1.
Asia
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Asia covers an area of 44,579,000 square kilometres, about 30% of Earths total land area and 8. 7% of the Earths total surface area. The continent, which has long been home to the majority of the population, was the site of many of the first civilizations. Asia is notable for not only its large size and population. In general terms, Asia is bounded on the east by the Pacific Ocean, on the south by the Indian Ocean, the western boundary with Europe is a historical and cultural construct, as there is no clear physical and geographical separation between them. The most commonly accepted boundaries place Asia to the east of the Suez Canal, the Ural River, and the Ural Mountains, and south of the Caucasus Mountains, China and India alternated in being the largest economies in the world from 1 to 1800 A. D. The accidental discovery of America by Columbus in search for India demonstrates this deep fascination, the Silk Road became the main East-West trading route in the Asian hitherland while the Straits of Malacca stood as a major sea route. Asia has exhibited economic dynamism as well as robust population growth during the 20th century, given its size and diversity, the concept of Asia—a name dating back to classical antiquity—may actually have more to do with human geography than physical geography. Asia varies greatly across and within its regions with regard to ethnic groups, cultures, environments, economics, historical ties, the boundary between Asia and Africa is the Red Sea, the Gulf of Suez, and the Suez Canal. This makes Egypt a transcontinental country, with the Sinai peninsula in Asia, the border between Asia and Europe was historically defined by European academics. In Sweden, five years after Peters death, in 1730 Philip Johan von Strahlenberg published a new atlas proposing the Urals as the border of Asia, the Russians were enthusiastic about the concept, which allowed them to keep their European identity in geography. Tatishchev announced that he had proposed the idea to von Strahlenberg, the latter had suggested the Emba River as the lower boundary. Over the next century various proposals were made until the Ural River prevailed in the mid-19th century, the border had been moved perforce from the Black Sea to the Caspian Sea into which the Ural River projects. The border between the Black Sea and the Caspian is usually placed along the crest of the Caucasus Mountains, the border between Asia and the loosely defined region of Oceania is usually placed somewhere in the Malay Archipelago. The terms Southeast Asia and Oceania, devised in the 19th century, have had several different geographic meanings since their inception. The chief factor in determining which islands of the Malay Archipelago are Asian has been the location of the possessions of the various empires there. Lewis and Wigen assert, The narrowing of Southeast Asia to its present boundaries was thus a gradual process, Asia is larger and more culturally diverse than Europe. It does not exactly correspond to the borders of its various types of constituents. From the time of Herodotus a minority of geographers have rejected the three-continent system on the grounds there is no or is no substantial physical separation between them

2.
Tissot's indicatrix
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It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map. A single indicatrix describes the distortion at a single point, because distortion varies across a map, generally Tissots indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians, there is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion. Tissots theory was developed in the context of cartographic analysis, generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid. The quotient is called the scale factor, unless the projection is conformal at the point being considered, the scale factor varies by direction around the point. A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection and this is expressed by an ellipse of distortion which is not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection and this is expressed by ellipses of distortion whose areas vary across the map. In conformal maps, where each point preserves angles projected from the geometric model, in equal-area projections, where area proportions between objects are conserved, the Tissots indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across the map, in the adjacent image, ABCD is a circle with unit area defined in a spherical or ellipsoidal model of the Earth, and A′B′C′D′ is the Tissots indicatrix that results from its projection on the plane. Segment OA is transformed in OA′, and segment OB is transformed in OB′, linear scale is not conserved along these two directions, since OA′ is not equal to OA and OB′ is not equal to OB. Angle MOA, in the unit circle, is transformed in angle M′OA′ in the distortion ellipse. Because M′OA′ ≠ MOA, we know there is an angular distortion. The area of circle ABCD is, by definition, equal to 1, because the area of ellipse A′B′ is less than 1, a distortion of area has occurred. In dealing with a Tissot indicatrix, different notions of radius come into play, the first is the infinitesimal radius of the original circle. The resulting ellipse of distortion will also have infinitesimal radius, but by the mathematics of differentials, so, for example, if the resulting ellipse of distortion is the same size of infinitesimal as on the sphere, then its radius is considered to be 1. Lastly, the size that the indicatrix gets drawn for human inspection on the map is arbitrary, when a network of indicatrices is drawn on a map, they are all scaled by the same arbitrary amount so that their sizes are proportionally correct. Like M in the diagram, the axes from O along the parallel and along the meridian may undergo a change of length and it is common in the literature to represent scale along the meridian as h and scale along the parallel as k, for a given point

3.
Map projection
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A map projection is a systematic transformation of the latitudes and longitudes of locations on the surface of a sphere or an ellipsoid into locations on a plane. Map projections are necessary for creating maps, all map projections distort the surface in some fashion. There is no limit to the number of map projections. More generally, the surfaces of bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Even more generally, projections are the subject of several mathematical fields, including differential geometry. However, map projection refers specifically to a cartographic projection and these useful traits of maps motivate the development of map projections. However, Carl Friedrich Gausss Theorema Egregium proved that a spheres surface cannot be represented on a plane without distortion, the same applies to other reference surfaces used as models for the Earth. Since any map projection is a representation of one of surfaces on a plane. Every distinct map projection distorts in a distinct way, the study of map projections is the characterization of these distortions. Projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, rather, any mathematical function transforming coordinates from the curved surface to the plane is a projection. Few projections in actual use are perspective, for simplicity, most of this article assumes that the surface to be mapped is that of a sphere. In reality, the Earth and other celestial bodies are generally better modeled as oblate spheroids. These other surfaces can be mapped as well, therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane. Many properties can be measured on the Earths surface independent of its geography, some of these properties are, Area Shape Direction Bearing Distance Scale Map projections can be constructed to preserve at least one of these properties, though only in a limited way for most. Each projection preserves or compromises or approximates basic metric properties in different ways, the purpose of the map determines which projection should form the base for the map. Because many purposes exist for maps, a diversity of projections have been created to suit those purposes, another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information, their collection depends on the datum of the Earth. Different datums assign slightly different coordinates to the location, so in large scale maps, such as those from national mapping systems

4.
Azimuthal equidistant projection
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The azimuthal equidistant projection is an azimuthal map projection. A useful application for this type of projection is a projection which shows all meridians as straight. The flag of the United Nations contains an example of an azimuthal equidistant projection. It is useful for showing airline distances from center point of projection and for seismic, while it may have been used by ancient Egyptians for star maps in some holy books, the earliest text describing the azimuthal equidistant projection is an 11th-century work by al-Biruni. The projection appears in many Renaissance maps, and Gerardus Mercator used it for an inset of the polar regions in sheet 13. In France and Russia this projection is named Postel projection after Guillaume Postel, many modern star chart planispheres use the polar azimuthal equidistant projection. A point on the globe is chosen to be special in the sense that mapped distances and that point, will project to the center of a circular projection, with φ referring to latitude and λ referring to longitude. All points along a given azimuth will project along a line from the center. The distance from the point to another projected point is given as ρ. An operator can point the antenna, usually by an electric rotator, simply locating the target in the map, the map should be centered as nearly as possible to the actual antenna location. GeographicLib provides a class for performing azimuthal equidistant projections centered at any point on the ellipsoid, animated US National Weather Service Wind Data for Azimuthal equidistant projection

5.
Locus (mathematics)
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In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex

6.
National Geographic Society
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The National Geographic Society, headquartered in Washington, D. C. United States, is one of the largest nonprofit scientific and educational institutions in the world and its interests include geography, archaeology and natural science, the promotion of environmental and historical conservation, and the study of world culture and history. In partnership with 21st Century Fox, the Society operates the magazine, TV channels, a website that features extra content and worldwide events, the National Geographic Society was founded in 1888 to increase and diffuse geographic knowledge. The Society believes in the power of science, exploration and storytelling to change the world, National Geographic is governed by a board of trustees, whose 21 members include distinguished educators, business executives, former government officials and conservationists. The organization sponsors and funds research and exploration. National Geographic maintains a museum for the public in its Washington and its Education Foundation gives grants to education organizations and individuals to improve geography education. Its Committee for Research and Exploration has awarded more than 11,000 grants for scientific research, National Geographic has retail stores in Washington, D. C. The locations outside of the United States are operated by Worldwide Retail Store S. L and it also publishes other magazines, books, school products, maps, and Web and film products in numerous languages and countries. National Geographics various media properties reach more than 280 million people monthly, the National Geographic Society began as a club for an elite group of academics and wealthy patrons interested in travel. After preparing a constitution and a plan of organization, the National Geographic Society was incorporated two weeks later on January 27, Gardiner Greene Hubbard became its first president and his son-in-law, Alexander Graham Bell, succeeded him in 1897. Bell and Gilbert Hovey Grosvenor devised the successful marketing notion of Society membership, the current National Geographic Society president and CEO is Gary E. Knell. The chairman of the board of trustees is John Fahey, the editor-in-chief of National Geographic magazine is Susan Goldberg. Gilbert Melville Grosvenor, a chairman of the Society board of trustees received the Presidential Medal of Freedom in 2005 for his leadership in geography education. In 2004, the National Geographic Society headquarters in Washington, D. C. was one of the first buildings to receive a Green certification from Global Green USA. The National Geographic received the prestigious Prince of Asturias Award for Communication and Humanities in October 2006 in Oviedo, in 2013 the society was investigated for possible violation of the Foreign Corrupt Practices Act relating to their close association with an Egyptian government official responsible for antiquities. This new, for-profit corporation, will own National Geographic and other magazines, as reported by The Guardian, a spokesman for National Geographic in a November 2,2015 e-mail statement, briefly discussed the rationale for the staff reductions as part of the. Process of reorganizing in order to move forward following the closing the National Geographic Partners deal. Additional specifics were provided to Photo District News by M. J. Jacobsen, National Geographic’s SVP of communications, similar to the contents of a formal announcement by the two companies

7.
Chamberlin trimetric projection
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The Chamberlin trimetric projection is a map projection where three points are fixed on the globe and the points on the sphere are mapped onto a plane by triangulation. It was developed in 1946 by Wellman Chamberlin for the National Geographic Society, Chamberlin was chief cartographer for the Society from 1964 to 1971. The projections principal feature is that it compromises between distortions of area, direction, and distance, a Chamberlin trimetric map therefore gives an excellent overall sense of the region being mapped. Many National Geographic Society maps of single continents use this projection, as originally implemented, the projection algorithm begins with the selection of three points near the outer boundary of the area to be mapped. From these three points, the true distances to a point on the mapping area are calculated. The distances from each of the three points are then drawn on the plane by compass circles. Unlike triangulation on a plane where three such compass circles will intersect at a point, the compass circles from a sphere do not intersect precisely at a point. A small triangle is generated from the intersections, and the center of triangle is calculated as the mapped point. Based on the principles of the projection, precise, but lengthy, the Chamberlin trimetric projection is neither conformal nor equal-area. Rather, the projection was conceived to minimize distortion of distances everywhere with the side-effect of balancing between areal equivalence and conformality and this projection is not appropriate for mapping the entire sphere because the outer boundary would loop and overlap itself in most configurations. Two-point equidistant projection The Chamberlin Trimetric Projection - Implementations of the projection using Matlab scripts, the Chamberlin Trimetric Projection - Notes on the projection from a cartography class at Colorado State University. National Geographic Map Collection - Many examples of National Geographic Society maps employing the Chamberlin Trimetric Projection can be seen here

8.
Waldo R. Tobler
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Waldo Tobler is an American-Swiss geographer and cartographer. Toblers idea that Everything is related to everything else, but near things are related to each other is referred to as the first law of geography. He has proposed a law as well, The phenomenon external to an area of interest affects what goes on inside. Tobler is a Professor Emeritus at the University of California, Santa Barbara Department of Geography, in 1961, Tobler received his Ph. D. in the Department of Geography at the University of Washington at Seattle. At Washington, he participated in geographys William Garrison-led quantitative revolution of the late 1950s, after graduating, Tobler spent several years at the University of Michigan. Until his retirement he held the positions of Professor of Geography and Professor of Statistics at the University of California, the University of Zurich, Switzerland, awarded him an honorary doctorate in 1988. Tobler was one of the investigators and a senior scientist in the National Science Foundation-sponsored National Center for Geographic Information. He has used computers in geographic research since the 1960s, with emphasis on mathematical modeling and he also invented a method for smooth two-dimensional mass-preserving areal data redistribution. In 1989, the American Geographical Society awarded Tobler with the Osborn Maitland Miller Medal, Tobler has been involved recently in building a global, latitude-longitude oriented, demographic information base with resolution two orders of magnitude better than was previously available. He also had been examining the development of finite element. In July 1999 he presented a speech, The World is Shriveling as it Shrinks, at the ESRI International User Conference. Taylor and Francis of London recently published a map projection book, co-authored with Q. Yang of China, current interests relate to ideas in computational geography including the analysis of geographical vector fields and the development of migration and of global trade models. Tobler has also been concerned with the representing flow, in 2003, Tobler released a freeware, Microsoft Windows-based version of his flow representation software Flow Mapper. In 2005, an ESRI ArcGIS version of the software, inspired by Tobler, was developed by Alan Glennon, Phi Beta Kappa, Sigma Xi, Phi Kappa Phi Tobler served on the National Research Council the Board on Earth Sciences. Until his retirement, he was a member of the Royal Geographical Society of Great Britain. Toblers hiking function First law of geography General W. Tobler,2002, “Ma Vie, Growing Up in America and Europe”, in Geographical Voices, W. Pitts and P. Gould, University of Syracuse Press, Syracuse, pages 292–322. En français, “Ma Vie, Grandir en Amérique et en Europe”, dans Mémoires de Géographes, P. Gould et A. Bailly, specific Waldo R. Tobler at UCSB CSISS/Flow Mapper Software UCSB Geography

9.
Root mean square
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In statistics and its applications, the root mean square is defined as the square root of mean square. The RMS is also known as the mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the values during a cycle. For a cyclically alternating electric current, RMS is equal to the value of the current that would produce the same average power dissipation in a resistive load. In Estimation theory the mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data. The RMS value of a set of values is the root of the arithmetic mean of the squares of the values. In Physics, the RMS current is the value of the current that dissipates power in a resistor. In the case of a set of n values, the RMS x r m s =1 n, the RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a function or signal can be approximated by taking the RMS of a sequence of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, in the case of the RMS statistic of a random process, the expected value is used instead of the mean. If the waveform is a sine wave, the relationships between amplitudes and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for a waveform which may or may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is, Peak-to-peak =22 × R M S ≈2.8 × R M S, for other waveforms the relationships are not the same as they are for sine waves. Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal. R M S Total = R M S12 + R M S22 + ⋯ + R M S n 2 A special case of this, another special case, useful in statistics, is given in #Relationship to other statistics. Electrical engineers often need to know the power, P, dissipated by an electrical resistance and it is easy to do the calculation when there is a constant current, I, through the resistance. Average power can also be using the same method that in the case of a time-varying voltage, V, with RMS value VRMS. This equation can be used for any waveform, such as a sinusoidal or sawtooth waveform

10.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

11.
History of cartography
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Cartography, or mapmaking, has been an integral part of the human history for thousands of years. Maps began as drawings but can also adopt three-dimensional shapes. The term cartography is modern, loaned into English from French cartographie in the 1840s, the earliest known maps are of the stars, not the earth. Dots dating to 16,500 BC found on the walls of the Lascaux caves map out part of the night sky, the Cuevas de El Castillo in Spain contain a dot map of the Corona Borealis constellation dating from 12,000 BC. Cave painting and rock carvings used simple visual elements that may have aided in recognizing landscape features, another ancient picture that resembles a map was created in the late 7th millennium BC in Çatalhöyük, Anatolia, modern Turkey. This wall painting may represent a plan of this Neolithic village, however, Maps in Ancient Babylonia were made by using accurate surveying techniques. For example, a 7.6 ×6.8 cm clay tablet found in 1930 at Ga-Sur, near contemporary Kirkuk, cuneiform inscriptions label the features on the map, including a plot of land described as 354 iku that was owned by a person called Azala. Most scholars date the tablet to the 25th to 24th century BC, Leo Bagrow dissents with a date of 7000 BC. Hills are shown by overlapping semicircles, rivers by lines, the map also is marked to show the cardinal directions. An engraved map from the Kassite period of Babylonian history shows walls, in contrast, the Babylonian World Map, the earliest surviving map of the world, is a symbolic, not a literal representation. It deliberately omits peoples such as the Persians and Egyptians, who were known to the Babylonians. The area shown is depicted as a circular shape surrounded by water, examples of maps from ancient Egypt are quite rare. Its originality can be seen in the inscriptions, its precise orientation. In reviewing the literature of early geography and early conceptions of the earth, all lead to Homer. Regardless of the doubts about Homers existence, one thing is certain, Homers knowledge of the Earth was very limited. He and his Greek contemporaries knew very little of the Earth beyond Egypt as far south as the Libyan desert, the south-west coast of Asia Minor, furthermore, the coast of the Black Sea was only known through myths and legends that circulated during his time. In his poems there is no mention of Europe and Asia as geographical concepts and that is why the big part of Homers world that is portrayed on this interpretive map represents lands that border on the Aegean Sea. Additional statements about ancient geography may be found in Hesiods poems, through the lyrics of Works and Days and Theogony he shows to his contemporaries some definite geographical knowledge. He introduces the names of rivers as Nile, Ister, the shores of the Bosporus, and the Euxine, the coast of Gaul, the island of Sicily

12.
Mercator projection
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The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are relative to land masses near the equator, Mercators 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, being a conformal projection, angles are preserved around all locations. At latitudes greater than 70° north or south the Mercator projection is practically unusable, a Mercator map can therefore never fully show the polar areas. All lines of constant bearing are represented by segments on a Mercator map. The name and explanations given by Mercator to his world map show that it was conceived for the use of marine navigation. The development of the Mercator projection represented a breakthrough in the nautical cartography of the 16th century. However, it was ahead of its time, since the old navigational. If these sheets were brought to the scale and assembled an approximation of the Mercator projection would be obtained. English mathematician Edward Wright, who published accurate tables for its construction, english mathematicians Thomas Harriot and Henry Bond who, independently, associated the Mercator projection with its modern logarithmic formula, later deduced by calculus. As on all map projections, shapes or sizes are distortions of the layout of the Earths surface. The Mercator projection exaggerates areas far from the equator, for example, Greenland appears larger than Africa, when in reality Africas area is 14 times greater and Greenlands is comparable to Algerias alone. Africa also appears to be roughly the size as Europe. Alaska takes as much area on the map as Brazil, when Brazils area is nearly five times that of Alaska, finland appears with a greater north-south extent than India, although Indias is greater. Antarctica appears as the biggest continent, although it is actually the fifth in area, the Mercator projection is still used commonly for navigation. On the other hand, because of land area distortions. Therefore, Mercator himself used the equal-area sinusoidal projection to show relative areas, 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 now called the Gall–Peters projection as the alternative to the Mercator

13.
Transverse Mercator projection
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The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is used in national and international mapping systems around the world. When paired with a geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent. The transverse Mercator projection is the aspect of the standard Mercator projection. For the transverse Mercator, the axis of the lies in the equatorial plane. Both projections may be modified to secant forms, which means the scale has been reduced so that the cylinder slices through the model globe, Both exist in spherical and ellipsoidal versions. Both projections are conformal, so that the point scale is independent of direction and local shapes are well preserved, since the central meridian of the transverse Mercator can be chosen at will, it may be used to construct highly accurate maps anywhere on the globe. The secant, ellipsoidal form of the transverse Mercator is the most widely applied of all projections for accurate large scale maps. In constructing a map on any projection, a sphere is normally chosen to model the Earth when the extent of the region exceeds a few hundred kilometers in length in both dimensions. For maps of regions, an ellipsoidal model must be chosen if greater accuracy is required. The spherical form of the transverse Mercator projection was one of the seven new projections presented, in 1772, Lambert did not name his projections, the name transverse Mercator dates from the second half of the nineteenth century. The principal properties of the projection are here presented in comparison with the properties of the normal projection. The ellipsoidal form of the transverse Mercator projection was developed by Carl Friedrich Gauss in 1825, the projection is known by several names, Gauss Conformal or Gauss-Krüger in Europe, the transverse Mercator in the US, or Gauss-Krüger transverse Mercator generally. The projection is conformal with a constant scale on the central meridian, the Gauss-Krüger projection is now the most widely used projection in accurate large scale mapping. The projection, as developed by Gauss and Krüger, was expressed in terms of low order power series which were assumed to diverge in the east-west direction and this was proved to be untrue by British cartographer E. H. Thompson, whose unpublished exact version of the projection, reported by L. P. Lee in 1976, near the central meridian the projection has low distortion and the shapes of Africa, western Europe, Britain, Greenland, Antarctica compare favourably with a globe. The central regions of the projections on sphere and ellipsoid are indistinguishable on the small scale projections shown here. The meridians at 90° east and west of the central meridian project to horizontal lines through the poles

14.
Cylindrical equal-area projection
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In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map 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 are mapped to horizontal lines. The mapping of meridians to vertical lines can be visualized by imagining a cylinder wrapped around the Earth and then projecting onto the cylinder, by the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections and this divides north-south distances by a factor equal to the secant of the latitude, preserving area but heavily distorting shapes. Any particular cylindrical equal-area map has a pair of identical latitudes of opposite sign at which the east–west scale matches the north–south scale, the stretch factor S is what distinguishes the variations of cylindric equal-area projection. The various specializations of the cylindric equal-area projection differ only in the ratio of the vertical to horizontal axis and this ratio determines the standard parallel of the projection, which is the parallel at which there is no distortion and along which distances match the stated scale. There are always two standard parallels on the cylindric equal-area projection, each at the distance north and south of the equator. The standard parallels of the Gall–Peters are 45° N and 45° S, several other specializations of the equal-area cylindric have been described, promoted, or otherwise named. The invention of the Lambert cylindrical equal-area projection is attributed to the Alsatian mathematician Johann Heinrich Lambert in 1772, variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios. The Tobler hyperelliptical projection, first described by Tobler in 1973, is a generalization of the cylindrical equal-area family

15.
Behrmann projection
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The Behrmann projection is a cylindrical map projection described by Walter Behrmann in 1910. It is a member of the cylindrical equal-area projection family, members of the family differ by their standard parallels, which are parallels along which the projection has no distortion. In the case of the Behrmann projection, the parallels are 30°N. While equal-area, distortion of shape increases in the Behrmann projection according to distance from the standard parallels

16.
Lambert cylindrical equal-area projection
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In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points, lamberts projection is the basis for the cylindrical equal-area projection family. Lambert chose the equator as the parallel of no distortion and these variations, particularly the Gall–Peters projection, are more commonly encountered in maps than Lambert’s original projection due to their lower distortion overall

17.
Cassini projection
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The Cassini projection is a map projection described by César-François Cassini de Thury in 1745. Considering the earth as a sphere, the projection is composed of operations, x = arcsin y = arctan where λ is the longitude from the central meridian. When programming these equations, the tangent function used is actually the atan2 function, with the first argument sin φ). Nevertheless, the use of the Cassini projection has largely superseded by the Transverse Mercator projection. Areas along the meridian, and at right angles to it, are not distorted. Elsewhere, the distortion is largely in a direction. As such, the greater the extent of the area. Due to this, the Cassini projection works best on long, narrow areas, Cassini is known as a spherical projection, but can be generalised as an elliptical form. net Ordnance Survey GeoFacts on the Cassini Projection Cassini dans proj4

18.
Central cylindrical projection
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The Central cylindrical projection is a perspective cylindrical map projection. It corresponds to projecting the Earths surface onto a tangent to the equator as if from a light source at Earths center. The cylinder is cut along one of the projected meridians. Distortion increases so rapidly away from the equator that the central cylindrical is only used as an easily understood illustration of projection and its vertical stretching is even worse than that of the Mercator projection, whose construction method is sometimes erroneously described equivalently to central cylindricals. As with any cylindrical projection, the construction can be generalized by positioning the cylinder to be tangent to a circle of the globe that is not the equator. This projection has prominent use in photography where it is usually called “cylindrical projection”. X = R, y = R tan φ R denotes the radius of the globe, φ is the latitude, λ is the longitude, λ₀ is the longitude of the central meridian. Gnomonic projection Map projection List of map projections

19.
Equirectangular projection
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The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD100. The projection maps meridians to vertical lines of constant spacing. The projection is neither equal area nor conformal, because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. The forward projection transforms spherical coordinates into planar coordinates, the reverse projection transforms from the plane back onto the sphere. X = cos φ1 y = The plate carrée, is the case where φ1 is zero. While a projection with equally spaced parallels is possible for an ellipsoidal model, more complex formulae can be used to create an equidistant map whose parallels reflect the true spacing. Table of examples and properties of all projections, from radicalcartography. net

20.
Gall stereographic projection
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The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection, Gall called the projection stereographic because the spacing of the parallels is the same as the spacing of the parallels along the central meridian of the equatorial stereographic projection. The reverse projection is defined as, λ = x 2 R, φ =2 arctan y R This later cylindrical projection by Carl Braun is similar and this yields a projection tangent to the sphere. Its formula is, x = R λ, y =2 R tan φ2 List of map projections James P. Snyder, Map Projections—A Working Manual, USGS Professional Paper 1395

21.
Space-oblique Mercator projection
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Space-oblique Mercator projection is a map projection. The space-oblique Mercator projection was developed by John P. Snyder, Alden Partridge Colvocoresses, Snyder had an interest in maps, originating back to his childhood and he regularly attended cartography conferences while on vacation. Snyder worked on the problem armed with his newly purchased pocket calculator and he submitted these to the USGS at no charge, starting off a new career at USGS. His formulas were used to maps from Landsat 4 images launched in the summer of 1978. The space-oblique Mercator projection provides continual conformal mapping of the swath sensed by a satellite, scale is true along the ground track, varying 0.01 percent within the normal sensing range of the satellite. Conformality is correct within a few parts per million for the sensing range, distortion is essentially constant along lines of constant distance parallel to the ground track. SOM is the only projection presented that takes the rotation of Earth into account, P2 = time required for revolution of satellite. P1 = length of Earth rotation, X, y = rectangular map coordinates. John Hessler, Projecting Time, John Parr Snyder and the Development of the Space Oblique Mercator Projection, Library of Congress,2003 Snyders 1981 Paper Detailing the Projections Derivation

22.
Web Mercator
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Web Mercator, Google Web Mercator, Spherical Mercator, WGS84 Web Mercator or WGS 84/Pseudo-Mercator is a variant of the Mercator projection and is the de facto standard for Web mapping applications. It rose to prominence when used in the first Google Maps in 2005 and it is used by virtually all major online map providers, including Google Maps, Bing Maps, OpenStreetMap, Mapquest, Esri, Mapbox, and many others. Its official EPSG identifier is EPSG,3857, although others have used historically. Web Mercator is a variant of the Mercator projection, one used primarily in Web-based mapping programs. It uses the same formulas as the standard Mercator as used for small-scale maps, however, the Web Mercator uses the spherical formulas at all scales whereas large-scale Mercator maps normally use the ellipsoidal form of the projection. The discrepancy is imperceptible at the scale but causes maps of local areas to deviate slightly from true ellipsoidal Mercator maps at the same scale. This deviation becomes more pronounced further from the equator, and can reach as high as 35 km on the ground, while the Web Mercators formulas are for the spherical form of the Mercator, geographical coordinates are required to be in the WGS84 ellipsoidal datum. This discrepancy causes the projection to be slightly non-conformal, general lack of understanding that the Web Mercator differs from standard Mercator usage has caused considerable confusion and misuse. For all these reasons, the United States Department of Defense through the National Geospatial-Intelligence Agency has declared this map projection to be unacceptable for any official use, because the Mercator projects the poles at infinity, Google Maps cannot show the poles. Instead it cuts off coverage at 85. 051129° north and south and this is not considered a limitation, given the purpose of the service. The value 85. 051129° is the latitude at which the map becomes a square. EPSGs definition says the projection uses spherical development of ellipsoidal coordinates, alastair Aitchison says the underlying geographic coordinates are defined using WGS84, but projected as if they were defined on a sphere. Unlike the ellipsoidal Mercator and spherical Mercator, the Web Mercator is not quite conformal due to its use of ellipsoidal datum geographical coordinates against a spherical projection, rhumb lines are not straight lines. The benefit is that the form is much simpler to calculate. And believe that it is technically flawed and we will not devalue the EPSG dataset by including such inappropriate geodesy and cartography. The unofficial code 900913 came to be used and it was originally defined by Christopher Schmidt in his Technical Ramblings blog. In 2008, EPSG provided the official identifier EPSG,3785 with the official name Popular Visualisation CRS / Mercator and this definition used a spherical model of the Earth. Later that year, EPSG provided an updated identifier, EPSG,3857 with the official name WGS84 / Pseudo-Mercator, the definition switched to using the WGS84 ellipsoid, rather than the sphere

23.
Eckert II projection
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The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect the network of longitude and latitude lines consists solely of straight lines, and it was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the shape. The pair to Eckert II is the Eckert I projection, the projection is symmetrical about the straight equator and straight central meridian. Parallels vary in spacing in order to preserve areas, as a pseudocylindric projection, spacing of meridians along any given parallel is constant. The poles are represented as lines, each half as long as the equator, the projection has correct scale only on the central meridian at latitudes 55°10′ north and south. Here, y assumes the sign of φ, Max Eckert-Greifendorff List of map projections Eckert IV projection Eckert VI projection Eckert II at Mapthematics radicalcartography. net Cartographic Projection Procedures by Gerald I

24.
Eckert IV projection
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The Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the lines is half that of the equator. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections, in each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. θ can be solved for numerically using Newtons method

25.
Eckert VI projection
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The Eckert VI projection is an equal-area pseudocylindrical map projection. The length of line is half that of the equator. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections, in each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to Eckert VI is the Eckert V projection, list of map projections Eckert II projection Eckert IV projection Max Eckert-Greifendorff Eckert VI projection at Mathworld

26.
Goode homolosine projection
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The Goode homolosine projection is a pseudocylindrical, equal-area, composite map projection used for world maps. Normally it is presented with multiple interruptions and its equal-area property makes it useful for presenting spatial distribution of phenomena. The projection was developed in 1923 by John Paul Goode to provide an alternative to the Mercator projection for portraying global areal relationships, Goode offered variations of the interruption scheme for emphasizing the world’s land masses and the world’s oceans. Some variants include extensions that repeat regions in two different lobes of the map in order to show Greenland or eastern Russia undivided. The homolosine evolved from Goode’s 1916 experiments in interrupting the Mollweide projection, because the Mollweide is sometimes called the homolographic projection, Goode fused the two names homolographic and sinusoidal to create the name “homolosine”. Common in the 1960s, the Goode homolosine projection is called an orange-peel map because of its resemblance to the flattened rind of a hand-peeled orange. In its most common form, the map interrupts the North Atlantic, the South Atlantic, the South Pacific, the Indian Ocean, up to latitudes 41°44′11. 8″N/S, the map is projected according to the Sinusoidal projection’s transformation. The higher latitudes are the top sections of a Mollweide projection and this grafting results in a kink in the meridians along the parallel of the graft. The projection’s equal-area property follows from the fact that its source projections are themselves both equal-area, list of map projections Simulating the Interrupted Goode Homolosine Projection With ArcInfo Table of examples and properties of all common projections, from radicalcartography. net

27.
Kavrayskiy VII projection
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The Kavrayskiy VII projection is a map projection invented by Vladimir V. Kavrayskiy in 1939 for use as a general purpose pseudocylindrical projection. Like the Robinson projection, it is a compromise intended to produce good quality maps with low distortion overall and it scores well in that respect compared to other popular projections, such as the Winkel Tripel, despite straight, evenly spaced parallels and a simple formulation. Regardless, it has not been used outside the former Soviet Union. The projection is defined as, x =3 λ213 −2 y = φ where λ is the longitude, list of map projections Cartography Wagner VI projection Curvature in Map Projections, quantification of overall distortion in projections. Mapthematics Kavrayskiy VII, bivariate distortion map, deducing the Kavrayskiy VII Projection, description of the properties of the Kavrayskiy VII projection

28.
Mollweide projection
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The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for global maps of the world or night sky. It is also known as the Babinet projection, homalographic projection, homolographic projection, the projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions. The projection was first published by mathematician and astronomer Karl Brandan Mollweide of Leipzig in 1805 and it was reinvented and popularized in 1857 by Jacques Babinet, who gave it the name homalographic projection. The variation homolographic arose from frequent nineteenth-century usage in star atlases, the Mollweide is a pseudocylindrical projection in which the equator is represented as a straight horizontal line perpendicular to a central meridian one-half its length. The other parallels compress near the poles, while the other meridians are equally spaced at the equator, the meridians at 90 degrees east and west form a perfect circle, and the whole earth is depicted in a proportional 2,1 ellipse. Shape distortion may be diminished by using an interrupted version, a sinusoidal interrupted Mollweide projection discards the central meridian in favor of alternating half-meridians which terminate at right angles to the equator. This has the effect of dividing the globe into lobes, in contrast, a parallel interrupted Mollweide projection uses multiple disjoint central meridians, giving the effect of multiple ellipses joined at the equator. More rarely, the project can be drawn obliquely to shift the areas of distortion to the oceans, the Mollweide, or its properties, has inspired the creation of several other projections, including the Goodes homolosine, van der Grinten and the Boggs eumorphic. The map has area 4πR2, conforming to the area of the generating globe. The x-coordinate has a range of, and the y-coordinate has a range of, if φ = ±π/2, then also θ = ±π/2. In that case the iteration should be bypassed, otherwise, division by zero may result, the inverse transformations allow one to find the latitude and longitude corresponding to the map coordinates x and y. List of map projections Aitoff projection Hammer projection An interactive Java applet to study deformations of the Mollweide Map Projection Mollweide Projection at Mathworld

29.
Sinusoidal projection
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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, the projection is defined by, x = cos φ y = φ where φ is the latitude, λ is the longitude, and λ0 is the central meridian. Scale is constant along the meridian, and east–west scale is constant throughout the map. Therefore the length of each parallel on the map is proportional to the cosine of the latitude and this makes the left and right bounding meridians of the map into half of a sine wave, each mirroring the other. Each meridian is half of a wave with only the amplitude differing, giving the projection its name. Each is shown on the map as longer than the central meridian, the true distance between two points on a meridian can be measured on the map as the vertical distance between the parallels that intersect the meridian at those points. With no distortion along the meridian and the equator, distances along those lines are correct. Distortion is lowest throughout the region of the map close to those lines, similar projections which wrap the east and west parts of the sinusoidal projection around the north pole are the Werner and the intermediate Bonne and Bottomley projections. The MODLAND Integerized Sinusoidal Grid, based on the projection, is a geodesic grid developed by the NASAs Moderate-Resolution Imaging Spectroradiometer science team. List of map projections Gerardus Mercator, Nicolas Sanson, and John Flamsteed - mathematicians who developed the technique, media related to Sinusoidal projection at Wikimedia Commons Pseudocylindrical Projections Table of examples and properties of all common projections, from radicalcartography. net

30.
Tobler hyperelliptical projection
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The Tobler hyperelliptical projection is a family of equal-area pseudocylindrical projections that may be used for world maps. Waldo R. Tobler introduced the construction in 1973 as the hyperelliptical projection, as with any pseudocylindrical projection, in the projection’s normal aspect, the parallels of latitude are parallel, straight lines. Their spacing is calculated to provide the equal-area property, the projection blends the cylindrical equal-area projection with meridians of longitude that follow a particular kind of curve known as superellipses or Lamé curves or sometimes as hyperellipses. The curve is described by xk + yk = γk, the relative weight of the cylindrical equal-area projection is given as α, ranging from all cylindrical equal-area with α =1 to all hyperellipses with α =0. When α =0 and k =1 the projection degenerates to the Collignon projection, when α =0, k =2, and γ ≈1.2731 the projection becomes the Mollweide projection. Tobler favored the parameterization shown with the illustration, that is, α =0, k =2.5, and γ =1.183136

31.
Wagner VI projection
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Wagner VI is a pseudocylindrical whole Earth map projection. Like the Robinson projection, it is a projection, not having any special attributes other than a pleasing. Wagner VI is equivalent to the Kavrayskiy VII horizontally elongated by a factor of 2⁄√3 and this elongation results in proper preservation of shapes near the equator but slightly more distortion overall. The aspect ratio of this projection is 2,1, as formed by the ratio of the equator to the central meridian and this matches the ratio of Earth’s equator to any meridian. The Wagner VI is defined by, x = λ1 −32 y = φ, list of map projections Kavrayskiy VII projection

32.
Albers projection
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The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. Although scale and shape are not preserved, distortion is minimal between the standard parallels, the Albers projection is one of the standard projections for British Columbia, and is the sole standard projection used by the government of Yukon. It is also used by the United States Geological Survey and the United States Census Bureau, snyder describes generating formulae for the projection, as well as the projections characteristics. net An interactive Java Applet to study the metric deformations of the Albers Projection

33.
Lambert conformal conic projection
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A Lambert conformal conic projection is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, conceptually, the projection seats a cone over the sphere of the Earth and projects the surface conformally onto the cone. The cone is unrolled, and the parallel that was touching the sphere is assigned unit scale and that parallel is called the reference parallel or standard parallel. By scaling the map, two parallels can be assigned unit scale, with scale decreasing between the two parallels and increasing outside them. This gives the map two standard parallels, in this way, deviation from unit scale can be minimized within a region of interest that lies largely between the two standard parallels. Unlike other conic projections, no true secant form of the projection exists because using a secant cone does not yield the same scale along both standard parallels. Pilots use aeronautical charts based on LCC because a line drawn on a Lambert conformal conic projection approximates a great-circle route between endpoints for typical flight distances. The US systems of VFR sectional charts and terminal area charts are drafted on the LCC with standard parallels at 33°N and 45°N. The European Environment Agency and the INSPIRE specification for coordinate systems using this projection for conformal pan-European mapping at scales smaller or equal to 1,500,000. In Metropolitan France, the projection is Lambert-93, a Lambert conic projection using RGF93 geodetic system. The National Spatial Framework for India uses Datum WGS84 with a LCC projection and is a recommended NNRMS standard, each state has its own set of reference parameters given in the standard. The projection as used in CCS83 yields maps in which errors are limited to 1 part in 10,000. The Lambert conformal conic is one of several map projection developed by Johann Heinrich Lambert, an 18th-century Swiss mathematician, physicist, philosopher

34.
Bonne projection
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A Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. Although named after Rigobert Bonne, the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De lIsle before 1700 and Coronelli in 1696. Both Sylvano and Honter’s usages were approximate, however, and it is not clear they intended to be the same projection, parallels of latitude are concentric circular arcs, and the scale is true along these arcs. On the central meridian and the standard latitude shapes are not distorted. The inverse projection is given by, φ = cot φ1 + φ1 − ρ λ = λ0 + ρ cos φ arctan where ρ = ± x 2 +2 taking the sign of φ1

35.
Polyconic projection
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Polyconic can refer either to a class of map projections or to a specific projection known less ambiguously as the American Polyconic. Polyconic as a class refers to those projections whose parallels are all non-concentric circular arcs, except for a straight equator, and this description applies to projections in equatorial aspect. Each parallel is an arc of true scale. The scale is true on the central meridian of the projection. The projection was in use by many map-making agencies of the United States from the time of its proposal by Ferdinand Rudolph Hassler in 1825 until the middle of the 20th century. To avoid division by zero, the formulas above are extended so that if φ =0 then x = λ − λ0, list of map projections Weisstein, Eric W. Polyconic projection. Table of examples and properties of all projections, from radicalcartography. net An interactive Java Applet to study the metric deformations of the Polyconic Projection

36.
Werner projection
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The Werner projection is a pseudoconic equal-area map projection sometimes called the Stab-Werner or Stabius-Werner projection. Like other heart-shaped projections, it is categorized as cordiform. Stab-Werner refers to two originators, Johannes Werner, a parish priest in Nuremberg, refined and promoted this projection that had developed earlier by Johannes Stabius of Vienna around 1500. The projection is a form of the Bonne projection, having its standard parallel at one of the poles. Distances along each parallel and along the meridian are correct. List of map projections Media related to Maps with Stab-Werner projection at Wikimedia Commons Table of examples and properties of all common projections, Radical Cartography

37.
General Perspective projection
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The General Perspective Projection is a map projection of cartography. When the Earth is photographed from space, the records the view as a perspective projection. If the camera precisely faces the center of the Earth, the projection is Vertical Perspective, otherwise, a Tilted Perspective projection is obtained. The Vertical Perspective is related to the Stereographic projection, Gnomonic projection and these are all true perspective projections, and are also azimuthal. The point of perspective for the General Perspective Projection is a finite distance and it depicts the earth as it appears from some relatively short distance above the surface, typically a few hundred to a few tens of thousands of kilometers. Tilted Perspective projections are not azimuthal, directions are not true from the point. Some forms of the projection were known to the Greeks and Egyptians 2,000 years ago and it was studied by several French and British scientists in the 18th and 19th centuries. But the projection had little value, computationally simpler nonperspective azimuthal projections could be used instead. Space exploration led to a renewed interest in the perspective projection, now the concern was for a pictorial view from space, not for minimal distortion. A picture taken with a camera from the window of a spacecraft has a tilted vertical perspective, so the manned Gemini. Some prominent Internet mapping tools also use the tilted perspective projection, for example, Google Earth and NASA World Wind show the globe as it appears from space. List of map projections Map projection

38.
Gnomonic projection
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This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the sphere passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it, less than half of the sphere can be projected onto a finite map. Consequently, a photographic lens, which is based on the gnomonic principle. The gnomonic projection is said to be the oldest map projection, the path of the shadow-tip or light-spot in a nodus-based sundial traces out the same hyperbolae formed by parallels on a gnomonic map. Since meridians and the equator are great circles, they are shown as straight lines on a gnomonic map. If the tangent point is one of the then the meridians are radial. The equator is at infinity in all directions, other parallels are depicted as concentric circles. If the tangent point is not on a pole or the equator, then the meridians are radially outward straight lines from a Pole, the equator is a straight line that is perpendicular to only one meridian, indicating that the projection is not conformal. Other parallels are depicted as conic sections, if the tangent point is on the equator then the meridians are parallel but not equally spaced. The equator is a line perpendicular to the meridians. Other parallels are depicted as hyperbolae, as with all azimuthal projections, angles from the tangent point are preserved. The map distance from that point is an r of the true distance d. The radial scale is r ′ =1 cos 2 d R and the transverse scale 1 cos d R so the scale increases outwardly. Gnomonic projections are used in work because seismic waves tend to travel along great circles. They are also used by navies in plotting direction finding bearings, meteors also travel along great circles, with the Gnomonic Atlas Brno 2000.0 being the IMOs recommended set of star charts for visual meteor observations. Aircraft and ship pilots use the projection to find the shortest route between start and destination, the gnomonic projection is used extensively in photography, where it is called rectilinear projection. The gnomonic projection is used in astronomy where the tangent point is centered on the object of interest, the sphere being projected in this case is the celestial sphere, R =1, and not the surface of the Earth. List of map projections Beltrami–Klein model, the mapping of the hyperbolic plane Snyder

39.
Orthographic projection in cartography
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The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, the point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, the shapes and areas are distorted, particularly near the edges. The orthographic projection has been known since antiquity, with its uses being well documented. Hipparchus used the projection in the 2nd century B. C. to determine the places of star-rise, in about 14 B. C. Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions. Vitruvius also seems to have devised the term orthographic for the projection, however, the name analemma, which also meant a sundial showing latitude and longitude, was the common name until François dAguilon of Antwerp promoted its present name in 1613. The earliest surviving maps on the projection appear as woodcut drawings of terrestrial globes of 1509,1533 and 1551, a highly refined map designed by Renaissance polymath Albrecht Dürer and executed by Johannes Stabius appeared in 1515. Photographs of the Earth and other planets from spacecraft have inspired renewed interest in the projection in astronomy. The formulas for the orthographic projection are derived using trigonometry. They are written in terms of longitude and latitude on the sphere, define the radius of the sphere R and the center point of the projection. This ensures that points on the opposite hemisphere are not plotted, the point should be clipped from the map if cos is negative. This ensures that the sign of the projection as written is correct in all quadrants. The inverse formulas are useful when trying to project a variable defined on a grid onto a rectilinear grid in. Direct application of the orthographic projection yields scattered points in, which creates problems for plotting, one solution is to start from the projection plane and construct the image from the values defined in by using the inverse formulas of the orthographic projection. See References for a version of the orthographic map projection. In a wide sense, all projections with the point of perspective at infinity are considered as orthographic and these kinds of projections distort angles and areas close to the poles. An example of a projection onto a cylinder is the Lambert cylindrical equal-area projection. List of map projections Orthographic Projection—from MathWorld

40.
Stereographic projection
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In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =

41.
Lambert azimuthal equal-area projection
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The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, and it is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. Zenithal being synonymous with azimuthal, the projection is known as the Lambert zenithal equal-area projection. The Lambert azimuthal projection is used as a map projection in cartography and it is also used in scientific disciplines such as geology for plotting the orientations of lines in three-dimensional space. This plotting is aided by a kind of graph paper called a Schmidt net. To define the Lambert azimuthal projection, imagine a plane set tangent to the sphere at some point S on the sphere, let P be any point on the sphere other than the antipode of S. Let d be the distance between S and P in three-dimensional space, then the projection sends P to a point P′ on the plane that is a distance d from S. To make this precise, there is a unique circle centered at S, passing through P. It intersects the plane in two points, let P′ be the one that is closer to P, the antipode of S is excluded from the projection because the required circle is not unique. The case of S is degenerate, S is projected to itself, explicit formulas are required for carrying out the projection on a computer. Consider the projection centered at S = on the unit sphere, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are then described by =, =. In spherical coordinates on the sphere and polar coordinates on the disk, in cylindrical coordinates on the sphere and polar coordinates on the plane, the map and its inverse are given by =, =. The projection can be centered at other points, and defined on spheres of other than 1. As defined in the section, the Lambert azimuthal projection of the unit sphere is undefined at. It sends the rest of the sphere to the disk of radius 2 centered at the origin in the plane. It sends the point to, the equator z =0 to the circle of radius √2 centered at, the projection is a diffeomorphism between the sphere and the open disk of radius 2. It is a map, which can be seen by computing the area element of the sphere when parametrized by the inverse of the projection. In Cartesian coordinates it is d A = d X d Y and this means that measuring the area of a region on the sphere is tantamount to measuring the area of the corresponding region on the disk

42.
Aitoff projection
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The Aitoff projection is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Expressed simply, x =2 azeq x , y = azeq y where azeqx, in all of these formulas, λ is the longitude from the central meridian and φ is the latitude. Three years later, Ernst Hermann Heinrich Hammer suggested the use of the Lambert azimuthal equal-area projection in the manner as Aitoff. While Hammer was careful to cite Aitoff, some authors have referred to the Hammer projection as the Aitoff projection. List of map projections Mollweide projection Hammer projection Table of common projections An interactive Java Applet to study the metric deformations of the Aitoff Projection