Mercator projection

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The Mercator projection of the world up to a latitude of 86° N and S

The Mercator projection is a cylindrical map projection devised by Gerardus Mercator in 1569. Its parallels and meridians are straight lines, and the unavoidable east-west stretching away from the equator is accompanied by a corresponding north-south stretching, so that at each location the east-west scale is the same as the north-south-scale. A Mercator map can never fully show the polar areas; it would be infinitely high.

It is a conformal map, that is, it preserves angles. Any straight line on a Mercator map is a line of constant bearing, that is, a loxodrome or rhumb line. This makes it particularly useful to navigators, even though the plotted route is usually not a great circle (shortest distance) route. In the era of sailing ships, time of travel was subject to the elements, hence the distance traveled was not as important as the direction taken—especially since longitude was hard to determine accurately.

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Gudermannian.png
relation between vertical position on the map (horizontal in the graph) and latitude (vertical in the graph)

The following equations determine the x and y coordinates of a point on a Mercator map from its latitude φ and longitude λ (with λ0 being the longitude in the center of map):

[itex]

\begin{matrix} x &=& \lambda - \lambda_0 \\ \\ y &=& \ln \left[ \tan \left( \frac {1} {4} \pi + \frac {1} {2} \phi \right) \right] \\ \\ \ & =& \frac {1} {2} \ln \left( \frac {1 + \sin \phi} {1 - \sin \phi} \right) \\ \\ \ & =& \sinh^{-1} \left( \tan \phi \right) \\ \\ \ & =& \tanh^{-1} \left( \sin \phi \right) \\ \\ \ & =& \ln \left( \tan \phi + \sec \phi \right). \end{matrix} [itex]

This is the inverse of the Gudermannian function:

[itex]

\begin{matrix} \phi &=& 2\tan^{-1} \left( e^y \right) - \frac{1} {2} \pi \\ \\ \ &=& \tan^{-1} \left( \sinh y \right) \\ \\ \lambda &=& x + \lambda_0. \end{matrix} [itex]

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Tissot_mercator.png
Tissot's Indicatrix

The scale is proportional to the secant of the latitude φ, getting arbitrarily large near the poles, where φ = plus or minus 90°. Moreover, as seen from the formulas, for the poles y is plus or minus infinity.

Controversy

Like all map projections attempting to fit a curved surface onto a flat sheet, the shape of the map is a distortion of the true layout of the Earth's surface. The Mercator projection exaggerates the size (and to a lesser extent, the shape) of areas far from the equator. For example, Greenland is presented as being roughly as large as Africa, when in fact Africa's area is approximately 13 times that of Greenland as shown by Tissot's indicatrix.

Although the Mercator projection is still in common use for navigation, critics argue that it is not suited to representing the entire world in publications and wall maps due to its distortion of land area. As a result of these criticisms, modern atlases no longer use the Mercator projection for world maps or for areas distant from the equator, preferring other cylindrical projections, or forms of equal-area projection. The Mercator projection is still commonly used for areas near the equator, however.

The equal-area Gall-Peters projection has also been proposed as an alternative to address these concerns. This presents a very different view of the world: the shape, rather than the size of areas is distorted. Areas near the equator are stretched vertically; areas far from the equator are squashed. A 1989 resolution by seven North American Geographical groups decried the use of all rectangular coordinate world maps, including the Gall-Peters projection.

Derivation of the projection

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The Mercator projection is a cylindrical projection.

Assume a spherical Earth. (It is actually slightly flattened, but for small-scale maps the difference is immaterial. For more precision, interpose conformal latitude.) We seek a transform of longitude-latitude (λ,φ) to Cartesian (x,y) that is "a cylinder tangent to the equator" (i.e. x=λ) and conformal (i.e. with [itex]\partial x/\partial\lambda=\cos\phi\,\partial y/\partial\phi[itex] and [itex]-\cos\phi\,\partial x/\partial\phi=\partial y/\partial\lambda[itex].)

From x = λ we get

[itex]\partial x/\partial\lambda=1[itex]
[itex]\partial x/\partial\phi=0[itex]

giving

[itex]1=\cos\phi\,\partial y/\partial\phi[itex]
[itex]0=\partial y/\partial\lambda.\,[itex]

Thus y is a function only of φ with [itex]y'=\sec\phi[itex] from which a table of integrals gives [itex]y=\ln|\sec\phi+\tan\phi|+C[itex]. It is convenient to map φ = 0 to y = 0, so take C = 0.

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