A New Algorithm for Continuous Area Cartogram Construction with Triangulation of Regions and Restriction on Bearing Changes of Edges
Inoue, Ryo, Shimizu, Eihan, Cartography and Geographic Information Science
In recent years, a variety of statistical datasets compiled by national and local governments have become generally available in the form of digital data at little or no cost. As a result, both analysts and ordinary citizens now have ready access to extensive databases. Many statistical datasets can be sourced within geographic frameworks provided by geographic information systems (GIS). One primary function of GIS is to allow the visual presentation of the statistical analyses of data. Typical visualization procedures provided by standard GIS software packages include such methods as choropleth and dot mappings. An alternative visualization procedure is a cartogram, which has been discussed in quantitative geography (Monmonier 1977; Dorling 1996; Tobler 2004). A cartogram is a transformed map on which areas or distances are proportional to the statistical data values. This is a powerful method for the visual representation of statistical data.
One of the most common cartograms is a continuous (or contiguous) area cartogram. A continuous area cartogram is a deformed map obtained by resizing its regions according to the statistical data of the regions; it preserves the boundary relationships of its regions on the geographical map. The deformation of the shapes of the regions assists map readers to intuitively recognize the distribution of statistical data. A continuous area cartogram is sometimes called a "cartogram" or "value-by-area map." In this study, we describe continuous area cartograms, hereafter referred to as area cartograms.
The advantage of using area cartograms for visualizing statistical data has resulted in the proposal of various methods for their construction. However, the existing area cartogram construction algorithms tend to be problematic with regard to the initial setting of their fundamental parameter values since the mathematical meaning of the parameters is not clear. Some existing algorithms often rail to create area cartograms that depict accurate statistical data; others create area cartograms that are highly deformed or not visually elegant, such that the transformed polygonal shapes do not resemble the original shapes. The mathematical limitations of these algorithms have resulted in their unavailability in software packages. As a result, they are not easily accessible to cartographers and, despite their utility in visual presentations, area cartograms are not yet widely used.
In this study, we provide solutions whereby GIS users may construct area cartograms from their own datasets. In order to achieve this objective, we document the basic requirements needed to prepare a visually elegant area cartogram based on a literature review of previous area cartogram construction methods; subsequently, we explain mathematically straightforward and user-friendly construction procedures.
Previous Studies on Area Cartogram Construction
Initially, area cartograms were manually created. Subsequently, numerous computational solutions have been proposed in order to facilitate their construction (Dorling 1996; Tobler 2004). However, some of the suggested methods require the setting of multiple parameters and long calculation times, or they fail to accurately represent the data. Furthermore, since only a few procedures (Du and Liu 1999) have been provided as a part of GIS packages, cartographers are unable to access them readily. We review previous area cartogram construction solutions below.
The first mathematical solution was proposed by Tobler (1963). Subsequently, he introduced the mathematical definition of area cartogram construction by a pair of nonlinear partial differential equations (Tobler 1973). He adopted the minimization of the Dirichlet integral in order to achieve the least distortion. Then, he used an approximate solution provided by tetragonal or hexagonal lattices; he calculated the movement of vertices for these lattices. …