General Relativity: An Introduction for Physicists · Problem 2.5

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · map-projections-mercator

Problem

Consider the surface of the Earth, which we assume for simplicity to be a 2-sphere of radius

a

. In terms of standard polar coordinates

(\theta, \phi)

, the longitude of a point, in radians, rather than the usual degrees, is simply

\phi

(measured eastwards from the Greenwich meridian), whereas its latitude

\lambda = \pi/2 - \theta

radians. Show that the line element on the Earth's surface in these coordinates is

ds^2 = a^2(d\lambda^2 + \cos^2\lambda\,d\phi^2).

To make a map of the Earth's surface, we introduce the functions

x = x(\lambda, \phi)

and

y = y(\lambda, \phi)

and use them as Cartesian coordinates on a flat rectangular piece of paper. Each choice of the two functions corresponds to a different map projection. The Mercator projection is defined by

x = \frac{W\phi}{2\pi}, \qquad y = \frac{H}{2\pi}\ln\left[\tan\left(\frac{\pi}{4} + \frac{\lambda}{2}\right)\right],

where

W

and

H

are the width and height of the map respectively. Find the line element for this projection.

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