# Mercator projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569, in a large planisphere measuring 202 by 124 cm, printed in eighteen separate sheets. Like in all cylindric projections, parallels and meridians are straight and perpendicular to each other. But the unavoidable east-west stretching away from the equator is here accompanied by a corresponding north-south stretching, so that at every location the east-west scale is the same as the north-south scale, making the projection conformal. A Mercator map can never fully show the polar areas, since linear scale becomes infinitely high at the poles. Being a conformal projection, the linear scale does not vary with direction and the angles are preserved around all locations. However, and like in any other map projection, scale varies from place to place, distorting the shapes of geographical objects. In particular, areas are strongly affected, transmitting a grossly distorted image of the geometry of our planet. All lines of constant bearing (rhumb lines or loxodromes), i. e., those making constant angles with the meridians, are represented by straight segments on a Mercator map. This is precisely the type of route usually employed by ships at sea, where compasses are used to indicate geographical directions and steer the ships. The two properties, conformality and straight rhumb lines, makes this projection uniquely suited to marine navigation: courses and bearings are measured using wind-roses or protractors, and the corresponding directions are easily transferred from point to point, on the map, with the help of a parallel ruler or a pair of navigational squares.

The abuse of the Mercator projection in world representations, as well as the controversy caused by the political promotion of the so-called Gall-Peters projection, lead several American geographic societies to approve, in 1989-90, a recommendation rejecting the use of rectangular world maps for general purposes or artistic displays.

Several authors are associated with the development of Mercator projection:

- German geographer Erhard Etzlaub (c. 1460-1532), who printed small-scale maps of Europe and Africa, in 1511, using a projection identical to Mercator’s. Nothing is known about the method used.

- Portuguese mathematician and cosmographer Pedro Nunes (1502-1578), who first described the loxodrome and its use in marine navigation, and suggested the construction of several large-scale nautical charts in the cylindrical equidistant projection to represent the world with minimum angle distortion (1537).

- English mathematician Edward Wright (c. 1558-1615), who formalized the mathematics of Mercator projection (1599), and published accurate tables for its construction (1599, 1610).

- English mathematicians Thomas Harriot (1560-1621) and Henry Bond (c.1600-1678) who, independently (c. 1600 and 1645), associated the Mercator projection with its modern logarithmic formula, later deduced by calculus.

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):

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

This is the inverse of the Gudermannian function:

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

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, the pole's y is plus or minus infinity.

## Controversy

Image:Tissot mercator.png 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 distorts 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. Mercator himself used the equal-area sinusoidal projection to show relative areas. 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.

Google Maps currently uses a Mercator projection for its map images, probably because a magnification of any small region of a Mercator map will appear undistorted in shape with north at the top. Despite its relative scale distortions, the Mercator is therefore well-suited to an interactive world map that can be panned and zoomed seamlessly to local maps. (Google Satellite, on the other hand, used a plate carrée projection until recently.)

## Derivation of the projection

Image:Usgs map mercator.PNG 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 $\partial x/\partial\lambda=\cos\phi\,\partial y/\partial\phi$ and $-\cos\phi\,\partial x/\partial\phi=\partial y/\partial\lambda$.)

From x = λ we get

$\partial x/\partial\lambda=1$
$\partial x/\partial\phi=0$

giving

$1=\cos\phi\,\partial y/\partial\phi$
$0=\partial y/\partial\lambda.\,$

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