1.
Cartography
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Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively, the fundamental problems of traditional cartography are to, Set the maps agenda and select traits of the object to be mapped. This is the concern of map editing, traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. Represent the terrain of the object on flat media. This is the concern of map projections, eliminate characteristics of the mapped object that are not relevant to the maps purpose. This is the concern of generalization, reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization, orchestrate the elements of the map to best convey its message to its audience. This is the concern of map design, modern cartography constitutes many theoretical and practical foundations of geographic information systems. The earliest known map is a matter of debate, both because the term map isnt well-defined and because some artifacts that might be maps might actually be something else. A wall painting that might depict the ancient Anatolian city of Çatalhöyük has been dated to the late 7th millennium BCE, the oldest surviving world maps are from 9th century BCE Babylonia. One shows Babylon on the Euphrates, surrounded by Assyria, Urartu and several cities, all, in turn, another depicts Babylon as being north of the world center. The ancient Greeks and Romans created maps since Anaximander in the 6th century BCE, in the 2nd century AD, Ptolemy wrote his treatise on cartography, Geographia. This contained Ptolemys world map – the world known to Western society. As early as the 8th century, Arab scholars were translating the works of the Greek geographers into Arabic, in ancient China, geographical literature dates to the 5th century BCE. The oldest extant Chinese maps come from the State of Qin, dated back to the 4th century BCE, in the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song, a star map on the equidistant cylindrical projection. Early forms of cartography of India included depictions of the pole star and these charts may have been used for navigation. Mappa mundi are the Medieval European maps of the world, approximately 1,100 mappae mundi are known to have survived from the Middle Ages. Of these, some 900 are found illustrating manuscripts and the remainder exist as stand-alone documents, the Arab geographer Muhammad al-Idrisi produced his medieval atlas Tabula Rogeriana in 1154

2.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen

3.
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

4.
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 =

5.
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

6.
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

7.
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

8.
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

9.
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

10.
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

11.
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

12.
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