Flat Map of a Sphere via Stress Minimization

TL;DR

This paper introduces a new map projection derived from stress minimization, involving a spherical rubber ball model of Earth that is stretched to flatten hemispheres, illustrating calculus of variations in geometric optimization.

Contribution

It presents a novel map projection obtained through stress minimization, offering a new approach to flattening spherical surfaces with mathematical rigor.

Findings

The projection closely resembles the GGV projection.

It demonstrates the application of calculus of variations in map design.

The method provides a mathematically interesting way to flatten spheres.

Abstract

In this paper we describe a mathematically interesting but relatively minor improvement to the Gott-Goldberg-Vanderbei (GGV) map projection. This new projection can be described as what one would get by making a spherical rubber ball representation of the Earth and then stretching the ball circularly around the equator until the Northern and Southern hemispheres flatten to a disk. It is interesting that this new projection is very similar to but not exactly the same as the GGV projection. And, the mathematics required to solve this flattening problem is a very nice example of using the calculus of variations to solve an infinite dimensional optimization problem.