Method for generating contiguous cartograms
First Claim
1. A method for generating a cartogram from a plurality of contiguous polygons having vertices, and from a vector containing values corresponding to the polygons, the method comprising the steps of:
- determining a first curvature function of a first polygon;
performing a Fourier transform of the first curvature function to calculate a first shape value;
repositioning one of the vertices of the polygon to produce an altered polygon;
determining an altered curvature function of the altered polygon;
performing a Fourier transform of the altered curvature function to calculate an altered shape value;
calculating a shape distortion by comparing the first and the altered shape values; and
deciding whether to accept the altered polygon based on the shape distortion.
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Abstract
The present invention is a method for generating cartograms using a base map of contiguous polygons and a vector containing values to which areas of corresponding polygons are scaled. The general problem is intractable, so an iterative heuristic is proposed. The heuristic is based on “scanlines.” The scanlines may be defined automatically (typically, by placing a grid over the map) or entered manually (to provide finer control over the results). At each step, one scanline is chosen and a new candidate map is made by adjusting the vertices of polygons intersected by the scanline, moving them orthogonally to the scanline. A candidate solution is accepted if it improves the solution and preserves the input mesh'"'"'s topology. Improvement depends on metrics for area and shape error. The solver is run until the improvement falls below some threshold, or a time limit or maximum number of iterations is reached
The method determines shape error created by a candidate iterative step by first estimating a curvature function of the polygon and then performing a Fourier transform on the function to yield a shape representation that is relatively independent of scale, translation and rotation of the polygon. The proposed iterative step is accepted or discarded based in part on the magnitude of the resulting shape error.
31 Citations
33 Claims
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1. A method for generating a cartogram from a plurality of contiguous polygons having vertices, and from a vector containing values corresponding to the polygons, the method comprising the steps of:
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determining a first curvature function of a first polygon;
performing a Fourier transform of the first curvature function to calculate a first shape value;
repositioning one of the vertices of the polygon to produce an altered polygon;
determining an altered curvature function of the altered polygon;
performing a Fourier transform of the altered curvature function to calculate an altered shape value;
calculating a shape distortion by comparing the first and the altered shape values; and
deciding whether to accept the altered polygon based on the shape distortion. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17)
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18. A method for generating a cartogram from a map having a plurality of contiguous map polygons with vertices, the cartogram having a plurality of cartogram polygons corresponding to the map polygons, the cartogram polygons having areas proportional to values contained in a data vector, the method comprising the steps of:
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selecting a vertex for repositioning, the vertex having adjacent polygons;
repositioning the selected vertex;
determining an area error of the adjacent polygons based on corresponding data vector values;
determining a shape error of the adjacent polygons based on a Fourier transformation of a curvature function of a perimeter of each polygon; and
deciding whether to accept or reject the repositioning step based on the shape error and the area error. - View Dependent Claims (19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33)
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Specification