NationMaster - Encyclopedia: Mercator projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. 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. While the direction and shapes are accurate on a Mercator projection, it distorts the size. Image File history File links Mercator_1569. ... Image File history File links Mercator_1569. ... The Mercator projection shows courses of constant bearing as straight lines. ... The term Flemings (Dutch: ) denotes the majority population in Flanders (the northern half of Belgium). ... Gerardus Mercator (March 5, 1512 â€“ December 2, 1594) was a Flemish cartographer. ... Events January 11 - First recorded lottery in England. ... In navigation, a bearing is the clockwise angle between a reference direction (or a datum line) and the direction to an object. ... A course, in navigation, is the direction of travel. ... Example of pole-to-pole loxodrome In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...

1 Properties and historical details
2 Mathematics of the projection
3 Derivation of the projection
4 Controversy
5 See also
6 External links
7 References

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. 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. 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 planisphere consists of a circular star chart attached at the center of the starchart to an opaque overlay that has a clear roundish window (or cutout hole) that is free to rotate about the pivot point. ... The Mercator projection shows courses of constant bearing as straight lines. ... On the Earth, a circle of latitude is an imaginary east-west circle that connects all locations with a given latitude. ... On the earth, a meridian is a north-south line between the North Pole and the South Pole. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses IlhÃ©u das Rolas, in SÃ£o TomÃ© and PrÃncipe. ... In mathematics, a conformal map is a function which preserves angles. ...

A star map with Mercator projection, from the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song.

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. 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. 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. Image File history File links Metadata Size of this preview: 800 Ã— 576 pixelsFull resolution (984 Ã— 708 pixel, file size: 358 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Metadata Size of this preview: 800 Ã— 576 pixelsFull resolution (984 Ã— 708 pixel, file size: 358 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Star Maps were ancient semi-sentient devices created during the reign of the Rakatan Infinite Empire. ... Su Song è˜‡é Œ (1020 â€“ 1101), style Zirong åå®¹, was a Chinese engineer. ... In navigation, a bearing is the clockwise angle between a reference direction (or a datum line) and the direction to an object. ... Example of pole-to-pole loxodrome In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i. ... Line crossing all meridians at the same angle. ... This article is about the navigational instrument. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... Line crossing all meridians at the same angle. ... Parallel rulers are a navigational 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. This article is about longitude and latitude; see also UTM coordinate system Map of Earth showing lines of latitude (horizontally) and longitude (vertically); large version (pdf) The geographic (earth-mapping) coordinate system expresses every horizontal position on Earth by two of the three coordinates of a spherical coordinate system which...

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. (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... A marine chronometer is a timekeeper precise enough to be used as a portable time standard, used to determine longitude by means of celestial navigation. ... Magnetic declination. ...

Several authors are associated with the development of Mercator projection:

German Erhard Etzlaub (c. 1460-1532), who had engraved miniature "compass maps" (about 10x8 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". This has since proven to be an error, tracing back to doubtable research in 1917.

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.

Erhard Etzlaub (born c. ... Pedro Nunes (latin, Petrus Nonius), (1502, AlcÃ¡cer do Sal â€“ August 11, 1578, Coimbra) was a Portuguese mathematician, maybe born from a New Christian (of Jewish origin) family. ... Edward Gordon Dundas Wright (born October 3, 1884 - died June 5, 1947) was a English amateur football (soccer) player who competed in the 1912 Summer Olympics. ... Thomas Harriot (ca. ...

Mathematics of the projection

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): Gudermannian function with asymptotes y = +- pi/2 marked in File links The following pages link to this file: Mercator projection Gudermannian function Categories: GFDL images ... Gudermannian function with asymptotes y = +- pi/2 marked in File links The following pages link to this file: Mercator projection Gudermannian function Categories: GFDL images ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... This article is about the geographical term. ... Longitude is the east-west geographic coordinate measurement most commonly utilized in cartography and global navigation. ...

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

This is the inverse of the Gudermannian function: The Gudermannian function, named after Christoph Gudermann (1798 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. ...

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

The scale is proportional to the secant of the latitude φ, getting arbitrarily large near the poles, where φ = plus or minus 90°. 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.

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, so that: Image File history File links Usgs_map_mercator. ... This article is about the geographical term. ...

$frac{partial x}{partial lambda} = cos(phi) frac{partial y}{partial phi}$

$frac{partial y}{partial lambda} = -cos(phi) frac{partial x}{partial phi}$

From x = λ we get

$frac{partial x}{partial lambda} = 1$

$frac{partial x}{partial phi} = 0$

giving

$1 = cos(phi) frac{partial y}{partial phi}$

$0 = frac{partial y}{partial lambda}$

Thus y is a function only of φ with $y' = secφ$ from which a table of integrals gives It has been suggested that this article or section be merged with List of integrals. ...

$y = ln(|sec(phi) + tan(phi)|) + C,$ .

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

Controversy

Tissot's Indicatrix

The above reprojected as sinusoidal

The world in Mercator Projection

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. The Mercator projection exaggerates the size of areas far from the equator. For example: Download high resolution version (777x720, 46 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (777x720, 46 KB) Wikipedia does not have an article with this exact name. ... Image File history File links Sinusiodal_earth_circles. ... Image File history File links Sinusiodal_earth_circles. ... Image File history File links Download high-resolution version (1500x1043, 418 KB) Description: World in Mercator projection Source: Kober-KÃ¼mmerly+Frey Media AG Date: 21. ... Image File history File links Download high-resolution version (1500x1043, 418 KB) Description: World in Mercator projection Source: Kober-KÃ¼mmerly+Frey Media AG Date: 21. ... The Mercator projection shows courses of constant bearing as straight lines. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses IlhÃ©u das Rolas, in SÃ£o TomÃ© and PrÃncipe. ...

Greenland is presented as being roughly as large as Africa, when in fact Africa's area is approximately 14 times that of Greenland.

Alaska is presented as being similar or even slightly larger in size than Brazil, when Brazil's area is actually more than 5 times that of Alaska.

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 or 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. A world map showing the continent of Africa Africa is the worlds second-largest and second most-populous continent, after Asia. ... Official language(s) None[1] Spoken language(s) English 85. ... Sinusoidal projection A sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection. ... For other uses, see Atlas (disambiguation). ... The Mercator projection shows courses of constant bearing as straight lines. ... A map projection is any of many methods used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a plane. ...

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. A 1989 resolution by seven North American geographical groups decried the use of all rectangular-coordinate world maps, including the Mercator and Gall-Peters.^[1] Arno Peters (May 22, 1916 - December 2, 2002) was the developer of the Peters Projection map. ... Gall-Peters projection of the Earth The Gall-Peters projection is one specialization of a configurable equal-area map projection known as the equal-area cylindric or cylindrical equal-area projection. ... Year 1989 (MCMLXXXIX) was a common year starting on Sunday (link displays 1989 Gregorian calendar). ...

Google Maps currently uses a Mercator projection for its map images. 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.) 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 including the Google Maps website, Google Ride Finder and embedded maps on third-party websites via the Google Maps API. It offers street maps... Equirectangular projection of the Globe Equirectangular projection of a composite satellite image (NASA) The plate carrÃ©e projection or geographic projection or equirectangular projection, is a very simple map projection that has been in use since the earliest days of spherical cartography. ... 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 including the Google Maps website, Google Ride Finder and embedded maps on third-party websites via the Google Maps API. It offers street maps...

The Google Maps maximum latitude occurs at +/- 85.0511287798066 degrees when the Mercator y value = PI. 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 including the Google Maps website, Google Ride Finder and embedded maps on third-party websites via the Google Maps API. It offers street maps...

External links

Ad maiorem Gerardi Mercatoris gloriam - contains high resolution images of the 1569 world map by Mercator.
Panoramic Image Projections - Three-dimensional illustrations of the Mercator projection for use in panoramic photography, in comparison to other projections.
Table of examples and properties of all common projections, from radicalcartography.net.
An interactive Java Applet to study the metric deformations of the Mercator Projection.

References

^ American Cartographer. 1989. 16(3): 222-223.

Snyder, John P. (1987). Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C. This paper can be downloaded from USGS pages.

Monmonier, Mark (2004). Rhumb Lines and Map Wars. Chicago: The University of Chicago Press.

Category: Cartographic projections

Results from FactBites:

OMC: supported projections (628 words)
	Projection number five (Azimuthal Equidistant Projection) returns a circular plot of the entire world with the area of interest as the center, distances measured from there are true.
	The Mercator projection has been used extensively for world maps im which the distortion towards the polar regions grows rather large, thus incorrectly giving the impression that, for example, Greenland is larger than South America.
	It is a useful projection for a global view of locations at various or identical distance from a given point (the map center).

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Encyclopedia > Mercator projection

Contents