we mean areal objects (points) whose dimensions (area) in this scale of research are negligible. Then, any territory can be represented as a finite number of such areal objects on a certain scale. A hit at any point of the area of the object is equivalent to falling into its center. Since it is necessary to cover with points the entire investigated territory, it is advisable to choose the corresponding figure, a hexagon, as an area object. Thus, the transport development of the territory will be at a maximum when all the centers of the hexagons are interconnected by faces (a linear object) (Figure 2). These line features are actually roads on a territory map. Figure 1: Determination of the Hausdorff–Besicovitch dimension using the \(\varepsilon\) coating. Since any line to a certain scale is a fractal, the transport provision of the territory can be understood as the desire of the roads to occupy the entire area on which they are located.