Mercator projection

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.

Related Topics:
Cylindrical map projection - Gerardus Mercator - 1569 - Parallel - Meridian - Equator

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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.

Related Topics:
Conformal map - Angle - Bearing - Loxodrome - Rhumb line - Great circle - Longitude

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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):

Related Topics:
Coordinates - Point - Latitude

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:

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egin{matrix}

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x &=& lambda - lambda_0

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\ \ y &=& ln left

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\ \ & =& rac {1} {2} ln left( rac {1 + sin phi} {1 - sin phi} ight)

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\ \ & =& sinh^{-1} left( an phi ight)

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\ \ & =& anh^{-1} left( sin phi ight)

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\ \ & =& ln left( an phi + sec phi ight).

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end{matrix}

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This is the inverse of the Gudermannian function:

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:

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egin{matrix}

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phi &=& 2 an^{-1} left( e^y ight) - rac{1} {2} pi

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\ \ &=& an^{-1} left( sinh y ight)

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\ \ lambda &=& x + lambda_0.

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end{matrix}

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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.

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