Mercator projection

# Mercator projection

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The Mercator projection is a cylindrical map projection presented by the Belgian geographer and cartographer Gerardus Mercator
Gerardus Mercator
thumb|right|200px|Gerardus MercatorGerardus Mercator was a cartographer, born in Rupelmonde in the Hapsburg County of Flanders, part of the Holy Roman Empire. He is remembered for the Mercator projection world map, which is named after him...

, in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course
In navigation, a vehicle's course is the angle that the intended path of the vehicle makes with a fixed reference object . Typically course is measured in degrees from 0° clockwise to 360° in compass convention . Course is customarily expressed in three digits, using preliminary zeros if needed,...

, known as rhumb line
Rhumb line
In navigation, a rhumb line is a line crossing all meridians of longitude at the same angle, i.e. a path derived from a defined initial bearing...

s or loxodromes, as straight segments. While the linear scale is constant in all directions around any point, thus preserving the angles and the shapes of small objects (which makes the projection conformal), the Mercator projection distorts the size and shape of large objects, as the scale increases from the Equator to the poles, where it becomes infinite.

## Properties and historical details

Mercator's 1569 edition was a large planisphere
Planisphere
A planisphere is a star chart analog computing instrument in the form of two adjustable disks that rotate on a common pivot. It can be adjusted to display the visible stars for any time and date. It is an instrument to assist in learning how to recognize stars and constellations...

measuring 202 by 124 cm, printed in eighteen separate sheets. As in all cylindrical projections, parallel
Circle of latitude
A circle of latitude, on the Earth, is an imaginary east-west circle connecting all locations that share a given latitude...

s and meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

s are straight and perpendicular to each other. In accomplishing this, the unavoidable east-west stretching of the map, which increases as distance away from the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

increases, is accompanied by a corresponding north-south stretching, so that at every point location, the east-west scale is the same as the north-south scale, making the projection conformal
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

. A Mercator map can never fully show the polar areas, since linear scale
Linear scale
A linear scale, also called a bar scale, scale bar, graphic scale, or graphical scale, is a means of visually showing the scale of a map, nautical chart, engineering drawing, or architectural drawing....

becomes infinitely high at the poles. Being a conformal projection, angles are preserved around all locations. However scale varies from place to place, distorting the size of geographical objects and conveying a distorted idea of the overall geometry of the planet. At latitudes greater than 70° north or south, the Mercator projection is practically unusable.

All lines of constant bearing
In marine navigation, a bearing is the direction one object is from another object, usually, the direction of an object from one's own vessel. In aircraft navigation, a bearing is the actual compass direction of the forward course of the aircraft...

(rhumb line
Rhumb line
In navigation, a rhumb line is a line crossing all meridians of longitude at the same angle, i.e. a path derived from a defined initial bearing...

s or loxodromes — 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 compass
Compass
A compass is a navigational instrument that shows directions in a frame of reference that is stationary relative to the surface of the earth. The frame of reference defines the four cardinal directions – north, south, east, and west. Intermediate directions are also defined...

es are used to indicate geographical directions and to steer the ships. The two properties, conformality and straight rhumb lines, make this projection uniquely suited to marine navigation: courses and bearings are measured using wind rose
Wind rose
A wind rose is a graphic tool used by meteorologists to give a succinct view of how wind speed and direction are typically distributed at a particular location. Historically, wind roses were predecessors of the compass rose , as there was no differentiation between a cardinal direction and the wind...

s 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 protractor triangles.

The name and explanations given by Mercator to his world map (Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata: "new and augmented description of Earth corrected for the use of sailors") show that it was expressly conceived for the use of marine navigation. Although the method of construction is not explained by the author, Mercator probably used a graphical method, transferring some rhumb lines previously plotted on a globe to a square graticule
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

, and then adjusting the spacing between parallels so that those lines became straight, making the same angle with the meridians as in the globe.

The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application: the impossibility of determining the longitude at sea with adequate accuracy and the fact that magnetic directions, instead of geographical directions, were used in navigation. Only in the middle of the 18th century, after the marine chronometer
Marine chronometer
A marine chronometer is a clock that is precise and accurate enough to be used as a portable time standard; it can therefore be used to determine longitude by means of celestial navigation...

was invented and the spatial distribution of magnetic declination
Magnetic declination
Magnetic declination is the angle between magnetic north and true north. The declination is positive when the magnetic north is east of true north. The term magnetic variation is a synonym, and is more often used in navigation...

was known, could the Mercator projection be fully adopted by navigators.

Several authors are associated with the development of Mercator projection:
• German Erhard Etzlaub
Erhard Etzlaub
Erhard Etzlaub , was an astronomer, geodesist, cartographer, instrument maker and physician.-Life:...

(c. 1460–1532), who had engraved miniature "compass maps" (about 10×8 cm) of Europe and parts of Africa, latitudes 67°–0°, to allow adjustment of his portable pocket-size sundials, was for decades declared to have designed "a projection identical to Mercator’s".

• Portuguese mathematician and cosmographer Pedro Nunes
Pedro Nunes
Pedro Nunes , was a Portuguese mathematician, cosmographer, and professor, from a New Christian family. Nunes, considered to be one of the greatest mathematicians of his time , is best known for his contributions in the technical field of navigation, which was crucial to the Portuguese period of...

(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
Edward Wright (mathematician)
Edward Wright was an English mathematician and cartographer noted for his book Certaine Errors in Navigation , which for the first time explained the mathematical basis of the Mercator projection, and set out a reference table giving the linear scale multiplication factor as a function of...

(c. 1558–1615), who formalized the mathematics of Mercator projection (1599), and published accurate tables for its construction (1599, 1610).

• English mathematicians Thomas Harriot
Thomas Harriot
Thomas Harriot was an English astronomer, mathematician, ethnographer, and translator. Some sources give his surname as Harriott or Hariot or Heriot. He is sometimes credited with the introduction of the potato to Great Britain and Ireland...

(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.

## Mathematics of the projection

Mathematically, the Mercator projection is completely characterized
Characterization (mathematics)
In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...

by the fact that bearings on the globe are everywhere equal to bearings on the map; for example, north on the globe is always upward on the map, and for any angle θ°, the direction that is θ° east of north on the globe is everywhere θ° clockwise from upward on the map.

That necessarily implies that meridians of longitude are vertical on the map, and therefore all parallels of latitude are equally long on the map, even though on the globe, parallels of latitude that are farther from the equator are shorter. This stretching in a horizontal direction then necessitates stretching in a vertical direction in order that bearings not be distorted. Consequently the scale is different at locations remote from the equator from what appears at the equator.

Specifically, the length of the parallel at θ° latitude (either north or south) is cos θ° times the length of the equator. The scale at θ° latitude is therefore multiplied by 1/cos θ° = sec θ°.

The following equations determine the x and y coordinates of a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

on a Mercator map from its latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

φ and longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

λ. The number λ0 is the longitude for x=0. R is the radius of the sphere of the Earth (6378.1 km) at the scale of the map as drawn, and φ and λ are given in radians.

This is the integral of the secant function
Integral of the secant function
The integral of the secant function of trigonometry was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums...

, and is the inverse of the Gudermannian function:

The scale is proportional to the secant
Secant
Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the secant variety, in algebraic geometry...

of the latitude φ, getting arbitrarily large near the pole
Geographical pole
A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...

s, where φ = ±90°. Moreover, as seen from the formulas, the pole's y is plus or minus infinity.

The scaling factor for distances measured along lines of constant latitude φ is sec(φ) – this gives a scaling factor that is 1 at the equator (φ = 0) and approaches infinity near the poles (φ = ±90 degrees). The vertical distance on the map between two points with the same longitude is more complex and depends on their respective latitudes – it is:

### Derivation of the 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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

.) We seek a transform of longitude-latitude (λφ) to Cartesian (xy) that is "a cylinder tangent to the equator" (i.e. x = λ) and conformal, so that:

From x = λ we get

giving

Thus y is a function only of φ with . By reasoning explained in detail at Integral of the secant function
Integral of the secant function
The integral of the secant function of trigonometry was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums...

, this implies:

It is convenient to map φ = 0 to y = 0, so take C = 0.

## Uses

As on all map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

s, shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

. For example:
• Greenland
Greenland
Greenland is an autonomous country within the Kingdom of Denmark, located between the Arctic and Atlantic Oceans, east of the Canadian Arctic Archipelago. Though physiographically a part of the continent of North America, Greenland has been politically and culturally associated with Europe for...

takes as much area on the map as Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...

, when in fact Africa's area is approximately 14 times greater than Greenland.
Alaska is the largest state in the United States by area. It is situated in the northwest extremity of the North American continent, with Canada to the east, the Arctic Ocean to the north, and the Pacific Ocean to the west and south, with Russia further west across the Bering Strait...

takes as much area on the map as Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...

, when Brazil's area is actually more than 5 times that of Alaska.
• Finland
Finland
Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...

appears with a greater north-south extent than India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...

, although India's is greater.
• Antarctica appears as the biggest continent, although it is actually the fifth in terms of area.

Although the Mercator projection is still used commonly for navigation, due to its unique properties, cartographers agree that it is not suited to general reference world maps due to its distortion of land area. Mercator himself used the equal-area sinusoidal projection
Sinusoidal projection
The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570...

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, where distortion is minimal.

Arno Peters
Arno Peters
Arno Peters developed the Peters world map, based on the Gall–Peters projection.Born in Berlin, Germany, he began his career as a filmmaker who studied American techniques of filmmaking during the late 1930s, and helped to revolutionize film production in Germany at the time...

stirred controversy when he proposed what is now usually called the Gall–Peters projection as the alternative to the Mercator. The projection is a specific parameterization of the cylindrical equal-area projection
Cylindrical equal-area projection
In cartography, the cylindrical equal-area projection is a family of cylindrical, equal area map projections.-Cylindrical projections:The term "normal cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude ...

. A 1989 resolution by seven North American geographical groups decried the use of all rectangular-coordinate world maps, including the Mercator and Gall–Peters.

Many major online street mapping services (Bing Maps, Google Maps
Google Maps is a web mapping service application and technology provided by Google, free , that powers many map-based services, including the Google Maps website, Google Ride Finder, Google Transit, and maps embedded on third-party websites via the Google Maps API...

, MapQuest
MapQuest
MapQuest is an American free online web mapping service owned by AOL. The company was founded in 1967 as Cartographic Services, a division of R.R. Donnelley & Sons in Chicago, Illinois, United States. It moved to Lancaster, Pennsylvania in 1969. When it became an independent company in 1994, it was...

, Yahoo Maps, and others) use a variant of the Mercator projection for their map images. Despite its obvious scale variation at small scales, the projection is well-suited as an interactive world map that can be zoomed seamlessly to large-scale (local) maps, where there is relatively little distortion due to the projection's near-conformal
Conformal
Conformal may refer to:* Conformal map, in mathematics* Conformal geometry, in mathematics* Conformal map projection, in cartography* Conformal film on a surface* Conformal coating in electronics* Conformal hypergraph, in mathematics...

ity.

The major online street mapping services tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions. Since the Mercator coordinate x varies over 2π, the other coordinate is limited to −π ≤ y ≤ π. Because

the corresponding latitude extrema are φ = ±85.05113°. Latitude values outside this range are mapped using a different relationship that doesn't diverge at φ = ±90°.

• Cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

• Transverse Mercator projection
Transverse Mercator projection
The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM...

• Universal Transverse Mercator coordinate system
Universal Transverse Mercator coordinate system
The Universal Transverse Mercator geographic coordinate system uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. It is a horizontal position representation, i.e...

• Cassini projection
Cassini projection
The Cassini projection is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied...

• Dymaxion map
Dymaxion map
The Dymaxion map or Fuller map is a projection of a world map onto the surface of a polyhedron, which can be unfolded and flattened to two dimensions. The projection depicts the earth's continents as "one island," or nearly contiguous land masses. The arrangement heavily interrupts the map in order...

• Equirectangular projection
Equirectangular projection
The equirectangular projection is a very simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100...

• Gall–Peters projection with resolution regarding the use of rectangular world maps
• Gnomonic projection
Gnomonic projection
A gnomonic map projection displays all great circles as straight lines. Thus the shortest route between two locations in reality corresponds to that on the map. This is achieved by projecting, with respect to the center of the Earth , the Earth's surface onto a tangent plane. The least distortion...

• Jordan Transverse Mercator
Jordan Transverse Mercator
Jordan Transverse Mercator is a grid system created by the Royal Jordan Geographic Center . This system is based on 6° belts with a Central Meridian of 37° East and a Scale Factor at Origin = 0.9998. The JTM is based on the International Hayford Ellipsoid 1924. No transformation parameters...

• Lambert conformal conic projection
Lambert conformal conic projection
A Lambert conformal conic projection is a conic map projection, which is often used for aeronautical charts. In essence, the projection superimposes a cone over the sphere of the Earth, with two reference parallels secant to the globe and intersecting it. This minimizes distortion from projecting...

(used extensively in aviation
Aviation
Aviation is the design, development, production, operation, and use of aircraft, especially heavier-than-air aircraft. Aviation is derived from avis, the Latin word for bird.-History:...

)
• Map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

• Mollweide projection
Mollweide projection
The Mollweide projection is a pseudocylindrical map projection generally used for global maps of the world . Also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection...

• Nautical chart
Nautical chart
A nautical chart is a graphic representation of a maritime area and adjacent coastal regions. Depending on the scale of the chart, it may show depths of water and heights of land , natural features of the seabed, details of the coastline, navigational hazards, locations of natural and man-made aids...

• Robinson projection
Robinson projection
The Robinson projection is a map projection of a world map, which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image....

• Reversed map
Reversed map
thumb|right|The [[The Blue Marble|Blue Marble]] photograph in its original orientationA reversed map, also known as an Upside-Down map or South-Up map, is a map where south is up, north is down, east is left and west is right. Thus the Southern Hemisphere at the top of the map instead of the bottom...

• Tissot's indicatrix
Tissot's Indicatrix
Tissot’s indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize distortions due to map projection...

• Winkel Tripel projection
Winkel tripel projection
The Winkel Tripel projection , a modified azimuthal map projection, is one of three projections proposed by Oswald Winkel in 1921...