P. D. Converse提出的城市断裂点理论被广泛用来确定城市的空间影响范围和城市经济区的划分。由于该理论仅给出了每两个城市间一个断裂点的计算公式,在实际应用中就出现了多种空间分割方法。分析表明,许多方法是不可行、不严密的。本文通过对比分析Voronoi图和城市断裂点理论的性质,提出了扩展断裂点理论和断裂弧的概念。并证明:在匀质平面区域内,如果两个城市点的权重相同,那么其吸引范围的分界线是这两个城市点连线的垂直平分线;如果它们的权重不同,那么其吸引范围的分界线是一个圆弧;平面内所有断裂点的轨迹分别构成常规Voronoi图和加权Voronoi图,并且每个城市点的权重分别等于其中心性强度值的平方根。最后,以河南省为例进行了城市空间影响范围和城市经济区划分的应用分析。

The Breaking Point Theory is widely used in delimiting urban abstracted regions and dividing up urban economic regions. Because the theory just gives only one breaking point between two cities, many methods for partitioning space are used, for instance, making vertical line through the breaking point on the linked line between the near two cities and linking the near breaking points with smooth lines. In fact, these are not feasible and not rigorous.
As a common method of dividing space, Voronoi diagram is usually used in cartography, meteorology, geognosy, archeology, chemistry, ecology, and computational science, etc. but it is rarely used in urban geography home and abroad because of many causes. Wang Xin-sheng et al. putted forward two methods of delimiting urban abstracted regions with ordinary Voronoi diagram and weighted Voronoi diagram which weights are the each city's central strength value, but some viewpoints should be discussed deeply. Okabe and Suzuki studied on the location's optimize of different level establishments in a continuous plane, and validated Christaller's Central Place Theory in urban geography, etc.
In this paper, the expanded Breaking Point Theory and the conception of breaking arcs were putted forward by contrasting the Voronoi diagram to the classical theory. It was proved that in a well proportioned plane, the boundary between two city's abstracted regions is the vertical bisector of the connected line with them if the cities' weights are equal; the boundary is an arc if the weights are not equal; the orbit of all the breaking points in the plane forms ordinary Voronoi diagram and weighted Voronoi diagram accordingly; and each city's weight equals to the square root of its central strength value in the second situation. As a demonstration, the expanded theory was used in Henan province, and the scheme of the urban economic regions was putted forward. It is noticed that because of the complexity and difficulty of making weighted Voronoi diagram, the wide applications in many fields are restricted in large extent, especially in urban geography. It should be pointed out that the expanded theory is still a theoretic model that must be verified in actual applications, and the authors expect farther discussion.