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Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569)
Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569)

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. A map projection is any method of representing the Surface of a sphere or other shape on a plane. The terms Fleming and Flemings ( Vlaming and Vlamingen in Dutch) denote respectively a person and people and the Flemings or A separate article is about the mathematician Nicholas Mercator. It became the standard map projection for nautical purposes because of its ability to represent lines of constant true bearing or true course, known as rhumb lines, as straight line segments. In Navigation, a bearing is the direction one object is from another object In Navigation, a course is the intended path of a vehicle over the surface of the Earth See also Great circle Small circle In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points While the direction and shapes are accurate on a Mercator projection, it distorts size, in an increasing degree away from the equator.

Contents

Properties and historical details

Mercator's 1569 edition was a large planisphere measuring 202 by 124 cm, printed in eighteen separate sheets. As in all cylindrical projections, parallels and meridians are straight and perpendicular to each other. A map projection is any method of representing the Surface of a sphere or other shape on a plane. A circle of latitude, on the Earth, is an imaginary East - West circle connecting all locations (not taking into account elevation that share a given This article is about the geographical concept For other uses of the word see Meridian. In accomplishing this, the unavoidable east-west stretching of the map, which increases as distance away from the equator 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. The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the 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 A Mercator map can never fully show the polar areas, since linear scale 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. In particular, areas closer to the poles are more affected, transmitting an image of the geometry of the planet which is more distorted the closer to the poles. At latitudes higher than 70° north or south, the Mercator projection is practically unusable.

A star map with cylindrical projection similar to Mercator projection, from the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song.
A star map with cylindrical projection similar to Mercator projection, from the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song. A star chart is a map of the Night sky. Astronomers divide these into grids to easily use them China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National Su Song ( style name: Zirong 子容 (1020&ndash1101 AD was a renowned Chinese statesman, astronomer, cartographer, [1][2]

All lines of constant bearing (rhumb lines or loxodromes — those making constant angles with the meridians), are represented by straight segments on a Mercator map. In Navigation, a bearing is the direction one object is from another object See also Great circle Small circle See also Great circle Small circle This is precisely the type of route usually employed by ships at sea, where compasses are used to indicate geographical directions and to steer the ships. A compass, magnetic compass or mariner's compass is a navigational instrument for determining direction relative to the earth's Magnetic poles It consists The two properties, conformality and straight rhumb lines, make 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. 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 See also Great circle Small circle Parallel rulers are a drafting instrument used by navigators to draw parallel lines on charts

The name and explanations given by Mercator to his world map (Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate: "new and augmented description of Earth corrected for the use of navigation") 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, and then adjusting the spacing between parallels so that those lines became straight, making the same angle with the meridians as in the globe. A geographic coordinate system enables every location on the Earth to be specified in three coordinates using mainly a spherical coordinate system.

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 was invented and the spatial distribution of magnetic declination was known, could the Mercator projection be fully adopted by navigators. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system A marine chronometer is a timekeeper precise enough to be used as a portable Time standard; it can therefore be used to determine Longitude by means of Celestial The magnetic declination (also known as grid magnetic angle in military circles at any point on the Earth is the angle between the local magnetic field -- the direction

Several authors are associated with the development of Mercator projection:

Mathematics of the projection

Relation between vertical position on the map (horizontal in the graph) and latitude (vertical in the graph).
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):

This is the inverse of the Gudermannian function:

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

This is the Gudermannian function:

\begin{align} \varphi & = 2\tan^{-1}(e^y) - \frac{\pi}{2} \\ & = \tan^{-1}(\sinh(y)) \\ \lambda & = x + \lambda_0. \end{align}

The scale is proportional to the secant of the latitude φ, getting arbitrarily large near the poles, where φ = ±90°. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement The Gudermannian function, named after Christoph Gudermann (1798 &ndash 1852 relates the circular and Hyperbolic trigonometric functions without using The Gudermannian function, named after Christoph Gudermann (1798 &ndash 1852 relates the circular and Hyperbolic trigonometric functions without using Secant is a term in mathematics It comes from the Latin secare (to cut A geographical pole, or geographic pole, is either of two fixed points on the surface of a spinning body or Planet, at 90 degrees from the Equator, based Moreover, as seen from the formulas, the pole's y is plus or minus infinity.

Derivation of the projection

The Mercator projection is a cylindrical projection.
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. An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane For more precision, interpose conformal latitude. Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the ) 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, so that:

\frac{\partial x}{\partial \lambda} = \cos(\varphi) \frac{\partial y}{\partial \varphi}
\frac{\partial y}{\partial \lambda} = -\cos(\varphi) \frac{\partial x}{\partial \varphi}

From x = λ we get

\frac{\partial x}{\partial \lambda} = 1
\frac{\partial x}{\partial \varphi} = 0

giving

1 = \cos(\varphi) \frac{\partial y}{\partial \varphi}
0 = \frac{\partial y}{\partial \lambda}

Thus y is a function only of φ with y'=\sec\varphi from which a table of integrals gives

y = \ln(|\sec(\varphi) + \tan(\varphi)|) + C.\,

It is convenient to map φ = 0 to y = 0, so take C = 0. See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions

Uses

Tissot's Indicatrix
Tissot's Indicatrix
The above reprojected as sinusoidal
The above reprojected as sinusoidal

Like all map projections that attempt 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. A map projection is any method of representing the Surface of a sphere or other shape on a plane. The Mercator projection exaggerates the size of areas far from the equator. The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the For example:

Although the Mercator projection is still in common use for navigation, due to its unique properties, cartographers agree that it is not suited to representing the entire world in publications or wall maps due to its distortion of land area. Finland, officially the Republic of Finland ( is a Nordic country situated in the Fennoscandian region of northern Europe. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country Mercator himself used the equal-area sinusoidal projection to show relative areas. The sinusoidal projection is a pseudocylindrical equal-area Map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection. 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. An atlas is a collection of Maps typically of Earth or a region of Earth but there are atlases of the other planets (and their satellites in the solar system A map projection is any method of representing the Surface of a sphere or other shape on a plane. A map projection is any method of representing the Surface of a sphere or other shape on a plane. The Mercator projection is still commonly used for areas near the equator, however, where distortion is minimal.

Arno Peters stirred controversy when he proposed what is known as the Gall-Peters projection, a slight modification of the Lambert Cylindrical Equal-Area projection, as the alternative to the Mercator. Arno Peters ( May 22, 1916 - December 2, 2002) developed the Peters world map based on the Gall-Peters projection. The Gall-Peters projection is one specialization of a configurable equal-area Map projection known as the equal-area cylindric or cylindric equal-area A 1989 resolution by seven North American geographical groups decried the use of all rectangular-coordinate world maps, including the Mercator and Gall-Peters. Year 1989 ( MCMLXXXIX) was a Common year starting on Sunday (link displays 1989 Gregorian calendar) [3]

Google Maps currently uses a Mercator projection for its map images. Google Maps (for a time named Google Local) is a free Web mapping service application and technology provided by Google that powers many map-based services Despite its relative scale distortions, the Mercator is well-suited as an interactive world map that can be panned and zoomed seamlessly to local maps. (Google Satellite Maps, on the other hand, used a plate carrée projection until 2005-07-22. The equirectangular projection (also called the equidistant cylindrical projection, geographic projection, plate carré or carte parallelogrammatique Google Maps (for a time named Google Local) is a free Web mapping service application and technology provided by Google that powers many map-based services )

The Google Maps maximum latitude φ occurs at ±85. Google Maps (for a time named Google Local) is a free Web mapping service application and technology provided by Google that powers many map-based services 05113 degrees when the Mercator y value = π. Or more precisely:

\frac{1}{2} \ln\bigg(\frac{1+\sin(\varphi)}{1-\sin(\varphi)}\bigg) = \pm\pi \Rightarrow \varphi = \pm\arcsin\bigg(\frac{\mathrm{e}^{2 \pi}-1}{\mathrm{e}^{2 \pi}+1}\bigg)

See also

External links

Notes

  1. ^ Needham, Volume 3, 227.
  2. ^ Needham, Volume 4, Part 3, 569.
  3. ^ American Cartographer. 1989. 16(3): 222-223.

References

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