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The Mercator projection is a map projection devised by Gerardus Mercator in 1569.

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 wildly distorts area: Greenland is presented as having roughly the same size as Africa, when in fact Africa is approximately 13 times larger than Greenland.

Mercator projection: public domain Online Map Creation

This is done for an important reason: the projection is a conformal map, that is, it preserves angle. Any straight line on a Mercator map is a line of constant bearing, 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, the time of travel was subject to the elements and hence the distance to travel was not as important as the direction to take especially since longitude could not be accurately determined.

To achieve this effect, the Mercator projection stretches East-West distances by an increasing amount as the distance from the equator increases. The extreme case of distorted area is at the poless, where the two geographical points have become lines at the top and bottom of the map.

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):

This is the inverse of the Gudermannian function.

And the inverse:

Although the Mercator projection is still in common use for navigation, some critics argue that is no longer suited to represent the entire world in publications and wall maps. The position of Europe and North America are much closer to the center of the map than they ought to be, and their increased size compared to Africa is seen as perpetuating the idea of the Third World's inferiority especially as it is largely relegated to the lower part of the map.

Furthermore, some Mercator maps omit most or all of Antarctica. This has the effect of placing Europe at the center of the map. Mercator projections are rarely used in atlases to display the world. They continue to be useful for navigation.

The Gall-Peters projection has been proposed as an alternative to address these concerns. This presents a very different view of the world: the shape of countries is highly distorted, especially away from the equator, but area is preserved. Nevertheless, a 1989 resolution by seven North American Geographical groups decries the use of all rectangular coordinate world maps including the Gall-Peters.

## Derivation of the Projection

The latitude and longitude are equivalent to the spherical coordinates φ and θ, respectively. The radius R can be ignored. A straight line on a Mercator projection represents movement (e.g. of a ship) on the Earth at a constant angle with respect to geographic North.

Assume φ=0 (due to symmetry, the following argument is similar for any φ). Imagine a plane tangent to the sphere at point . Such plane can be described by a pair of unit basis vectors and , where

Let us say a ship moves at a (clockwise) angle β away from the North (eθ), then the ship's movement is
where eφ points towards the East.

Now let us say the ship moves at a 45 ° angle from the North (). Then, if the ship moves a certain distance in the eφ direction, it will move an equal distance in the eθ direction.

However, movement in the eθ direction is along a geodesic of circumference

where R is the Earth's radius.

Therefore if the ship moves a distance ΔSθ in the eθ direction, this is equivalent to moving an angle Δθ, where

(the Earth's radius is the conversion factor).

Movement in the eφ direction is along a circle of circumference

When the ship moves a distance ΔSφ in the eφ direction, this is equivalent to movement by an angle Δφ, where

Letting β=45°, then , so
Divide both sides by R and convert differences to differentials,
.
Group together the thetas,
Integrate,

And, since φ is directly proportional to lengths on a map (x = φ), equation (1) means (since x and y have uniform units of the same length) that

Quod Erat Demonstrandum.  