1.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
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Circumscribed cylinder to a sphere
Sphere
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A two-dimensional perspective projection of a sphere
Sphere
Sphere
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Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres
2.
Geographic coordinate system
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A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are chosen such that one of the numbers represents a vertical position. A common choice of coordinates is latitude, longitude and elevation, to specify a location on a two-dimensional map requires a map projection. The invention of a coordinate system is generally credited to Eratosthenes of Cyrene. Ptolemy credited him with the adoption of longitude and latitude. Ptolemys 2nd-century Geography used the prime meridian but measured latitude from the equator instead. Mathematical cartography resumed in Europe following Maximus Planudes recovery of Ptolemys text a little before 1300, in 1884, the United States hosted the International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt the longitude of the Royal Observatory in Greenwich, the Dominican Republic voted against the motion, while France and Brazil abstained. France adopted Greenwich Mean Time in place of local determinations by the Paris Observatory in 1911, the latitude of a point on Earths surface is the angle between the equatorial plane and the straight line that passes through that point and through the center of the Earth. Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they are parallel to the equator, the north pole is 90° N, the south pole is 90° S. The 0° parallel of latitude is designated the equator, the plane of all geographic coordinate systems. The equator divides the globe into Northern and Southern Hemispheres, the longitude of a point on Earths surface is the angle east or west of a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses, which converge at the north and south poles, the prime meridian determines the proper Eastern and Western Hemispheres, although maps often divide these hemispheres further west in order to keep the Old World on a single side. The antipodal meridian of Greenwich is both 180°W and 180°E, the combination of these two components specifies the position of any location on the surface of Earth, without consideration of altitude or depth. The grid formed by lines of latitude and longitude is known as a graticule, the origin/zero point of this system is located in the Gulf of Guinea about 625 km south of Tema, Ghana. To completely specify a location of a feature on, in, or above Earth. Earth is not a sphere, but a shape approximating a biaxial ellipsoid. It is nearly spherical, but has an equatorial bulge making the radius at the equator about 0. 3% larger than the radius measured through the poles, the shorter axis approximately coincides with the axis of rotation
Geographic coordinate system
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Longitude lines are perpendicular and latitude lines are parallel to the equator.
3.
Geoid
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The geoid is the shape that the surface of the oceans would take under the influence of Earths gravity and rotation alone, in the absence of other influences such as winds and tides. This surface is extended through the continents, all points on a geoid surface have the same gravity potential energy. The geoid can be defined at any value of gravitational potential such as within the earths crust or far out in space and it does not correspond to the actual surface of Earths crust, but to a surface which can only be known through extensive gravitational measurements and calculations. It is often described as the true figure of the Earth. The surface of the geoid is higher than the reference ellipsoid wherever there is a gravity anomaly. The geoid surface is irregular, unlike the ellipsoid which is a mathematical idealized representation of the physical Earth. Although the physical Earth has excursions of +8,848 m and −429 m, If the ocean surface were isopycnic and undisturbed by tides, currents, or weather, it would closely approximate the geoid. The permanent deviation between the geoid and mean sea level is called ocean surface topography, If the continental land masses were criss-crossed by a series of tunnels or canals, the sea level in these canals would also very nearly coincide with the geoid. This means that when traveling by ship, one does not notice the undulations of the geoid, the vertical is always perpendicular to the geoid. Likewise, spirit levels will always be parallel to the geoid, a long voyage, indicate height variations, even though the ship will always be at sea level. This is because GPS satellites, orbiting about the center of gravity of the Earth, to obtain ones geoidal height, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a tidal measurement station, as in land surveying. Modern GPS receivers have a grid implemented inside where they obtain the height over the World Geodetic System ellipsoid from the current position. Then they are able to correct the height above WGS ellipsoid to the height above WGS84 geoid, in that case when the height is not zero on a ship it is due to various other factors such as ocean tides, atmospheric pressure and local sea surface topography. The gravitational field of the earth is neither perfect n If that perfect sphere were then covered in water, instead, the water level would be higher or lower depending on the particular strength of gravity in that location. Spherical harmonics are used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM96, the geoid is a particular equipotential surface, and is somewhat involved to compute. The gradient of this also provides a model of the gravitational acceleration
Geoid
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Three-dimensional visualization of geoid undulations, using units of gravity.
Geoid
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Geodesy
4.
Sea level
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Mean sea level is an average level of the surface of one or more of Earths oceans from which heights such as elevations may be measured. A common and relatively straightforward mean sea-level standard is the midpoint between a low and mean high tide at a particular location. Sea levels can be affected by factors and are known to have varied greatly over geological time scales. The careful measurement of variations in MSL can offer insights into ongoing climate change, the term above sea level generally refers to above mean sea level. Precise determination of a sea level is a difficult problem because of the many factors that affect sea level. Sea level varies quite a lot on several scales of time and this is because the sea is in constant motion, affected by the tides, wind, atmospheric pressure, local gravitational differences, temperature, salinity and so forth. The easiest way this may be calculated is by selecting a location and calculating the mean sea level at that point, for example, a period of 19 years of hourly level observations may be averaged and used to determine the mean sea level at some measurement point. One measures the values of MSL in respect to the land, hence a change in MSL can result from a real change in sea level, or from a change in the height of the land on which the tide gauge operates. In the UK, the Ordnance Datum is the sea level measured at Newlyn in Cornwall between 1915 and 1921. Prior to 1921, the datum was MSL at the Victoria Dock, in Hong Kong, mPD is a surveying term meaning metres above Principal Datum and refers to height of 1. 230m below the average sea level. In France, the Marégraphe in Marseilles measures continuously the sea level since 1883 and it is used for a part of continental Europe and main part of Africa as official sea level. Elsewhere in Europe vertical elevation references are made to the Amsterdam Peil elevation, satellite altimeters have been making precise measurements of sea level since the launch of TOPEX/Poseidon in 1992. A joint mission of NASA and CNES, TOPEX/Poseidon was followed by Jason-1 in 2001, height above mean sea level is the elevation or altitude of an object, relative to the average sea level datum. It is also used in aviation, where some heights are recorded and reported with respect to sea level, and in the atmospheric sciences. An alternative is to base height measurements on an ellipsoid of the entire Earth, in aviation, the ellipsoid known as World Geodetic System 84 is increasingly used to define heights, however, differences up to 100 metres exist between this ellipsoid height and mean tidal height. The alternative is to use a vertical datum such as NAVD88. When referring to geographic features such as mountains on a topographic map, the elevation of a mountain denotes the highest point or summit and is typically illustrated as a small circle on a topographic map with the AMSL height shown in metres, feet or both. In the rare case that a location is below sea level, for one such case, see Amsterdam Airport Schiphol
Sea level
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This marker indicating sea level is situated between Jerusalem and the Dead Sea.
Sea level
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Sea Level sign seen on cliff (circled in red) at Badwater Basin, Death Valley National Park
5.
Normal (geometry)
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In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a normal, or simply normal. The word normal is used as an adjective, a line normal to a plane, the normal component of a force. The concept of normality generalizes to orthogonality, the concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at a point P is the set of the vectors which are orthogonal to the tangent space at P, in the case of differential curves, the curvature vector is a normal vector of special interest. For a convex polygon, a surface normal can be calculated as the cross product of two edges of the polygon. For a plane given by the equation a x + b y + c z + d =0, the vector is a normal. For a hyperplane in n+1 dimensions, given by the equation r = a 0 + α1 a 1 + ⋯ + α n a n, where a0 is a point on the hyperplane and ai for i =1. N are non-parallel vectors lying on the hyperplane, a normal to the hyperplane is any vector in the space of A where A is given by A =. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. If a surface S is parameterized by a system of coordinates x, with s and t real variables. For a surface S given explicitly as a function f of the independent variables x, y, the first one is obtaining its implicit form F = z − f =0, from which the normal follows readily as the gradient ∇ F. The second way of obtaining the normal follows directly from the gradient of the form, ∇ f, by inspection, ∇ F = k ^ − ∇ f. Note that this is equal to ∇ F = k ^ − ∂ f ∂ x i ^ − ∂ f ∂ y j ^, if a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base, however, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous, a normal to a surface does not have a unique direction, the vector pointing in the opposite direction of a surface normal is also a surface normal. For an oriented surface, the normal is usually determined by the right-hand rule
Normal (geometry)
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A polygon and two of its normal vectors
6.
Reference ellipsoid
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In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Current practice uses the word alone in preference to the full term oblate ellipsoid of revolution or the older term oblate spheroid. In the rare instances where a more general shape is required as a model the term used is triaxial ellipsoid. A great many ellipsoids have been used with various sizes and centres, the shape of an ellipsoid is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated about its minor axis. The semi-major axis of the ellipse, a, is identified as the radius of the ellipsoid. For the Earth, f is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21 km, some precise values are given in the table below and also in Figure of the Earth. A great many other parameters are used in geodesy but they can all be related to one or two of the set a, b and f, a primary use of reference ellipsoids is to serve as a basis for a coordinate system of latitude, longitude, and elevation. For this purpose it is necessary to identify a zero meridian, for other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater Airy-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid, the longitude measures the rotational angle between the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed in degrees ranging from −180° to +180° For other bodies a range of 0° to 360° is used. The latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, the common or geodetic latitude is the angle between the equatorial plane and a line that is normal to the reference ellipsoid. Depending on the flattening, it may be different from the geocentric latitude. For non-Earth bodies the terms planetographic and planetocentric are used instead, see geodetic system for more detail. If these coordinates, i. e. N is the radius of curvature in the prime vertical, in contrast, extracting φ, λ and h from the rectangular coordinates usually requires iteration. A straightforward method is given in an OSGB publication and also in web notes, more sophisticated methods are outlined in geodetic system. Currently the most common reference used, and that used in the context of the Global Positioning System, is the one defined by WGS84. Traditional reference ellipsoids or geodetic datums are defined regionally and therefore non-geocentric, modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e. g. WGS84. Reference ellipsoids are also useful for mapping of other planetary bodies including planets, their satellites, asteroids
Reference ellipsoid
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Flattened sphere
7.
Phi (letter)
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Phi is the 21st letter of the Greek alphabet. In Ancient Greek, it represented a voiceless bilabial plosive. In modern Greek, it represents a voiceless fricative and is correspondingly romanized as f. Its origin is uncertain but it may be that phi originated as the letter qoppa, in traditional Greek numerals, phi has a value of 500 or 500 000. The Cyrillic letter Ef descends from phi, phi is also used as a symbol for the golden ratio and on other occasions in math and science. This use is separately encoded as the Unicode glyph ϕ, the modern Greek pronunciation of the letter is sometimes encountered in English when the letter is being used in this sense. The lower-case letter φ is often used to represent the following, Magnetic flux in physics The golden ratio 1 +52 ≈1.618033988749894848204586834. in mathematics, art, Eulers totient function φ in number theory, also called Eulers phi function. The cyclotomic polynomial functions Φn of algebra, in algebra, group or ring homomorphisms In probability theory, ϕ = −½e−x2/2 is the probability density function of the normal distribution. In probability theory, φX = E is the function of a random variable X. An angle, typically the second angle mentioned, after θ, especially, The argument of a complex number. The phase of a wave in signal processing, in spherical coordinates, mathematicians usually refer to phi as the polar angle. The convention in physics is to use phi as the azimuthal angle, one of the dihedral angles in the backbones of proteins in a Ramachandran plot Internal or effective angle of friction. The work function of a surface, in solid-state physics, a shorthand representation for an aromatic functional group in organic chemistry. The ratio of free energy destabilizations of protein mutants in phi value analysis, in cartography, geodesy and navigation, latitude. In aircraft flight mechanics as the symbol for bank angle, in combustion engineering, fuel–air equivalence ratio. The ratio between the fuel air ratio to the stoichiometric fuel air ratio. The Veblen function in set theory Porosity in geology and hydrology, strength reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods. The symbol for a voiceless fricative in the International Phonetic Alphabet In economics
Phi (letter)
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Greek alphabet
8.
Theodolite
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A theodolite /θiːˈɒdəlaɪt/ is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are used mainly for surveying applications, and have adapted for specialized purposes such as meteorology. A modern theodolite consists of a telescope mounted within two perpendicular axes, the horizontal or trunnion axis and the zenith axis. A theodolite measures vertical angles as angles between the zenith, forwards or plunged—typically approximately 90 and 270 degrees, when the telescope is pointed at a target object, the angle of each of these axes can be measured with great precision, typically to seconds of arc. A theodolite may be either transit or non-transit, in a transit theodolite, the telescope can be inverted in the vertical plane, whereas the rotation in the same plane is restricted to a semi-circle in a non-transit theodolite. Some types of transit theodolites do not allow the measurement of vertical angles, the builders level is sometimes mistaken for a transit theodolite, but it measures neither horizontal nor vertical angles. It uses a level to set a telescope level to define a line of sight along a horizontal plane. A theodolite is mounted on its head by means of a forced centering plate or tribrach containing four thumbscrews, or in modern theodolites. Before use, a theodolite must be placed vertically above the point to be measured using a plumb bob. The instrument is then set level using leveling footscrews and circular, both axes of a theodolite are equipped with graduated circles that can be read through magnifying lenses. The vertical circle which transits about the horizontal axis should read 90° when the axis is horizontal, or 270° when the instrument is in its second position. Half of the difference between the two positions is called the index error, the horizontal and vertical axes of a theodolite must be perpendicular, if not then a horizontal axis error exists. This can be tested by aligning the tubular spirit bubble parallel to a line between two footscrews and setting the bubble central, a horizontal axis error is present if the bubble runs off central when the tubular spirit bubble is reversed. To adjust, the operator removes 1/2 the amount the bubble has run off using the screw, then re-level, test. If not, then a collimation error exists, index error, horizontal axis error and collimation error are regularly determined by calibration and are removed by mechanical adjustment. Their existence is taken account in the choice of measurement procedure in order to eliminate their effect on the measurement results of the theodolite. The term diopter was used in old texts as a synonym for theodolite. This derives from an astronomical instrument called a dioptra
Theodolite
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An optical theodolite, manufactured in the Soviet Union in 1958 and used for topographic surveying
Theodolite
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Diagram of an Optical Theodolite
Theodolite
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Sectioned theodolite showing the complexity of the optical paths
Theodolite
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Eight-inch theodolite, c. 1898
9.
Geodesy
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Geodesists also study geodynamical phenomena such as crustal motion, tides, and polar motion. For this they design global and national networks, using space and terrestrial techniques while relying on datums. Geodesy — from the Ancient Greek word γεωδαισία geodaisia — is primarily concerned with positioning within the temporally varying gravity field, such geodetic operations are also applied to other astronomical bodies in the solar system. It is also the science of measuring and understanding the earths geometric shape, orientation in space and this applies to the solid surface, the liquid surface and the Earths atmosphere. For this reason, the study of the Earths gravity field is called physical geodesy by some, the geoid is essentially the figure of the Earth abstracted from its topographical features. It is an idealized surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. The geoid, unlike the ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between the geoid and the ellipsoid is called the geoidal undulation. It varies globally between ±110 m, when referred to the GRS80 ellipsoid, a reference ellipsoid, customarily chosen to be the same size as the geoid, is described by its semi-major axis a and flattening f. The quantity f = a − b/a, where b is the axis, is a purely geometrical one. The mechanical ellipticity of the Earth can be determined to high precision by observation of satellite orbit perturbations and its relationship with the geometrical flattening is indirect. The relationship depends on the density distribution, or, in simplest terms. The 1980 Geodetic Reference System posited a 6,378,137 m semi-major axis and this system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics. It is essentially the basis for geodetic positioning by the Global Positioning System and is also in widespread use outside the geodetic community. The locations of points in space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such systems are typically geocentric. The X-axis lies within the Greenwich observatorys meridian plane, the coordinate transformation between these two systems is described to good approximation by sidereal time, which takes into account variations in the Earths axial rotation. A more accurate description also takes polar motion into account, a closely monitored by geodesists
Geodesy
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An old geodetic pillar (1855) at Ostend, Belgium
Geodesy
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Geodesy
Geodesy
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A Munich archive with lithography plates of maps of Bavaria
Geodesy
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Geodetic Control Mark (example of a deep benchmark)
10.
Prime Meridian
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A prime meridian is a meridian in a geographical coordinate system at which longitude is defined to be 0°. Together, a meridian and its antimeridian form a great circle. This great circle divides the sphere, e. g. the Earth, if one uses directions of East and West from a defined prime meridian, then they can be called Eastern Hemisphere and Western Hemisphere. The most widely used modern meridian is the IERS Reference Meridian and it is derived but deviates slightly from the Greenwich Meridian, which was selected as an international standard in 1884. The notion of longitude was developed by the Greek Eratosthenes in Alexandria, and Hipparchus in Rhodes, but it was Ptolemy who first used a consistent meridian for a world map in his Geographia. The main point is to be comfortably west of the tip of Africa as negative numbers were not yet in use. His prime meridian corresponds to 18°40 west of Winchester today, at that time the chief method of determining longitude was by using the reported times of lunar eclipses in different countries. Ptolemys Geographia was first printed with maps at Bologna in 1477, but there was still a hope that a natural basis for a prime meridian existed. The Tordesillas line was settled at 370 leagues west of Cape Verde. This is shown in Diogo Ribeiros 1529 map, in 1541, Mercator produced his famous 41 cm terrestrial globe and drew his prime meridian precisely through Fuertaventura in the Canaries. His later maps used the Azores, following the magnetic hypothesis, but by the time that Ortelius produced the first modern atlas in 1570, other islands such as Cape Verde were coming into use. In his atlas longitudes were counted from 0° to 360°, not 180°W to 180°E as is usual today and this practice was followed by navigators well into the 18th century. In 1634, Cardinal Richelieu used the westernmost island of the Canaries, Ferro, 19°55 west of Paris, the geographer Delisle decided to round this off to 20°, so that it simply became the meridian of Paris disguised. In the early 18th century the battle was on to improve the determination of longitude at sea, between 1765 and 1811, Nevil Maskelyne published 49 issues of the Nautical Almanac based on the meridian of the Royal Observatory, Greenwich. Maskelynes tables not only made the lunar method practicable, they made the Greenwich meridian the universal reference point. In 1884, at the International Meridian Conference in Washington, D. C.22 countries voted to adopt the Greenwich meridian as the meridian of the world. The French argued for a line, mentioning the Azores and the Bering Strait. In October 1884 the Greenwich Meridian was selected by delegates to the International Meridian Conference held in Washington, united States to be the common zero of longitude and standard of time reckoning throughout the world
Prime Meridian
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Gerardus Mercator in his Atlas Cosmographicae (1595) uses a prime meridian somewhere close to 25°W, passing just to the west of Santa Maria Island in the Atlantic. His 180th meridian runs along the Strait of Anián (Bering Strait)
Prime Meridian
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Ptolemy's 1st projection, redrawn under Maximus Planudes around 1300, using a prime meridian west of Africa
Prime Meridian
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Diogo Ribeiro's map of 1529, now in the Vatican library
11.
North Pole
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The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is defined as the point in the Northern Hemisphere where the Earths axis of rotation meets its surface. The North Pole is the northernmost point on the Earth, lying diametrically opposite the South Pole and it defines geodetic latitude 90° North, as well as the direction of true north. At the North Pole all directions point south, all lines of longitude converge there, along tight latitude circles, counterclockwise is east and clockwise is west. The North Pole is at the center of the Northern Hemisphere, while the South Pole lies on a continental land mass, the North Pole is located in the middle of the Arctic Ocean amid waters that are almost permanently covered with constantly shifting sea ice. This makes it impractical to construct a permanent station at the North Pole, however, the Soviet Union, and later Russia, constructed a number of manned drifting stations on a generally annual basis since 1937, some of which have passed over or very close to the Pole. Since 2002, the Russians have also established a base, Barneo. This operates for a few weeks during early spring, studies in the 2000s predicted that the North Pole may become seasonally ice-free because of Arctic ice shrinkage, with timescales varying from 2016 to the late 21st century or later. The sea depth at the North Pole has been measured at 4,261 m by the Russian Mir submersible in 2007 and at 4,087 m by USS Nautilus in 1958. The nearest land is said to be Kaffeklubben Island, off the northern coast of Greenland about 700 km away. The nearest permanently inhabited place is Alert in the Qikiqtaaluk Region, Nunavut, Canada, around the beginning of the 20th century astronomers noticed a small apparent variation of latitude, as determined for a fixed point on Earth from the observation of stars. Part of this variation could be attributed to a wandering of the Pole across the Earths surface, the wandering has several periodic components and an irregular component. The component with a period of about 435 days is identified with the eight-month wandering predicted by Euler and is now called the Chandler wobble after its discoverer and it is desirable to tie the system of Earth coordinates to fixed landforms. Of course, given plate tectonics and isostasy, there is no system in all geographic features are fixed. Yet the International Earth Rotation and Reference Systems Service and the International Astronomical Union have defined a framework called the International Terrestrial Reference System. As early as the 16th century, many eminent people correctly believed that the North Pole was in a sea and it was therefore hoped that passage could be found through ice floes at favorable times of the year. Several expeditions set out to find the way, generally with whaling ships, one of the earliest expeditions to set out with the explicit intention of reaching the North Pole was that of British naval officer William Edward Parry, who in 1827 reached latitude 82°45′ North. In 1871 the Polaris expedition, a US attempt on the Pole led by Charles Francis Hall, another British Royal Navy attempt on the pole, part of the British Arctic Expedition, by Commander Albert H. Markham reached a then-record 83°2026 North in May 1876 before turning back. An 1879–1881 expedition commanded by US naval officer George W. DeLong ended tragically when their ship, over half the crew, including DeLong, were lost
North Pole
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North Pole scenery
North Pole
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An Azimuthal projection showing the Arctic Ocean and the North Pole. The map also shows the 75th parallel north and 60th parallel north.
North Pole
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Gerardus Mercator 's map of the North Pole from 1595
North Pole
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C.G. Zorgdragers map of the North Pole from 1720
12.
South Pole
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The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earths axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth, situated on the continent of Antarctica, it is the site of the United States Amundsen–Scott South Pole Station, which was established in 1956 and has been permanently staffed since that year. The Geographic South Pole should not be confused with the South Magnetic Pole, the South Pole is at the center of the Southern Hemisphere. For most purposes, the Geographic South Pole is defined as the point of the two points where the Earths axis of rotation intersects its surface. However, the Earths axis of rotation is actually subject to very small wobbles, the geographic coordinates of the South Pole are usually given simply as 90°S, since its longitude is geometrically undefined and irrelevant. When a longitude is desired, it may be given as 0°, at the South Pole, all directions face north. For this reason, directions at the Pole are given relative to grid north, along tight latitude circles, clockwise is east, and counterclockwise is west, opposite to the North Pole. The Geographic South Pole is located on the continent of Antarctica. It sits atop a featureless, barren, windswept and icy plateau at an altitude of 2,835 metres above sea level, and is located about 1,300 km from the nearest open sea at Bay of Whales. The ice is estimated to be about 2,700 metres thick at the Pole, the polar ice sheet is moving at a rate of roughly 10 metres per year in a direction between 37° and 40° west of grid north, down towards the Weddell Sea. Therefore, the position of the station and other artificial features relative to the geographic pole gradually shift over time. The Geographic South Pole is marked by a stake in the ice alongside a small sign, these are repositioned each year in a ceremony on New Years Day to compensate for the movement of the ice. The sign records the respective dates that Roald Amundsen and Robert F. Scott reached the Pole, followed by a quotation from each man. A new marker stake is designed and fabricated each year by staff at the site, the Ceremonial South Pole is an area set aside for photo opportunities at the South Pole Station. It is located around 180 metres from the Geographic South Pole, Amundsens Tent, The tent was erected by the Norwegian expedition led by Roald Amundsen on its arrival on 14 December 1911. It is currently buried beneath the snow and ice in the vicinity of the Pole and it has been designated a Historic Site or Monument, following a proposal by Norway to the Antarctic Treaty Consultative Meeting. In 1820, several expeditions claimed to have been the first to have sighted Antarctica, with the very first being the Russian expedition led by Faddey Bellingshausen and Mikhail Lazarev. The first landing was probably just over a year later when American Captain John Davis, the basic geography of the Antarctic coastline was not understood until the mid-to-late 19th century
South Pole
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The Geographic South Pole. (The flag used on the flagpole is interchangeable.)
South Pole
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1. South Geographic Pole 2. South Magnetic Pole (2007) 3. South Geomagnetic Pole (2005) 4. South Pole of Inaccessibility
South Pole
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The Ceremonial South Pole in 1998. (Background structures shown have since been replaced or altered.)
South Pole
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Argentine soldiers saluting the flag after erecting the pole in 1965
13.
Arctic Circle
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The Arctic Circle is the most northerly of the abstract five major circles of latitude as shown on maps of the Earth. The region north of this circle is known as the Arctic, the position of the Arctic Circle is not fixed, as of 7 April 2017, it runs 66°33′46. 6″ north of the Equator. Its latitude depends on the Earths axial tilt, which fluctuates within a margin of 2° over a 40, 000-year period, consequently, the Arctic Circle is currently drifting northwards at a speed of about 15 m per year. The word arctic comes from the Greek word ἀρκτικός and that from the word ἄρκτος, directly on the Arctic Circle these events occur, in principle, exactly once per year, at the June and December solstices, respectively. That is true at sea level, those limits increase with elevation above sea level, tens of thousands of years ago, waves of people migrated from eastern Siberia across the Bering Strait into North America to settle. Much later, in the period, there has been migration into some Arctic areas by Europeans. The largest communities north of the Arctic Circle are situated in Russia and Norway, Murmansk, Norilsk, Tromsø, rovaniemi in Finland is the largest settlement in the immediate vicinity of the Arctic Circle, lying slightly south of the line. In contrast, the largest North American community north of the Arctic Circle, of the Canadian and United States Arctic communities, Barrow, Alaska is the largest settlement with about 4,000 inhabitants. The Arctic Circle is roughly 16,000 kilometres, the area north of the Circle is about 20,000,000 km2 and covers roughly 4% of Earths surface. The Arctic Circle passes through the Arctic Ocean, the Scandinavian Peninsula, North Asia, Northern America, the land within the Arctic Circle is divided among eight countries, Norway, Sweden, Finland, Russia, the United States, Canada, Denmark, and Iceland. In the interior, summers can be warm, while winters are extremely cold
Arctic Circle
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A sign along the Dalton Highway marking the location of the Arctic Circle in Alaska.
Arctic Circle
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Map of the Arctic, with the Arctic Circle in blue and the July 10 °C mean isotherm in red
Arctic Circle
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Arctic Circle line in Rovaniemi, Finland
Arctic Circle
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Aurora Borealis above Arctic Circle sign along the Dempster Highway in Yukon, Canada
14.
Tropic of Capricorn
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The Tropic of Capricorn is the circle of latitude that contains the subsolar point on the December solstice. It is thus the southernmost latitude where the Sun can be directly overhead and its northern equivalent is the Tropic of Cancer. The Tropic of Capricorn is one of the five major circles of latitude that mark maps of the Earth. As of 3 April 2017, its latitude is 23°26′13. 4″ south of the equator, the Tropic of Capricorn is the dividing line between the Southern Temperate Zone to the south and the tropics to the north. The northern hemisphere equivalent of the Tropic of Capricorn is the Tropic of Cancer, the position of the Tropic of Capricorn is not fixed, but rather it varies in a complex manner over time, see under circles of latitude for information. In southern Africa, where rainfall is more reliable, farming is possible, vegetation here is almost non-existent, though on the eastern slopes of the Andes rainfall is adequate for rainfed agriculture. In modern times the sun appears in the constellation Sagittarius during this time, the change is due to precession of the equinoxes. The word tropic itself comes from the Greek trope, meaning turn, change in direction or circumstances, referring to the fact that the sun appears to turn back at the solstices
Tropic of Capricorn
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Tropic of Capricorn in 1794 Dunn Map of the World
Tropic of Capricorn
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World map showing the Tropic of Capricorn
Tropic of Capricorn
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Monument marking the Tropic of Capricorn just north of Antofagasta, Chile
Tropic of Capricorn
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Longreach, Queensland, Australia
15.
Antarctic Circle
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The Antarctic Circle is the most southerly of the five major circles of latitude that mark maps of the Earth. The region south of this circle is known as the Antarctic, the position of the Antarctic Circle is not fixed, as of 9 April 2017, it runs 66°33′46. 6″ south of the Equator. Its latitude depends on the Earths axial tilt, which fluctuates within a margin of 2° over a 40, 000-year period, consequently, the Antarctic Circle is currently drifting southwards at a speed of about 15 m per year. Directly on the Antarctic Circle these events occur, in principle, exactly once per year, at the December and June solstices, respectively. That is true at sea level, those limits increase with elevation above sea level, in previous centuries some semi-permanent whaling stations were established on the continent, and some whalers would live there for a year or more. At least three children have been born in Antarctica, albeit in stations north of the Antarctic Circle, the Antarctic Circle is roughly 17,662 kilometres long. The area south of the Circle is about 20,000,000 km2, the continent of Antarctica covers much of the area within the Antarctic Circle
Antarctic Circle
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An iceberg near the Antarctic Circle north of Detaille Island
Antarctic Circle
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Map of the Antarctic with the Antarctic Circle in blue.
16.
Ecliptic
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The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. The ecliptic is actually the apparent path of the Sun throughout the course of a year, because Earth takes one year to orbit the Sun, the apparent position of the Sun also takes the same length of time to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day, again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, for example, the Sun is north of the celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time, if the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes. The Sun, in its apparent motion along the ecliptic, crosses the equator at these points, one from south to north. The crossing from south to north is known as the equinox, also known as the first point of Aries. The crossing from north to south is the equinox or descending node. Likewise, the ecliptic itself is not fixed, the gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earths orbit, and hence of the ecliptic, known as planetary precession. The combined action of two motions is called general precession, and changes the position of the equinoxes by about 50 arc seconds per year. Once again, this is a simplification, periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earths axis, and hence the celestial equator, known as nutation. Obliquity of the ecliptic is the used by astronomers for the inclination of Earths equator with respect to the ecliptic. It is about 23. 4° and is currently decreasing 0.013 degrees per hundred years due to planetary perturbations, the angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. From 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated, jPLs fundamental ephemerides have been continually updated. J. Laskar computed an expression to order T10 good to 0″. 04/1000 years over 10,000 years, all of these expressions are for the mean obliquity, that is, without the nutation of the equator included
Ecliptic
Ecliptic
17.
Axial tilt
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In astronomy, axial tilt, also known as obliquity, is the angle between an objects rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. At an obliquity of zero, the two axes point in the direction, i. e. the rotational axis is perpendicular to the orbital plane. Over the course of an orbit, the obliquity usually does not change considerably, and this causes one pole to be directed more toward the Sun on one side of the orbit, and the other pole on the other side — the cause of the seasons on the Earth. Earths obliquity oscillates between 22.1 and 24.5 degrees on a 41, 000-year cycle, the mean obliquity is currently 23°26′13. 4″. There are two methods of specifying tilt. The IAU also uses the rule to define a positive pole for the purpose of determining orientation. Using this convention, Venus is tilted 177° and it is denoted by the Greek letter ε. Earth currently has a tilt of about 23. 4°. This value remains about the relative to a stationary orbital plane throughout the cycles of axial precession. But the ecliptic due to planetary perturbations, and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about 47″ per century, Earths obliquity may have been reasonably accurately measured as early as 1100 BC in India and China. The ancient Greeks had good measurements of the obliquity since about 350 BC, about 830 AD, the Caliph Al-Mamun of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years. It was widely believed, during the Middle Ages, that both precession and Earths obliquity oscillated around a value, with a period of 672 years. Earths axis remains tilted in the direction with reference to the background stars throughout a year. This means that one pole will be directed away from the Sun at one side of the orbit and this is the cause of Earths seasons. Summer occurs in the Northern hemisphere when the pole is directed toward the Sun. Variations in Earths axial tilt can influence the seasons and is likely a factor in climate change. The exact angular value of the obliquity is found by observation of the motions of Earth, from 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac
Axial tilt
Axial tilt
18.
Cross section (geometry)
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In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans
Cross section (geometry)
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Pinus taeda cross section showing annual rings, Cheraw, South Carolina.
19.
Tropics
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The tropics are a region of the Earth surrounding the equator. The tropics are also referred to as the zone and the torrid zone. The tropics include all the areas on the Earth where the Sun is at a point directly overhead at least once during the solar year. The tropics are distinguished from the climatic and biomatic regions of Earth, which are the middle latitudes. Tropical is sometimes used in a sense for a tropical climate to mean warm to hot and moist year-round. Many tropical areas have a dry and wet season, the wet season, rainy season or green season, is the time of year, ranging from one or more months, when most of the average annual rainfall in a region falls. Areas with wet seasons are disseminated across portions of the tropics and subtropics, under the Köppen climate classification, for tropical climates, a wet season month is defined as a month where average precipitation is 60 millimetres or more. Tropical rainforests technically do not have dry or wet seasons, since their rainfall is distributed through the year. When the wet season occurs during the season, or summer, precipitation falls mainly during the late afternoon. The wet season is a time when air quality improves, freshwater quality improves and vegetation grows significantly, floods cause rivers to overflow their banks, and some animals to retreat to higher ground. Soil nutrients diminish and erosion increases, the incidence of malaria increases in areas where the rainy season coincides with high temperatures. Animals have adaptation and survival strategies for the wetter regime, unfortunately, the previous dry season leads to food shortages into the wet season, as the crops have yet to mature. Regions within the tropics may well not have a tropical climate, there are alpine tundra and snow-capped peaks, including Mauna Kea, Mount Kilimanjaro, and the Andes as far south as the northernmost parts of Chile and Argentina. Under the Köppen climate classification, much of the area within the tropics is classed not as tropical but as dry including the Sahara Desert. Tropical plants and animals are those native to the tropics. Tropical ecosystems may consist of rainforests, dry forests, spiny forests, desert. There are often significant areas of biodiversity, and species present, particularly in rainforests. In biogeography, the tropics are divided into Paleotropics and Neotropics, together, they are sometimes referred to as the Pantropic
Tropics
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Tropical sunset over the sea in Kota Kinabalu, Malaysia
Tropics
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World map with the intertropical zone highlighted in red
Tropics
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Coconut palms in the warm, tropical climate of Northern Brazil
Tropics
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Tropical forest near Fonds-Saint-Denis, Martinique
20.
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
Transverse Mercator projection
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A transverse Mercator projection
21.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
Isaac Newton
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Portrait of Isaac Newton in 1689 (age 46) by Godfrey Kneller
Isaac Newton
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Newton in a 1702 portrait by Godfrey Kneller
Isaac Newton
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Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)
Isaac Newton
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Replica of Newton's second Reflecting telescope that he presented to the Royal Society in 1672
22.
GPS
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The Global Positioning System is a space-based radionavigation system owned by the United States government and operated by the United States Air Force. The GPS system operates independently of any telephonic or internet reception, the GPS system provides critical positioning capabilities to military, civil, and commercial users around the world. The United States government created the system, maintains it, however, the US government can selectively deny access to the system, as happened to the Indian military in 1999 during the Kargil War. The U. S. Department of Defense developed the system and it became fully operational in 1995. Roger L. Easton of the Naval Research Laboratory, Ivan A, getting of The Aerospace Corporation, and Bradford Parkinson of the Applied Physics Laboratory are credited with inventing it. Announcements from Vice President Al Gore and the White House in 1998 initiated these changes, in 2000, the U. S. Congress authorized the modernization effort, GPS III. In addition to GPS, other systems are in use or under development, mainly because of a denial of access. The Russian Global Navigation Satellite System was developed contemporaneously with GPS, GLONASS can be added to GPS devices, making more satellites available and enabling positions to be fixed more quickly and accurately, to within two meters. There are also the European Union Galileo positioning system and Chinas BeiDou Navigation Satellite System, special and general relativity predict that the clocks on the GPS satellites would be seen by the Earths observers to run 38 microseconds faster per day than the clocks on the Earth. The GPS calculated positions would quickly drift into error, accumulating to 10 kilometers per day, the relativistic time effect of the GPS clocks running faster than the clocks on earth was corrected for in the design of GPS. The Soviet Union launched the first man-made satellite, Sputnik 1, two American physicists, William Guier and George Weiffenbach, at Johns Hopkinss Applied Physics Laboratory, decided to monitor Sputniks radio transmissions. Within hours they realized that, because of the Doppler effect, the Director of the APL gave them access to their UNIVAC to do the heavy calculations required. The next spring, Frank McClure, the deputy director of the APL, asked Guier and Weiffenbach to investigate the inverse problem — pinpointing the users location and this led them and APL to develop the TRANSIT system. In 1959, ARPA also played a role in TRANSIT, the first satellite navigation system, TRANSIT, used by the United States Navy, was first successfully tested in 1960. It used a constellation of five satellites and could provide a navigational fix approximately once per hour, in 1967, the U. S. Navy developed the Timation satellite, which proved the feasibility of placing accurate clocks in space, a technology required by GPS. In the 1970s, the ground-based OMEGA navigation system, based on comparison of signal transmission from pairs of stations. Limitations of these systems drove the need for a more universal navigation solution with greater accuracy, during the Cold War arms race, the nuclear threat to the existence of the United States was the one need that did justify this cost in the view of the United States Congress. This deterrent effect is why GPS was funded and it is also the reason for the ultra secrecy at that time
GPS
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Artist's conception of GPS Block II-F satellite in Earth orbit.
GPS
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Geodesy
GPS
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Civilian GPS receivers (" GPS navigation device ") in a marine application.
GPS
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Automotive navigation system in a taxicab.
23.
Datum (geodesy)
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A geodetic datum or geodetic system is a coordinate system, and a set of reference points, used to locate places on the Earth. An approximate definition of sea level is the datum WGS84, other datums are defined for other areas or at other times, ED50 was defined in 1950 over Europe and differs from WGS84 by a few hundred meters depending on where in Europe you look. Mars has no oceans and so no sea level, but at least two martian datums have been used to locate places there. Datums are used in geodesy, navigation, and surveying by cartographers, each starts with an ellipsoid, and then defines latitude, longitude and altitude coordinates. One or more locations on the Earths surface are chosen as anchor base-points, the difference in co-ordinates between datums is commonly referred to as datum shift. The datum shift between two particular datums can vary from one place to another within one country or region, the North Pole, South Pole and Equator will be in different positions on different datums, so True North will be slightly different. Different datums use different interpolations for the shape and size of the Earth. Because the Earth is an ellipsoid, localised datums can give a more accurate representation of the area of coverage than WGS84. OSGB36, for example, is an approximation to the geoid covering the British Isles than the global WGS84 ellipsoid. However, as the benefits of a global system outweigh the greater accuracy, horizontal datums are used for describing a point on the Earths surface, in latitude and longitude or another coordinate system. Vertical datums measure elevations or depths, in surveying and geodesy, a datum is a reference system or an approximation of the Earths surface against which positional measurements are made for computing locations. Horizontal datums are used for describing a point on the Earths surface, vertical datums are used to measure elevations or underwater depths. The horizontal datum is the used to measure positions on the Earth. A specific point on the Earth can have different coordinates. There are hundreds of local horizontal datums around the world, usually referenced to some convenient local reference point, contemporary datums, based on increasingly accurate measurements of the shape of the Earth, are intended to cover larger areas. The WGS84 datum, which is almost identical to the NAD83 datum used in North America, a vertical datum is used as a reference point for elevations of surfaces and features on the Earth including terrain, bathymetry, water levels, and man-made structures. Vertical datums are either, tidal, based on sea levels, gravimetric, based on a geoid, or geodetic, for the purpose of measuring the height of objects on land, the usual datum used is mean sea level. This is a datum which is described as the arithmetic mean of the hourly water elevation taken over a specific 19 years cycle
Datum (geodesy)
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City of Chicago Datum Benchmark
Datum (geodesy)
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Geodesy
24.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
Ellipse
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Drawing an ellipse with two pins, a loop, and a pen
Ellipse
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An ellipse obtained as the intersection of a cone with an inclined plane.
25.
ED50
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ED50 is a geodetic datum which was defined after World War II for the international connection of geodetic networks. This led to the setting up of ED50 as a consistent mapping datum for much of Western Europe and it was, and still is, used in much of Western Europe apart from Great Britain, Ireland, Sweden and Switzerland, which have their own datums. It used the International Ellipsoid of 1924 and that spheroid was an early attempt to model the whole Earth and was widely used around the world until the 1980s when GRS80 and WGS84 were established. Many national coordinate systems of Gauss–Krüger are defined by ED50 and oriented by means of geodetic astronomy, up to now it has been used in databases of gravity fields, cadastre, small surveying networks in Europe and America, and by some developing countries with no modern baselines. ED50 was also part of the fundamentals of the NATO coordinates up to the 1980s, the adjustments for later versions of the datum used the Munich Frauenkirche as starting point. The longitude and latitude lines on the two datums are the same in the Archangel region of north-west Russia, as one moves westwards across Europe, the longitude lines on ED50 gradually become further west than their WGS84 equivalents, and are around 100 metres west in Spain and Portugal. Moving southwards, the lines on ED50 gradually become further south than the WGS84 lines. This means that if the lines are further west, the value of any given point becomes more easterly. Similarly, if the lines are south, the values become northerly, the datum shift for the Universal Transverse Mercator grid is different. It should be noted that UTM northings are measured from the Equator, the French datum RGF93 The Great Britain datum OSGB36. For Ireland, the Irish Transverse Mercator
ED50
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Geodesy
26.
Length of a degree of longitude
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Longitude, is a geographic coordinate that specifies the east-west position of a point on the Earths surface. It is a measurement, usually expressed in degrees and denoted by the Greek letter lambda. Meridians connect points with the same longitude, by convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of zero degrees longitude. The longitude of places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward. Specifically, it is the angle between a plane containing the Prime Meridian and a plane containing the North Pole, South Pole and the location in question. A locations north–south position along a meridian is given by its latitude, everywhere on Earth the vertical north–south plane would contain the Earths axis. But the Earth is not homogeneous, and has mountains—which have gravity, the vertical north–south plane still intersects the plane of the Greenwich meridian at some angle, that angle is the astronomical longitude, calculated from star observations. The measurement of longitude is important both to cartography and for ocean navigation, mariners and explorers for most of history struggled to determine longitude. Finding a method of determining longitude took centuries, resulting in the history of recording the effort of some of the greatest scientific minds. Latitude was calculated by observing with quadrant or astrolabe the altitude of the sun or of charted stars above the horizon, I compared my observations with an almanac. I found that. at midnight Marss position was three and a half degrees to the east, by comparing the positions of the moon and Mars with their anticipated positions, Vespucci was able to crudely deduce his longitude. But this method had several limitations, First, it required the occurrence of an astronomical event. One needed also to know the time, which was difficult to ascertain in foreign lands. Finally, it required a stable viewing platform, rendering the technique useless on the deck of a ship at sea. In 1714 the British government passed the Longitude Act which offered financial rewards to the first person to demonstrate a practical method for determining the longitude of a ship at sea. These rewards motivated many to search for a solution, though the Board of Longitude rewarded John Harrison for his marine chronometer in 1773, chronometers remained very expensive and the lunar distance method continued to be used for decades. Finally, the combination of the availability of chronometers and wireless telegraph time signals put an end to the use of lunars in the 20th century. Unlike latitude, which has the equator as a starting position
Length of a degree of longitude
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John Harrison solved the greatest problem of his day.
Length of a degree of longitude
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The equator divides the planet into a Northern Hemisphere and a Southern Hemisphere, and has a latitude of 0°.
Length of a degree of longitude
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Key concepts
27.
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
Equirectangular projection
–
Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).
28.
Map projection
–
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
Map projection
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A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's Geography and using his second map projection
Map projection
–
Tobler hyperelliptical
Map projection
–
Mollweide
Map projection
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Goode homolosine
29.
Albers projection
–
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
Albers projection
–
Albers projection of the world with standard parallels 20°N and 50°N.
30.
Conformal map
–
In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
Conformal map
–
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
31.
Gudermannian function
–
The Gudermannian function, named after Christoph Gudermann, relates the circular functions and hyperbolic functions without explicitly using complex numbers. Some related formula, such as arccot , doesnt quite work as definition. Sin = tanh x, csc = coth x, cos = sech x, sec = cosh x, tan = sinh x, cot = csch x, tan = tanh . D d x gd x = sech x, d d x gd −1 x = sec x, the function was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions. Gudermann had published articles in Crelles Journal that were collected in Theorie der potenzial- oder cyklisch-hyperbolischen Functionen, the notation gd was introduced by Cayley where he starts by calling gd. The Gudermannian may be used in the definition of the transverse Mercator projection, the Gudermannian appears in a non-periodic solution of the inverted pendulum. The Gudermannian also appears in a moving mirror solution of the dynamical Casimir effect, hyperbolic secant distribution Mercator projection Tangent half-angle formula Tractrix Trigonometric identity
Gudermannian function
–
Graph of the Gudermannian function
32.
Geographic coordinate conversion
–
In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion comprises a number of different types of conversion, format change of coordinates, conversion of coordinate systems. Geographic coordinate conversion has applications in cartography, surveying, navigation, in geodesy, geographic coordinate conversion is defined as translation among different coordinate formats or map projections all referenced to the same geodetic datum. A geographic coordinate transformation is a translation among different geodetic datums, both geographic coordinate conversion and transformation will be considered in this article. This article assumes readers are already familiar with the content in the geographic coordinate system. Informally, specifying a location usually means giving the locations latitude and longitude. A coordinate system conversion is a conversion from one system to another. Common conversion tasks include conversion between geodetic and ECEF coordinates and conversion from one type of map projection to another, the normal N is the distance from the surface to the Z-axis along the ellipsoid normal. The orthogonality of the coordinates is confirmed via differentiation, =, =, there are several methods that solve the equation, two are shown. The following Bowrings irrational geodetic-latitude equation is efficient to be solved by Newton–Raphson iteration method, κ −1 − e 2 a κ p 2 + z 2 κ2 =0, where κ = p z tan ϕ. The height is calculated as, h = e −2 p 2 + z 2 κ2, κ0 = −1, the constant κ0 is a good starter value for the iteration when h ≈0. Bowring showed that the single iteration produces a sufficiently accurate solution and he used extra trigonometric functions in his original formulation. A number of techniques and algorithms are available but the most accurate according to Zhu, is the following 15 step procedure summarised by Kaplan. If a radar is located at and an aircraft at then the vector pointing from the radar to the aircraft in the ENU frame is = Note, a prior version of this page showed use of the geocentric latitude. The geocentric latitude is not the appropriate up direction for the tangent plane. The geocentric and geodetic longitude have the same value and this is true for the Earth and other similar shaped planets because their latitude lines can be considered in much more degree perfect circles when compared to their longitude lines. Tan λ = Y r X r Note, Unambiguous determination of ϕ and λ requires knowledge of which quadrant the coordinates lie in. The formulas involved can be complex and in cases, such as in the ECEF to geodetic conversion above
Geographic coordinate conversion
–
The length PQ is called Normal (). The length IQ is equal to. R =.
33.
Vertical line
–
The vertical bar is a computer character and glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings, Sheffer stroke, verti-bar, vbar, stick, broken bar, vertical line, vertical slash, bar, glidus, obelisk, or pipe. The vertical bar is used as a symbol in absolute value, | x |, read the absolute value of x. set-builder notation. Often a colon, is used instead of a vertical bar, sometimes a vertical bar following a function, with sub- and super-script limits a and b is used when evaluating definite integrals to mean f from a to b, or f-f. A vertical bar can be used to separate variables from fixed parameters in a function, examples, | ψ ⟩, The quantum physical state ψ. ⟨ ψ |, The dual state corresponding to the state above, ⟨ ψ | ρ ⟩, The inner product of states ψ and ρ. A pipe is a communication mechanism originating in Unix, which allows the output of one process to be used as input to another. In this way, a series of commands can be piped together, in most Unix shells, this is represented by the vertical bar character. For example, grep -i blair filename. log | more where the output from the process is piped to the more process. The same pipe feature is found in later versions of DOS. This usage has led to the character itself being called pipe, in many programming languages, the vertical bar is used to designate the logic operation or, either bitwise or or logical or. Specifically, in C and other languages following C syntax conventions, such as C++, Perl, Java and C#, since the character was originally not available in all code pages and keyboard layouts, ANSI C can transcribe it in form of the trigraph. Which, outside string literals, is equivalent to the | character, in regular expression syntax, the vertical bar again indicates logical or. For example, the Unix command grep -E fu|bar matches lines containing fu or bar, the double vertical bar operator || denotes string concatenation in PL/I, standard ANSI SQL, and theoretical computer science. Although not as common as commas or tabs, the bar can be used as a delimiter in a flat file. Examples of a standard data format are LEDES 1998B and HL7. It is frequently used because vertical bars are typically uncommon in the data itself, similarly, the vertical bar may see use as a delimiter for regular expression operations. This is useful when the expression contains instances of the more common forward slash delimiter
Vertical line
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The code point 124 (7C hexadecimal) is occupied by a broken bar in a dot matrix printer of the late 1980s, which apparently lacks a solid vertical bar. Due to this, broken bar is also used for vertical line approximation. See the full picture (3,000 × 2,500 pixels).
34.
Plumb line
–
A plumb bob, or plummet, is a weight, usually with a pointed tip on the bottom, suspended from a string and used as a vertical reference line, or plumb-line. It is essentially the equivalent of a water level. The instrument has been used since at least the time of ancient Egypt to ensure that constructions are plumb and it is also used in surveying, to establish the nadir with respect to gravity of a point in space. It is used with a variety of instruments to set the instrument exactly over a survey marker or to transcribe positions onto the ground for placing a marker. The plumb in plumb-bob comes from the fact that tools were originally made of lead. The adjective plumb developed by extension, as did the noun aplomb, until the modern age, plumb-bobs were used on most tall structures to provide vertical datum lines for the building measurements. A section of the scaffolding would hold a line, which was centered over a datum mark on the floor. As the building proceeded upward, the line would also be taken higher. Many cathedral spires, domes and towers still have brass datum marks inlaid into their floors, which signify the center of the structure above. Although a plumb-bob and line alone can determine only a vertical if they are mounted on a suitable scale, the early skyscrapers used heavy plumb-bobs, hung on wire in their elevator shafts. A plumb bob may be in a container of water, molasses, very viscous oils or other liquids to dampen any swinging movement, functioning as a shock absorber. Students of figure drawing will also use of a plumb line to find the vertical axis through the center of gravity of their subject. The device used may be purpose-made plumb lines, or simply makeshift devices made from a piece of string and this plumb line is important for lining up anatomical geometries and visualizing the subjects center of balance. Bob Centre of mass – used to find the centre of mass on a 2D shape which has uniform density Chalk line Vertical direction 60 oz. Plumb Bob
Plumb line
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A plumb-bob
Plumb line
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Plumb-bob with scale as an inclinometer
35.
Astronomer
–
An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
Astronomer
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The Astronomer by Johannes Vermeer
Astronomer
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Galileo is often referred to as the Father of modern astronomy
Astronomer
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Guy Consolmagno (Vatikan observatory), analyzing a meteorite, 2014
Astronomer
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Emily Lakdawalla at the Planetary Conference 2013
36.
Celestial equator
–
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earths equator. In other words, it is a projection of the terrestrial equator out into space, as a result of the Earths axial tilt, the celestial equator is inclined by 23. 4° with respect to the ecliptic plane. An observer standing on the Earths equator visualizes the celestial equator as a semicircle passing directly overhead through the zenith, as the observer moves north, the celestial equator tilts towards the opposite horizon. Celestial objects near the equator are visible worldwide, but they culminate the highest in the sky in the tropics. The celestial equator currently passes through these constellations, Celestial bodies other than Earth also have similarly defined celestial equators, Celestial pole Celestial sphere Declination Equatorial coordinate system
Celestial equator
–
The celestial equator is inclined by 23.4° to the ecliptic plane. The image shows the relations between Earth's axial tilt (or obliquity), rotation axis and plane of orbit.
37.
Equatorial coordinates
–
The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. The origin at the center of the Earth means the coordinates are geocentric, a right-handed convention means that coordinates are positive toward the north and toward the east in the fundamental plane. This description of the orientation of the frame is somewhat simplified. A slow motion of Earths axis, precession, causes a slow, continuous turning of the system westward about the poles of the ecliptic. Superimposed on this is a motion of the ecliptic. In order to fix the primary direction, these motions necessitate the specification of the equinox of a particular date, known as an epoch. The three most commonly used are, Mean equinox of an epoch is a fixed standard direction. Mean equinox of date is the intersection of the ecliptic of date with the mean equator, commonly used in planetary orbit calculation. True equinox of date is the intersection of the ecliptic of date with the true equator and this is the actual intersection of the two planes at any particular moment, with all motions accounted for. A position in the coordinate system is thus typically specified true equinox and equator of date, mean equinox and equator of J2000.0. Note that there is no mean ecliptic, as the ecliptic is not subject to small periodic oscillations, a stars spherical coordinates are often expressed as a pair, right ascension and declination, without a distance coordinate. The direction of distant objects is the same for all observers. Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects, setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on the celestial sphere. The declination symbol δ, measures the distance of an object perpendicular to the celestial equator, positive to the north. For example, the celestial pole has a declination of +90°. The origin for declination is the equator, which is the projection of the Earths equator onto the celestial sphere. Declination is analogous to terrestrial latitude, the right ascension symbol α, measures the angular distance of an object eastward along the celestial equator from the vernal equinox to the hour circle passing through the object. The vernal equinox point is one of the two where the ecliptic intersects the celestial equator, there are = 15° in one hour of right ascension, 24h of right ascension around the entire celestial equator
Equatorial coordinates
–
As seen from above the Earth 's north pole, a star's local hour angle (LHA) for an observer near New York (red). Also depicted are the star's right ascension and Greenwich hour angle (GHA), the local mean sidereal time (LMST) and Greenwich mean sidereal time (GMST). The symbol ʏ identifies the vernal equinox direction.
38.
Ecliptic latitude
–
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the positions and orbits of Solar System objects. Because most planets, and many small Solar System bodies have orbits with small inclinations to the ecliptic, using it as the fundamental plane is convenient. The systems origin can be either the center of the Sun or the center of the Earth, its direction is towards the vernal equinox. It may be implemented in spherical coordinates or rectangular coordinates, a slow motion of Earths axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years. Superimposed on this is a motion of the ecliptic. The three most commonly used are, Mean equinox of an epoch is a fixed standard direction. Mean equinox of date is the intersection of the ecliptic of date with the mean equator, commonly used in planetary orbit calculation. True equinox of date is the intersection of the ecliptic of date with the true equator and this is the actual intersection of the two planes at any particular moment, with all motions accounted for. A position in the coordinate system is thus typically specified true equinox and ecliptic of date, mean equinox and ecliptic of J2000.0. Note that there is no mean ecliptic, as the ecliptic is not subject to small periodic oscillations, ecliptic longitude or celestial longitude measures the angular distance of an object along the ecliptic from the primary direction. Like right ascension in the coordinate system, the primary direction points from the Earth towards the Sun at the vernal equinox of the Northern Hemisphere. Because it is a system, ecliptic longitude is measured positive eastwards in the fundamental plane from 0° to 360°. Because of axial precession, the longitude of most fixed stars increases by about 50.3 arcseconds per year, or 83.8 arcminutes per century. Ecliptic latitude or celestial latitude, measures the distance of an object from the ecliptic towards the north or south ecliptic pole. For example, the ecliptic pole has a celestial latitude of +90°. Ecliptic latitude for fixed stars is not affected by precession, distance is also necessary for a complete spherical position. Different distance units are used for different objects, within the Solar System, astronomical units are used, and for objects near the Earth, Earth radii or kilometers are used. From antiquity through the 18th century, ecliptic longitude was measured using twelve zodiacal signs, each of 30° longitude
Ecliptic latitude
–
Earth-centered ecliptic coordinates as seen from outside the celestial sphere. Ecliptic longitude (red) is measured along the ecliptic from the vernal equinox. Ecliptic latitude (yellow) is measured perpendicular to the ecliptic. A full globe is shown here, although high-latitude coordinates are seldom seen except for certain comets and asteroids.
39.
Ecliptic coordinates
–
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the positions and orbits of Solar System objects. Because most planets, and many small Solar System bodies have orbits with small inclinations to the ecliptic, using it as the fundamental plane is convenient. The systems origin can be either the center of the Sun or the center of the Earth, its direction is towards the vernal equinox. It may be implemented in spherical coordinates or rectangular coordinates, a slow motion of Earths axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years. Superimposed on this is a motion of the ecliptic. The three most commonly used are, Mean equinox of an epoch is a fixed standard direction. Mean equinox of date is the intersection of the ecliptic of date with the mean equator, commonly used in planetary orbit calculation. True equinox of date is the intersection of the ecliptic of date with the true equator and this is the actual intersection of the two planes at any particular moment, with all motions accounted for. A position in the coordinate system is thus typically specified true equinox and ecliptic of date, mean equinox and ecliptic of J2000.0. Note that there is no mean ecliptic, as the ecliptic is not subject to small periodic oscillations, ecliptic longitude or celestial longitude measures the angular distance of an object along the ecliptic from the primary direction. Like right ascension in the coordinate system, the primary direction points from the Earth towards the Sun at the vernal equinox of the Northern Hemisphere. Because it is a system, ecliptic longitude is measured positive eastwards in the fundamental plane from 0° to 360°. Because of axial precession, the longitude of most fixed stars increases by about 50.3 arcseconds per year, or 83.8 arcminutes per century. Ecliptic latitude or celestial latitude, measures the distance of an object from the ecliptic towards the north or south ecliptic pole. For example, the ecliptic pole has a celestial latitude of +90°. Ecliptic latitude for fixed stars is not affected by precession, distance is also necessary for a complete spherical position. Different distance units are used for different objects, within the Solar System, astronomical units are used, and for objects near the Earth, Earth radii or kilometers are used. From antiquity through the 18th century, ecliptic longitude was measured using twelve zodiacal signs, each of 30° longitude
Ecliptic coordinates
–
Earth-centered ecliptic coordinates as seen from outside the celestial sphere. Ecliptic longitude (red) is measured along the ecliptic from the vernal equinox. Ecliptic latitude (yellow) is measured perpendicular to the ecliptic. A full globe is shown here, although high-latitude coordinates are seldom seen except for certain comets and asteroids.
40.
Celestial sphere
–
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. All objects in the sky can be thought of as projected upon the inside surface of the celestial sphere. The celestial sphere is a tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant. Because astronomical objects are at such distances, casual observation of the sky offers no information on the actual distances. All objects seem equally far away, as if fixed to the inside of a sphere of large but unknown radius, which rotates from east to west overhead while underfoot, the celestial sphere can be considered to be infinite in radius. This means any point within it, including that occupied by the observer, all parallel planes will seem to intersect the sphere in a coincident great circle. On an infinite-radius celestial sphere, all observers see the things in the same direction. For some objects, this is over-simplified, objects which are relatively near to the observer will seem to change position against the distant celestial sphere if the observer moves far enough, say, from one side of the Earth to the other. This effect, known as parallax, can be represented as an offset from a mean position. The celestial sphere can be considered to be centered at the Earths center, the Suns center, or any convenient location. Individual observers can work out their own small offsets from the mean positions, in many cases in astronomy, the offsets are insignificant. The celestial sphere can thus be thought of as a kind of astronomical shorthand, for many rough uses, this position, as seen from the Earths center, is adequate. This greatly abbreviates the amount of detail necessary in such almanacs and these concepts are important for understanding celestial coordinate systems – frameworks for measuring the positions of objects in the sky. Certain reference lines and planes on Earth, when projected onto the celestial sphere and these include the Earths equator, axis, and the Earths orbit. At their intersections with the sphere, these form the celestial equator, the north and south celestial poles. As the celestial sphere is considered infinite in radius, all observers see the celestial equator, celestial poles, from these bases, directions toward objects in the sky can be quantified by constructing celestial coordinate systems. Similar to terrestrial longitude and latitude, the coordinate system specifies positions relative to the celestial equator and celestial poles. The ecliptic coordinate system specifies positions relative to the Earths orbit, besides the equatorial and ecliptic systems, some other celestial coordinate systems, such as the galactic coordinate system, are more appropriate for particular purposes
Celestial sphere
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Celestial Sphere, 18th century. Brooklyn Museum.
Celestial sphere
–
The Earth rotating within a relatively small-diameter Earth-centered celestial sphere. Depicted here are stars (white), the ecliptic (red), and lines of right ascension and declination (green) of the equatorial coordinate system.
Celestial sphere
–
Celestial globe by Jost Bürgi (1594)
41.
Geographical distance
–
Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude and this distance is an element in solving the second geodetic problem. Common abstractions for the surface between two points are, Flat surface, Spherical surface, Ellipsoidal surface. All abstractions above ignore changes in elevation, calculation of distances which account for changes in elevation relative to the idealized surface are not discussed in this article. Distance, D, is calculated between two points, P1 and P2, the geographical coordinates of the two points, as pairs, are and, respectively. Which of the two points is designated as P1 is not important for the calculation of distance, latitude and longitude coordinates on maps are usually expressed in degrees. In the given forms of the formulae below, one or more values must be expressed in the units to obtain the correct result. Many electronic calculators allow calculations of trigonometric functions in either degrees or radians, the calculator mode must be compatible with the units used for geometric coordinates. Differences in latitude and longitude are labeled and calculated as follows and it is not important whether the result is positive or negative when used in the formulae below. Mean latitude is labeled and calculated as follows, ϕ m = ϕ1 + ϕ22. Colatitude is labeled and calculated as follows, For latitudes expressed in radians, θ = π2 − ϕ, For latitudes expressed in degrees, θ =90 ∘ − ϕ. Unless specified otherwise, the radius of the earth for the calculations below is, D = Distance between the two points, as measured along the surface of the earth and in the same units as the value used for radius unless specified otherwise. Longitude has singularities at the Poles and a discontinuity at the ±180° meridian, also, planar projections of the circles of constant latitude are highly curved near the Poles. Hence, the equations for delta latitude/longitude and mean latitude may not give the expected answer for positions near the Poles or the ±180° meridian. Consider e. g. the value of Δ λ when λ1 and λ2 are on side of the ±180° meridian. If a calculation based on latitude/longitude should be valid for all Earth positions, it should be verified that the discontinuity, another solution is to use n-vector instead of latitude/longitude, since this representation does not have discontinuities or singularities. A planar approximation for the surface of the earth may be useful over small distances, the accuracy of distance calculations using this approximation become increasingly inaccurate as, The separation between the points becomes greater, A point becomes closer to a geographic pole. The shortest distance between two points in plane is a straight line, the Pythagorean theorem is used to calculate the distance between points in a plane
Geographical distance
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Geodesy
42.
Geotagging
–
This data usually consists of latitude and longitude coordinates, though they can also include altitude, bearing, distance, accuracy data, and place names, and perhaps a time stamp. Geotagging can help find a wide variety of location-specific information from a device. For instance, someone can find images taken near a location by entering latitude and longitude coordinates into a suitable image search engine. Geotagging-enabled information services can also potentially be used to find location-based news, websites, geotagging can tell users the location of the content of a given picture or other media or the point of view, and conversely on some media platforms show media relevant to a given location. The related term geocoding refers to the process of taking non-coordinate based geographical identifiers, such as a street address, such techniques can be used together with geotagging to provide alternative search techniques. Geotagging has become a feature on several social media platforms, such as Facebook. Facebook users can geotag photos that can be added to the page of the location they are tagging, users may also use a feature that allows them to find nearby Facebook friends, by generating a list of people according to the location tracker in their mobile devices. Instagram uses a map feature allows users to geotag photos. The map layout pin points specific photos that the user has taken on a world map, there are two main options for geotagging photos, capturing GPS information at the time the photo is taken or attaching the photograph to a map after the picture is taken. In order to capture GPS data at the time the photograph is captured, most smart phones already use a GPS chip along with built-in cameras to allow users to automatically geotag photos. Others may have the GPS chip and camera but do not have internal software needed to embed the GPS information within the picture, a few digital cameras also have built-on or built-in GPS that allow for automatic geotagging. Devices use GPS, A-GPS or both, A-GPS can be faster getting an initial fix if within range of a cell phone tower, and may work better inside buildings. Traditional GPS does not need cell phone towers and uses standard GPS signals outside of urban areas, traditional GPS tends to use more battery power. Almost any digital camera can be coupled with a stand-alone GPS and post processed with photo mapping software and these data are not visible in the picture itself but are read and written by special programs and most digital cameras and modern scanners. Latitude and longitude are stored in units of degrees with decimals and this geotag information can be read by many programs, such as the cross-platform open source ExifTool. One approach is used with the orthophotos where we store coordinates of four corners, the four corners are stored using GeoTIFF or World file standards. Audio/video files can be geotagged via, metadata, audio encoding, overlay, metadata records the geospatial data in the encoded video file to be decoded for later analysis. One of the standards used with unmanned aerial vehicle is MISB Standard 0601 which allows geocoding of corner points, audio encoding involves a process of converting gps data into audio data such as modem squawk
Geotagging
–
Geotag information stamped onto a JPEG photo by the software GPStamper
Geotagging
–
Geotag information in a JPEG photo, shown by the software gThumb
Geotagging
–
Geotagger "Solmeta N2" for Nikon D5000 DSLR
43.
International Standard Book Number
–
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
International Standard Book Number
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
44.
Sourceforge
–
SourceForge is a Web-based service that offers software developers a centralized online location to control and manage free and open-source software projects. SourceForge was one of the first to offer this service free of charge to open source projects, since 2012 the website runs on Apache Allura software. As of March 2014, the SourceForge repository claimed to host more than 430,000 projects and had more than 3.7 million registered users, the domain sourceforge. net attracted at least 33 million visitors by August 2009 according to a Compete. com survey. Negative community reactions to the program led to review of the program, nonetheless. The program was cancelled by new owners BizX on February 9,2016, on May 17,2016 they announced that it would scan all projects for malware, SourceForge is a web-based source code repository. It acts as a location for free and open-source software projects. It was the first to offer service for free to open-source projects. Project developers have access to centralized storage and tools for managing projects, though it is best known for providing revision control systems such as CVS, SVN, Bazaar, Git and Mercurial. Major features include project wikis, metrics and analysis, access to a MySQL database, the vast number of users at SourceForge. net exposes prominent projects to a variety of developers and can create a positive feedback loop. As a projects activity rises, SourceForge. nets internal ranking system makes it visible to other developers through SourceForge directory. Given that many projects fail due to lack of developer support. SourceForges traditional revenue model is through advertising sales on their site. In 2006 SourceForge Inc. reported quarterly takings of US$6.5 million, in 2009 SourceForge reported a gross quarterly income of US$23 million through media and e-commerce streams. In 2011 a revenue of 20 million USD was reported for the value of the SourceForge, slashdot and freecode holdings. Since 2013 additional revenue generation schemes, such as models, were trialled. The result has in some cases been the appearance of malware bundled with SourceForge downloads, on February 9,2016, SourceForge announced they had eliminated their DevShare program practice of bundling installers with project downloads. The software running the SourceForge site was released as software in January 2000 and was later named SourceForge Alexandria. In September 2002 SourceForge was temporarily banned in China, the site was banned again in China, for about a month, in July 2008
Sourceforge
–
Screenshot of SourceForge main page in 2014
45.
180th meridian
–
The 180th meridian or antimeridian is the meridian 180° east or west of the Prime Meridian, with which it forms a great circle dividing the earth into the Western and Eastern Hemispheres. It is common to both east longitude and west longitude and it is used as the basis for the International Date Line because it for the most part passes through the open waters of the Pacific Ocean. However, the passes through Russia and Fiji as well as Antarctica. The only place where roads cross this meridian, and where there are very close to it, is in Fiji. Many geographic software libraries or data formats project the world to a rectangle and this often makes it non-trivial to do simple tasks over the 180th meridian. Some examples, The GeoJSON specification strongly suggests splitting geometries so that neither of their parts cross the antimeridian, in OpenStreetMap, areas are split at the 180th meridian. 179th meridian east 179th meridian west Prime meridian
180th meridian
–
180th meridian
180th meridian
–
180° Meridian, Taveuni, Fiji
46.
40th parallel north
–
The 40th parallel north is a circle of latitude that is 40 degrees north of the Earths equatorial plane. It crosses Europe, the Mediterranean Sea, Asia, the Pacific Ocean, North America, at this latitude the sun is visible for 15 hours,1 minute during the summer solstice and 9 hours,20 minutes during the winter solstice. On 21 June, the altitude of the sun is 73.83 degrees and 26.17 degrees on 21 December. Starting in Spain at the Prime Meridian and heading eastwards, the parallel 40° north passes through, on 30 May 1854, the Kansas–Nebraska Act created the Territory of Kansas and the Territory of Nebraska divided by the parallel 40° north. Both territories were required to determine for themselves whether to permit slavery, open conflict between free-state and pro-slavery forces in the Kansas Territory was one of the root causes of the American Civil War. The parallel 40° north formed the northern boundary of the British Colony of Maryland. A subsequent royal grant gave the Colony of Pennsylvania land north of the 40th parallel but mistakenly assumed it would intersect the Twelve Mile Circle, pennsylvanias border was thus unclear and the colony pushed for a border far south of the 40th parallel. The Mason–Dixon Line was drawn between 1763 and 1767 as the boundary between the overlapping claims of these two colonies. The parallel 40° north passes through the cities of Philadelphia, Pennsylvania and Columbus, Ohio, as well as suburbs of Indianapolis, Indiana and Denver. Baseline Road in Boulder, Colorado, traces the parallel 40° north, thistle, Utah, a ghost town since 1983, is slightly below 40° north. 39th parallel north 41st parallel north Baseline Geological Exploration of the Fortieth Parallel
40th parallel north
–
Survey marker on the Kansas/Nebraska state line
47.
50th parallel north
–
The 50th parallel north is a circle of latitude that is 50 degrees north of the Earths equatorial plane. It crosses Europe, Asia, the Pacific Ocean, North America, at this latitude the sun is visible for 16 hours,22 minutes during the summer solstice and 8 hours,4 minutes during the winter solstice. The maximum altitude of the sun on the solstice is 63.5 degrees. At this latitude, the sea surface temperature between 1982 and 2011 was about 8. 5°C. The entire island came under Soviet control after World War II, 49th parallel north 51st parallel north
50th parallel north
–
50th latitude mark in central Mainz, Germany
50th parallel north
–
50th parallel north memorial at Kharkiv, Ukraine
48.
Arctic Ocean
–
The Arctic Ocean is the smallest and shallowest of the worlds five major oceans. Alternatively, the Arctic Ocean can be seen as the northernmost part of the all-encompassing World Ocean, located mostly in the Arctic north polar region in the middle of the Northern Hemisphere, the Arctic Ocean is almost completely surrounded by Eurasia and North America. It is partly covered by sea ice throughout the year and almost completely in winter, the summer shrinking of the ice has been quoted at 50%. The US National Snow and Ice Data Center uses satellite data to provide a record of Arctic sea ice cover. The Arctic may become ice free for the first time in human history within a few years or by 2040, for much of European history, the north polar regions remained largely unexplored and their geography conjectural. He was probably describing loose sea ice known today as growlers or bergy bits, his Thule was probably Norway, early cartographers were unsure whether to draw the region around the North Pole as land or water. The makers of navigational charts, more conservative than some of the more fanciful cartographers, tended to leave the region blank and this lack of knowledge of what lay north of the shifting barrier of ice gave rise to a number of conjectures. In England and other European nations, the myth of an Open Polar Sea was persistent, john Barrow, longtime Second Secretary of the British Admiralty, promoted exploration of the region from 1818 to 1845 in search of this. In the United States in the 1850s and 1860s, the explorers Elisha Kane, even quite late in the century, the eminent authority Matthew Fontaine Maury included a description of the Open Polar Sea in his textbook The Physical Geography of the Sea. Nevertheless, as all the explorers who travelled closer and closer to the reported, the polar ice cap is quite thick. Fridtjof Nansen was the first to make a crossing of the Arctic Ocean. The first surface crossing of the ocean was led by Wally Herbert in 1969, in a dog sled expedition from Alaska to Svalbard, with air support. The first nautical transit of the pole was made in 1958 by the submarine USS Nautilus. Since 1937, Soviet and Russian manned drifting ice stations have extensively monitored the Arctic Ocean, scientific settlements were established on the drift ice and carried thousands of kilometres by ice floes. In World War II, the European region of the Arctic Ocean was heavily contested, the Arctic Ocean occupies a roughly circular basin and covers an area of about 14,056,000 km2, almost the size of Antarctica. The coastline is 45,390 km long and it is surrounded by the land masses of Eurasia, North America, Greenland, and by several islands. It is connected to the Pacific Ocean by the Bering Strait and to the Atlantic Ocean through the Greenland Sea, countries bordering the Arctic Ocean are, Russia, Norway, Iceland, Greenland, Canada and the United States. There are several ports and harbours around the Arctic Ocean In Alaska, in Canada, ships may anchor at Churchill in Manitoba, Nanisivik in Nunavut, Tuktoyaktuk or Inuvik in the Northwest territories
Arctic Ocean
–
A bathymetric / topographic of the Arctic Ocean and the surrounding lands.
Arctic Ocean
–
Emanuel Bowen 's 1780s map of the Arctic features a "Northern Ocean".
Arctic Ocean
–
Sea cover in the Arctic Ocean, showing the median, 2005 and 2007 coverage
Arctic Ocean
–
Three polar bears approach USS Honolulu near the North Pole.
49.
45th parallel south
–
The 45th parallel south is a circle of latitude that is 45 degrees south of the Earths equatorial plane. It is the line marks the theoretical halfway point between the equator and the South Pole. The true halfway point is 16.2 kilometres south of this parallel because the Earth is not a perfect sphere, unlike its northern counterpart almost all of it passes over open ocean. It crosses the Atlantic Ocean, the Indian Ocean, Australasia, at this latitude the sun is visible for 15 hours,37 minutes during the December solstice and 8 hours,46 minutes during the June solstice. Starting at the Prime Meridian and heading eastwards, the parallel 45° south passes through, 44th parallel south 46th parallel south
45th parallel south
–
Highway sign marking the 45th parallel in New Zealand.
50.
Antarctica
–
It contains the geographic South Pole and is situated in the Antarctic region of the Southern Hemisphere, almost entirely south of the Antarctic Circle, and is surrounded by the Southern Ocean. At 14,000,000 square kilometres, it is the fifth-largest continent, for comparison, Antarctica is nearly twice the size of Australia. About 98% of Antarctica is covered by ice that averages 1.9 km in thickness, Antarctica, on average, is the coldest, driest, and windiest continent, and has the highest average elevation of all the continents. Antarctica is a desert, with precipitation of only 200 mm along the coast. The temperature in Antarctica has reached −89.2 °C, though the average for the quarter is −63 °C. Anywhere from 1,000 to 5,000 people reside throughout the year at the research stations scattered across the continent. Organisms native to Antarctica include many types of algae, bacteria, fungi, plants, protista, vegetation, where it occurs, is tundra. The continent, however, remained neglected for the rest of the 19th century because of its hostile environment, lack of easily accessible resources. In 1895, the first confirmed landing was conducted by a team of Norwegians, Antarctica is a de facto condominium, governed by parties to the Antarctic Treaty System that have consulting status. Twelve countries signed the Antarctic Treaty in 1959, and thirty-eight have signed it since then, the treaty prohibits military activities and mineral mining, prohibits nuclear explosions and nuclear waste disposal, supports scientific research, and protects the continents ecozone. Ongoing experiments are conducted by more than 4,000 scientists from many nations, the name Antarctica is the romanised version of the Greek compound word ἀνταρκτική, feminine of ἀνταρκτικός, meaning opposite to the Arctic, opposite to the north. Aristotle wrote in his book Meteorology about an Antarctic region in c.350 B. C, marinus of Tyre reportedly used the name in his unpreserved world map from the 2nd century A. D. Before acquiring its present geographical connotations, the term was used for locations that could be defined as opposite to the north. For example, the short-lived French colony established in Brazil in the 16th century was called France Antarctique, the first formal use of the name Antarctica as a continental name in the 1890s is attributed to the Scottish cartographer John George Bartholomew. Antarctica has no population and there is no evidence that it was seen by humans until the 19th century. Explorer Matthew Flinders, in particular, has credited with popularising the transfer of the name Terra Australis to Australia. Cook came within about 120 km of the Antarctic coast before retreating in the face of ice in January 1773. The first confirmed sighting of Antarctica can be narrowed down to the crews of ships captained by three individuals, according to various organisations, ships captained by three men sighted Antarctica or its ice shelf in 1820, von Bellingshausen, Edward Bransfield, and Nathaniel Palmer
Antarctica
–
Adelie penguins in Antarctica
Antarctica
–
Antarctica
Antarctica
–
Painting of James Weddell 's second expedition in 1823, depicting the brig Jane and the cutter Beaufroy.
Antarctica
–
Nimrod Expedition South Pole Party (left to right): Wild, Shackleton, Marshall and Adams
51.
List of map projections
–
This list provides an overview of some of the significant or common map projections. Because there is no limit to the number of map projections. The designation deployed means popularisers/users rather than necessarily creators, pseudocylindrical, In standard presentation, these map the central meridian and parallels as straight lines. Other meridians are curves, regularly spaced along parallels, pseudoazimuthal, In standard presentation, pseudoazimuthal projections map the equator and central meridian to perpendicular, intersecting straight lines. They map parallels to complex curves bowing away from the equator, listed here after pseudocylindrical as generally similar to them in shape and purpose. Conic, In standard presentation, conic projections map meridians as straight lines, pseudoconical, In standard presentation, pseudoconical projections represent the central meridian as a straight line, other meridians as complex curves, and parallels as circular arcs. Azimuthal, In standard presentation, azimuthal projections map meridians as straight lines and parallels as complete, in any presentation, they preserve directions from the center point. This means great circles through the point are represented by straight lines on the map. Retroazimuthal, Direction to a fixed location B corresponds to the direction on the map from A to B, conformal, Preserves angles locally, implying that local shapes are not distorted. Compromise, Neither conformal nor equal-area, but a balance intended to reduce overall distortion, equidistant, All distances from one points are correct. Other equidistant properties are mentioned in the notes, gnomonic, All great circles are straight lines. Snyder, John P. Map projections, A working manual, Professional Paper 1395, Washington, D. C
List of map projections
–
Equirectangular = equidistant cylindrical = rectangular = la carte parallélogrammatique
List of map projections
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Cassini = Cassini-Soldner
List of map projections
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Mercator = Wright
List of map projections
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Gall stereographic similar to Braun
52.
Behrmann projection
–
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
Behrmann projection
–
Behrmann projection of the world
53.
Lambert cylindrical equal-area projection
–
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
Lambert cylindrical equal-area projection
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Lambert cylindrical equal-area projection of the world.
Lambert cylindrical equal-area projection
–
How the Earth is projected onto a cylinder
54.
Gall stereographic projection
–
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
Gall stereographic projection
–
Gall stereographic projection of the world. 15° graticule.
Gall stereographic projection
55.
Space-oblique Mercator projection
–
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
Space-oblique Mercator projection
–
Space-oblique Mercator projection.
56.
Web Mercator
–
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
Web Mercator
–
OpenStreetMap, like most web maps, uses the Web Mercator projection.
57.
Goode homolosine projection
–
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
Goode homolosine projection
–
Goode homolosine projection of the world.
58.
Kavrayskiy VII projection
–
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
Kavrayskiy VII projection
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Kavrayskiy VII projection of the Earth.
Kavrayskiy VII projection
–
The Kavrayskiy VII projection with Tissot's indicatrix of deformation
59.
Tobler hyperelliptical projection
–
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
Tobler hyperelliptical projection
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Tobler hyperelliptical projection of the world, α = 0, γ = 1.18314, k = 2.5
60.
Wagner VI projection
–
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
Wagner VI projection
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Wagner VI projection of the world.
61.
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
Lambert conformal conic projection
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Lambert conformal conic projection with standard parallels at 20°N and 50°N. Projection extends toward infinity southward and so has been cut off at 30°S.
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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
Werner projection
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Werner projection of the world
Werner projection
63.
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
General Perspective projection
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Vertical perspective from an altitude of 35,786 km over (0°, 90°W), corresponding to a view from geostationary orbit. 10° graticule.
General Perspective projection
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Geometric projection of the parallels of the polar Perspective projections, Vertical and Tilted.
64.
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
Gnomonic projection
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Gnomonic projection of a portion of the north hemisphere centered on the geographic North Pole
Gnomonic projection
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Great circles transform to straight lines via gnomonic projection
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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
Azimuthal equidistant projection
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Polar azimuthal equidistant projection
66.
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
Lambert azimuthal equal-area projection
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Lambert azimuthal equal-area projection of the world. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary.
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Hammer projection
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The Hammer projection is an equal-area map projection described by Ernst Hammer in 1892. Using the same 2,1 elliptical outer shape as the Mollweide projection, Hammer intended to reduce distortion in the regions of the outer meridians, visually, the Aitoff and Hammer projections are very similar. The Hammer has seen more use because of its equal-area property, the Mollweide projection is another equal-area projection of similar aspect, though with straight parallels of latitude, unlike the Hammers curved parallels. William A. Briesemeister presented a variant of the Hammer in 1953, the purpose is to present the land masses more centrally and with lower distortion. Before projecting to Hammer, John Bartholomew rotated the coordinate system to bring the 45° north parallel to the center and he called this variant the Nordic projection. List of map projections Mollweide projection Aitoff projection Table of common projections An interactive Java Applet to study the metric deformations of the Hammer–Aitoff Projection
Hammer projection
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Hammer projection of the world
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Loximuthal projection
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In cartography, the loximuthal projection is a map projection introduced by Karl Siemon in 1935, and independently by Waldo R. Tobler, who named it. It is neither an equal-area projection nor conformal, a loxodrome on the surface of the earth is a curve of constant bearing, it meets every parallel of latitude at the same angle. Suppose its bearing is θ north of east, so, for example, due east is θ =0, due north is θ = a right angle, let a loxodrome pass through the point whose longitude and latitude are both 0, call this the central point. Suppose one starts at the point and travels a certain distance in a certain direction along this loxodrome. Let f be the point in the -plane reached by going that same distance in that direction from the origin. That point f is the image of p on the map, more than one loxodrome goes from the central point to p, but there is a unique shortest one, the one that does not cross the 180° meridian on its way from the central point to p. If one were to include loxodromes crossing the 180° meridian, one would get many images of the whole earth. Using only the unique shortest loxodrome from the point to each point p gives only one copy. List of map projections Loximuthal projection
Loximuthal projection
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Loximuthal projection of the world, central point = 0°E, 30°N. 15° graticule.
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Craig retroazimuthal projection
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The Craig retroazimuthal map projection was created by James Ireland Craig in 1909. It is a cylindrical projection. As a retroazimuthal projection, it preserves directions from everywhere to one location of interest that is configured during construction of the projection. The projection is known as the Mecca projection because Craig. In such maps, Mecca is the location of interest
Craig retroazimuthal projection
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Craig retroazimuthal projection centered on Mecca
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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
Chamberlin trimetric projection
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A map of Africa using the Chamberlin trimetric projection. The three red dots indicate the selected "base" locations: (22°N, 0°), (22°N, 45°E), (22°S, 22.5°E). 10° graticule.
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Robinson projection
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The Robinson projection is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a compromise to the problem of readily showing the whole globe as a flat image. The Robinson projection was devised by Arthur H. Robinson in 1963 in response to an appeal from the Rand McNally company, Robinson published details of the projections construction in 1974. The National Geographic Society began using the Robinson projection for general world maps in 1988. In 1998 NGS abandoned the Robinson projection for that use in favor of the Winkel tripel projection, the Robinson projection is neither equal-area nor conformal, abandoning both for a compromise. The creator felt this produced an overall view than could be achieved by adhering to either. The meridians curve gently, avoiding extremes, but thereby stretch the poles into long lines instead of leaving them as points, hence, distortion close to the poles is severe, but quickly declines to moderate levels moving away from them. The straight parallels imply severe angular distortion at the high latitudes toward the edges of the map. However, at the time it was developed, the projection effectively met Rand McNallys goal to produce appealing depictions of the entire world, I decided to go about it backwards. … I started with a kind of artistic approach, I visualized the best-looking shapes and sizes. I worked with the variables until it got to the point where, if I changed one of them, then I figured out the mathematical formula to produce that effect. Most mapmakers start with the mathematics, the projection is defined by the table, The table is indexed by latitude at 5 degree intervals, intermediate values are calculated using interpolation. Robinson did not specify any particular method, but it is reported that he used Aitken interpolation himself. The PLEN column is the length of the parallel of latitude, meridians of longitude are equally spaced on each parallel of latitude. List of map projections Winkel Tripel — projection currently used by the National Geographic, a New Map Projection, Its Development and Characteristics. In, National Geographic, December 1988, pp. 911–913, flattening The Earth—2000 Years of Map Projections, The University of Chicago Press. Table of examples and properties of all projections, from radicalcartography. net An interactive Java Applet to study the metric deformations of the Robinson Projection. Numerical evaluation of the Robinson projection, from Cartography and Geographic Information Science, April,2004 by Cengizhan Ipbuker
Robinson projection
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Robinson projection of the world
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Bernard J. S. Cahill
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His Butterfly World Map, like Buckminster Fullers later Dymaxion map of 1943 and 1954, enabled all continents to be uninterrupted, and with reasonable fidelity to a globe. Cahill demonstrated this principle by also inventing a rubber-ball globe which could be flattened under a pane of glass in the Butterfly form, a variant was developed by Gene Keyes in 1975, the Cahill–Keyes projection. World map Waterman butterfly projection Octants projection Cahill–Keyes projection About Cahill Parry, David, Architects Profiles, Bernard J. S. Cahill Collection, ca. J. S. Cahill Butterfly Map Resource Page By Cahill An Account of a New Land Map of the World pp. 449–469 The first publication and exposition of the Butterfly Map, one Base Map in Place of Five Monthly Weather Review, 68/2, 1940-02, p.4,1 illus
Bernard J. S. Cahill
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From cover of 1919 pamphlet by Cahill, "The Butterfly Map", 8 p.
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Dymaxion map
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The Dymaxion map or Fuller map is a projection of a world map onto the surface of an icosahedron, which can be unfolded and flattened to two dimensions. The flat map is heavily interrupted in order to preserve shapes and sizes, the projection was invented by Buckminster Fuller. The March 1,1943 edition of Life magazine included an essay titled Life Presents R. Buckminster Fullers Dymaxion World. Fuller applied for a patent in the United States in February 1944, the patent was issued in January 1946. This version depicts the Earths continents as one island, or nearly contiguous land masses, the Dymaxion projection is intended only for representations of the entire globe. It is not a projection, whereby global data expands from the center point of a tangent facet outward to the edges. The name Dymaxion was applied by Fuller to several of his inventions, Fuller claimed that his map had several advantages over other projections for world maps. It has less distortion of size of areas, most notably when compared to the Mercator projection. Other compromise projections attempt a similar trade-off, more unusually, the Dymaxion map does not have any right way up. Fuller argued that in the universe there is no up and down, or north and south, only in, gravitational forces of the stars and planets created in, meaning towards the gravitational center, and out, meaning away from the gravitational center. He attributed the north-up-superior/south-down-inferior presentation of most other world maps to cultural bias, Fuller intended the map to be unfolded in different ways to emphasize different aspects of the world. Peeling the solid apart in a different way presents a view of the world dominated by connected oceans surrounded by land, a 1967 Jasper Johns painting, Map, depicting a Dymaxion map, hangs in the permanent collection of the Museum Ludwig in Cologne. The World Game, a simulation game in which players attempt to solve world problems, is played on a 70-by-35-foot Dymaxion map. The competition received over 300 entries from 42 countries
Dymaxion map
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Example of use illustrating early human migrations according to mitochondrial population genetics (numbers are millennia before present)
Dymaxion map
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Map of the world in a Fuller projection with Tissot's Indicatrix of deformation
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Waterman butterfly projection
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The Waterman Butterfly World Map is a map arrangement created by Steve Waterman. Waterman first published a map in this arrangement in 1996, the arrangement is an unfolding of a globe treated as a truncated octahedron, evoking the butterfly map principle first developed by Bernard J. S. Cahill in 1909. Cahill and Waterman maps can be shown in various profiles, typically linked at the north Pacific or north Atlantic oceans, as Cahill was an architect, his approach tended toward forms that could be demonstrated physically, such as by his flattenable rubber-ball map. Waterman, on the hand, derived his design from his work on close-packing of spheres. This involves connecting the sphere centers from cubic closest-packed spheres into a convex hull. These illustrate the W5 sphere cluster, W5 convex hull, to project the polyhedron to the plane, straight lines are used to define each 5 ×5 section onto this convex hull. According to annotations on modern versions of the map, the projection divides the equator equally amongst the meridians, popko notes the projection can be gnomonic too. The two methods yield similar results. Parallels of latitude are drawn as three straight-line sections in each octant, from pole to fold-line, from fold-line to longest line parallel to equator, the longest line parallel to the equator also has equal-length delineations. Waterman chose a specific Waterman polyhedron and central meridian to minimize interrupting major land masses, list of map projections Waterman polyhedron Bernard J. S. Cahill World map Rotating Waterman as globe. Real-time winds and temperature on Waterman projection, interactive Tissot indicatrix of Waterman projection. Description of Waterman polyhedra and projection, explanation of equal-line delineation for projection
Waterman butterfly projection
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Waterman sphere cluster W5
Waterman butterfly projection
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Waterman polyhedron w5
Waterman butterfly projection
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Waterman projection (Pacific centered)