The Geometric Properties of Numerical Generalization

Abstract

Over the past twenty years, cartographers have become increasingly concerned with the nature and quality of cartographic data in digital format. One significant area of research has been in the generalization of these digital data. Specifically, algorithms have been developed to (1) weed out unnecessary detail, (2) smooth sharp angularity, (3) displace two features coalescing due to scale change, and (4) enhance certain characteristics of the data. This paper presents an analysis of nine algorithms developed for simplifying the superfluous detail in digital cartographic lines. A series of geometric measures were designed to evaluate the changes produced by simplification. These included both single attribute measures, which evaluate changes such as line length and angularity, and displacement measures, which evaluate geometric displacement such as area. The results of this analysis, using four digitized naturally occurring lines and their simplifications, indicates that four algorithms: DOUGLAS, OPHEIM, REUMANN, and LANG are mathematically superior. Although the Douglas routine was slightly better—in terms of area displacement—than the other three, it is the most computationally complex algorithm. Thus for certain mapping tasks, such as in thematic cartography, other routines such as LANG tolerancing appear more appropriate.

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