Continuous Generalization of Administrative Boundaries Based on Compatible Triangulations

Abstract

Continuous generalization aims to produce maps at arbitrary scales without abrupt changes. This helps users keep their foci when working with digital maps interactively, e.g., zooming in and out. Topological consistency is a key issue in cartographic generalization. Our aim is to ensure topological consistency during continuous generalization. In this paper, we present a five-step method for continuously generalizing between two maps of administrative boundaries at different scales, where the larger-scale map has not only more details but also an additional level of administrative regions.

Our main contribution is the proposal of a workflow for generalizing hierarchical administrative boundaries in a continuous and topologically consistent way. First, we identify corresponding boundaries between the two maps.We call the remaining boundary pieces (in the larger-scale map) unmatched boundaries. Second, we use a method based on so-called compatible triangulations to generate additional boundaries for the smaller-scale map that correspond to the unmatched boundaries. Third, we simplify the resulting additional boundaries. Fourth, we determine corresponding points for each pair of corresponding boundaries using a variant of an existing dynamic programming algorithm. Fifth, we interpolate between the corresponding points to generate the boundaries at intermediate scales.

We do a thorough case study based on the provinces and counties of Mainland China. Although topologically consistent algorithms for the third step and the fifth step exist, we have implemented simpler algorithms for our case study.

Animation

We tested our method on the administrative boundaries of Mainland China, which are from the National Fundamental Geographic Information System, and based on the projected coordinate system Krasovsky 1940 Lambert Conformal Conic. The tested data sets are obtained from the complete data sets of China by removing the only enclave in Gansu province and all the islands. We use a data set of county boundaries with scale 1 : 5,000,000 and 493,630 vertices, and a data set of provincial boundaries with scale 1 : 30,000,000 and 7,527 vertices. Here we show the results of our algorithm for three provinces: Tianjin, Fujian, and Shanghai.

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Tianjin	Fujian	Shanghai