An Exploration in Mathematical Cartography: Comparative Distortion

Abstract

The purpose of this paper is to find a notion of comparative distortion (abbreviated CD in this paper) between two map projections. In historical mathematical cartography, and emphasis has been placed on (1) projecting the globe onto a flat surface and (2) quantifying the necessary distortion that came from it. This paper forms a method to compute some notion of comparative distortion, which is a way to quantify how "different" two maps are in terms of distortion. The premise of the methodology is to form a mathematical model that most closely follows human intuition about when graphs are different and how different they are, because "difference" between maps is something inherently subjective. Procedurally, this is done by dissecting each map into a finite number of points, and then quantifying distortion at each of those points. Each map can then be thought of a matrix or vector with a finite number of distortion entries, and two maps can be compared by comparing their vectors. The method of comparison involves taking the difference of a group of maps' distortion vectors, and then taking the magnitude of that difference vector. This represents some Euclidean notion of "distance" between the two maps and some intuitive notion of how "different" they are. It is consistent with qualitative intuition about map differences and it preserves ordinality, so any maps that are closer together than any other maps will be reflected quantitatively. Overall, this method is successful in modeling intuition about comparative distortion.