Reference: Hobson, M.P., Efstathiou, G. P. & Lasenby, A. N. (2006), General Relativity: An Introduction for Physicists; Cambridge University Press. Problem 2.6.
The Mercator projection is one of many map projections used to represent a spherical surface on a plane. As is obvious from looking at a Mercator map, however, it does not preserve relative areas. Regions nearer to the poles are disproportionately larger than regions nearer the equator.
Suppose we have a general projection given by mapping the spherical coordinates (the latitude, where , with the usual polar angle) and (the longitude) to a rectangular coordinate system using the general formulas and . First, suppose we want the projection to preserve the angle between two vectors originating at some point . To see how to manage this, consider an infinitesimal displacement given in terms of the spherical coordinates ( is the radius of the sphere):
When mapping this to the rectangular system, we must transform the metric in the usual way (both coordinates are orthogonal systems, so only diagonal terms appear):
Now suppose we have two displacements that are perpendicular to each other; one () has and the other () has . In spherical coordinates their scalar product is
Since this is a scalar, it’s invariant between coordinates, so is true in the system as well. Thus perpendicular lines in the spherical system map into perpendicular lines in the rectangular system.
Given this, we can take a general line segment in the spherical system and transform its two perpendicular components separately, giving two corresponding perpendicular segments in the rectangular system. If the ratio of the lengths of these two components is the same in the rectangular system as it was in the spherical system, then the transformed line segment must have the same angles to the and axes as the original segment had to the and axes. If this is true, then it is true for any two different line segments, which means that these two segments must have the same angle between them in both systems.
The transformation of the component parallel to the axis is, since :
We can write the LHS as
where is some function that is determined by the requirement 7. All we’ve done is extract the dependence on the point into the function . The quantity is ultimately just a number, as is so the function maps one set of numbers into the other.
Now look at the transformation of the component parallel to the axis:
We can write the LHS as
where is some other function that maps the RHS onto the LHS. [Note that it’s possible to map the two components and into and components that are perpendicular, but where one axis is scaled up or down relative to the other. For example, if and , we are stretching the component by a factor of 2 relative to the component.]
Now in order for the transformed components to have the same length ratio as the original, we require
The Mercator projection has a line element of
Thus if (that is, it’s a square map), the condition 13 is satisfied.
Now suppose we want a projection that preserves relative areas instead of angles. The area element in an orthogonal system has sides and , so the area element is
For the original spherical system this comes out to
To preserve areas, the coefficient of should be a constant, since there is no preferred location on the plane. Thus the condition is
One projection that satisfies this condition is the Lambert cylindrical equal-area projection, where the transformations are
where is the central line of longitude in the map. In this projection, the area element can be calculated from
Plugging these in, we get
[Note that here and are dimensionless as they are defined in terms of angles, not distances. Thus to get a map that fits into a book, we need to reduce the radius of the sphere from that of the Earth to something approximating a book page.]
It doesn’t appear that it’s possible to preserve both equal areas and relative angles (if it were, I’m sure that would be the most popular projection around). To do so, we’d need to give equal relative angles and from 18 this would require
Thus we need to get rid of the while still retaining some dependence on (otherwise, we’d have no map). However, since every term in this equation is non-negative, and can contain no reference to , there’s no way of cancelling off the terms, so it would seem that we can’t satisfy this equation. [I’m sure there’s a more rigorous way of proving this, but I’m not a mathematician.]