Construction of Areas of Central-places

Abstract

The purpose of this paper is to discover the law concerning the construction of service areas of central-places.
1. How to determine the centrality-index
The centrality-index is determined by following ways. In Fig. 1, A in the vertical is total central functions, B is functions of basic activities, and F in the horizontal is functional units. Fix M, the intersection A and F, and combine A and M. The gradient AM shows the strength of the central-place. If central functions of basic activities of other central-place are smaller than B, as shown in B', the gradient changes AM to AM'. Or if functional units are smaller than F, as shown in F', the gradient changes AM to AM". Now the gradient is determined in the formula
B=A-nF where n is the number of the gradient.
If the number of n grows larger, the gradient makes steeper slopes, and means the strength of the central-place is weak. If the number of n grows smaller, the gradient makes more gentle slopes, and means the strength of the central-place is strong. However, the number of n is not always suitable to express the strength of the central-place. It is more suitable to express the strength of the central-place when the converse number of n (1/n) is caluculated, as the strength of the central-place is proportional to the converse number of n.
2. How to construct the area of the central-place
Consider there are two central-places P and Q, and the centrality-index of P is nine times larger than that of Q. Then the service area of P should be nine times larger than that of Q; that is, πγ²:πγ'²=3:1. This results in γ;γ'=3:1, and means that the ratio is equal to the ratio of the roots of the centrality-indexes concerning P and Q (√9:√1). Now, from this radius ratio, we are able to make the equilibrium circle with the diameter XY (Fig. 4). The inner sphere of the circle is the service area of Q, and the outer sphere of the circle is the service area of P. However, there are small central-places along the line of PQ, and each of them has the centrality-index. The strength of P and Q decrease whenever they met such central-places on the line PQ (Fig. 5). Then, the equilibrium point between P and Q must be determined where the cumulative indexes (roots of indexes) of small central-places from P is equal to that of Q (Fig. 3). If the ratio of cumulative indexes P:Q is larger than 3:1 at a central-place, she belongs to the area of Q. On the contrary, if the ratio is smaller than 3:1 at a central-place, she belongs to the area of P. And as a matter of course, the cumulative indexes must be caluculated on the actual traffic way.