Circular-Arc Cartograms

TL;DR

This paper introduces a novel circular-arc cartogram model that uses circular arcs to distort map regions proportionally to data values, offering an aesthetically pleasing and intuitive visualization, along with a heuristic construction method.

Contribution

The paper proposes a new circular-arc cartogram model and a heuristic algorithm for its construction, addressing the NP-hardness of the problem.

Findings

The model produces visually appealing cartograms resembling clouds or snowflakes.

The heuristic method effectively approximates desired areas in real-world examples.

Constructed cartograms are computationally feasible despite the NP-hardness of the problem.

Abstract

We present a new circular-arc cartogram model in which countries are drawn as polygons with circular arcs instead of straight-line segments. Given a political map and values associated with each country in the map, a cartogram is a distorted map in which the areas of the countries are proportional to the corresponding values. In the circular-arc cartogram model straight-line segments can be replaced by circular arcs in order to modify the areas of the polygons, while the corners of the polygons remain fixed. The countries in circular-arc cartograms have the aesthetically pleasing appearance of clouds or snowflakes, depending on whether their edges are bent outwards or inwards. This makes it easy to determine whether a country has grown or shrunk, just by its overall shape. We show that determining whether a given map and given area-values can be realized as a circular-arc cartogram is an NP-hard problem. Next we describe a heuristic method for constructing circular-arc cartograms, which uses a max-flow computation on the dual graph of the map, along with a computation of the straight skeleton of the underlying polygonal decomposition. Our method is implemented and produces cartograms that, while not yet perfectly accurate, achieve many of the desired areas in our real-world examples.