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.