Bisectors, Buffers, and Base Maps

Abstract

In 1911, Thiessen and Alter [21] wrote on the analysis of rainfall using polygons surrounding rain gauges. Given a scatter of rain gauges, represented abstractly as dots, partition the underlying plane into polygons containing the dots in such a way that all points within any given polygon are closer to the rain gauge dot within that polygon than they are to any other gauge-dot. The geometric construction usually associated with performing this partition of the plane into a mutually exclusive, yet exhaustive, set of polygons is performed by joining the gauge-dots with line segments, finding the perpendicular bisectors of those segments, and extracting a set of polygons with sides formed by perpendicular bisectors. It is this latter set of polygons that has come to be referred to as "Thiessen polygons" (and earlier names such as Dirichlet region or Voronoi polygon, see Coxeter [4]). The construction using bisectors is tedious and difficult to execute with precision when performed by hand. Kopec (1963) [11] noted that an equivalent construction results when circles of radius the distance between adjacent points are used. Indeed, that construction is but one case of a general construction of Euclid. Like Kopec, Rhynsburger (1973) [20] also sought easier ways to construct Thiessen polygons: Kopec through knowledgeable use of geometry and Rhynsburger through the development of computer algorithms. The world of the Geographical Information System (GIS) software affords an opportunity to combine both.