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, now usually known as the Tobler hyperelliptical projection. Overview 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, which has straight, vertical meridians, with meridians that follow a particular kind of curve known as ''superellipses'' or '' Lamé curves'' or sometimes as ''hyperellipses''. A hyperellipse is described by x^k + y^k = \gamma^k, where \gamma and k are free parameters. Tobler's hyperelliptical projection is given as: :\begin &x = \lambda alpha + (1 - \alpha) \frac\\ \alpha &y = \sin \varphi + \frac \int_0^y (\gamma^k - z^k)^ dz \end w ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Tobler Hyperelliptical Projection SW
Tobler may refer to: * Tobler (name), a surname originating in Germany (and list of people with the name) * Tom Brownlees or Tobler, an Australian rules football player * Tobler hyperelliptical projection, a family of map projections * Chocolat Tobler, a chocolate more commonly known as Toblerone * Villa Tobler Villa Tober, also known as the Theater an der Winkelwiese, is a protected building in Zürich-Hottingen that comprises the mansion built in 1853, and a public park. Location The villa (or theater) is situated in Zürich-Hottingen, between K ..., a listed building in Zürich, Switzerland See also * Tobler's crow (''Euploea tobleri''), a species of nymphalid butterfly * Tobler's first law of geography, according to Waldo Tobler {{disambiguation ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Cylindrical Equal-area Projection
In cartography, the normal cylindrical equal-area projection is a family of Map projection#Normal cylindrical, normal cylindrical, equal-area projection, equal-area map projections. History The invention of the Lambert cylindrical equal-area projection is attributed to the Switzerland, Swiss mathematician Johann Heinrich Lambert in 1772. Variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios. Description The projection: * is Map_projection#Cylindrical, cylindrical, that means it has a cylindrical projection surface * is normal, that means it has a normal Map projection#Aspect, aspect * is an equal-area projection, that means any two areas in the map have the same relative size compared to their size on the sphere. The term "normal cylindrical projection" is used to refer to any projection in which Meridian (geography), meridians are mapped to equally spaced vertical lines and cir ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Mollweide Projection
400px, Mollweide projection of the world 400px, The Mollweide projection with Tissot's indicatrix of deformation The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions. The projection was first published by mathematician and astronomer Karl (or Carl) Brandan Mollweide (1774–1825) of Leipzig in 1805. It was reinvented and popularized in 1857 by Jacques Babinet, who gave it the name homalographic projection. The variation homolographic arose from frequent nineteenth-century usage in star atlases. Properties The Mollweide is a pseudocylindrical projection in which the equator is represented as ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Collignon Projection
The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. For the smallest choices of the parameters chosen for this projection, the sphere may be mapped either to a single diamond, a pair of squares, or a triangle. The projection is used in the polar areas as part of the HEALPix spherical projection, which is widely used in physical cosmology in making maps of the cosmic microwave background, in particular by the WMAP and Planck space missions. Formulae Let ''R'' be the radius of the sphere, ''φ'' the latitude, ''λ'' the longitude, and ''λ0'' the longitude of the central meridian (chosen as desired). Also, define s = \sqrt = \sqrt \sin\left(\frac - \frac\right), where the two forms are equivalent for ''φ'' in the range of possible latitudes. Then the Collignon projection is given by: :\begin x &= \fracR \left( \lambda - \lambda_0 \right) s, \\ y &= ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar to the cas ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity and finite strain theory elaborate the mathematical abstractions). Biography Lamé was born in Tours, in today's ''département'' of Indre-et-Loire. He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves or superellipses, and defined by the equation: : \left, \,\,\^n + \left, \,\,\^n = 1 where ''n'' is any positive real number. He is also known for his running time analysis of the Euclidean algorithm, marking the beginning of computational complexity theory. In 1844, using Fibonacci numbers, he proved that when finding the greatest common divisor of integers ''a'' and ''b'', the algorithm runs in no more than 5''k'' steps, where ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Superellipse
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse. In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points (x,y) on the curve that satisfy the equation\left, \frac\^n\!\! + \left, \frac\^n\! = 1,where a and b are positive numbers referred to as Semidiameter, semi-diameters or Semi-major and semi-minor axes, semi-axes of the superellipse, and n is a positive parameter that defines the shape. When n=2, the superellipse is an ordinary ellipse. For n>2, the shape is more rectangular with rounded corners, and for 00, where m either equals to or differs from ''n''. If m=n, it is the Lamé's superellipses. If m\neq n, the curve possesses more flexibility of behavior, and is better possible fit to des ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Meridian (geography)
In geography and geodesy, a meridian is the locus connecting points of equal longitude, which is the angle (in degrees or other units) east or west of a given prime meridian (currently, the IERS Reference Meridian). In other words, it is a coordinate line for longitudes, a line of longitude. The position of a point along the meridian at a given longitude is given by its latitude, measured in angular degrees north or south of the Equator. On a Mercator projection or on a Gall-Peters projection, each meridian is perpendicular to all circles of latitude. Assuming a spherical Earth, a meridian is a great semicircle on Earth's surface. Adopting instead a spheroidal or ellipsoid model of Earth, the meridian is half of a north-south great ellipse. The length of a meridian is twice the length of an Earth quadrant, equal to on a modern ellipsoid ( WGS 84). Pre-Greenwich The first prime meridian was set by Eratosthenes in 200 BC. This prime meridian was used to provide mea ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Straight Lines
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its ''endpoints''). Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. Properties In the Greek deductive geometry of Euclid's ''Elements'', a ge ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Tobler Hyperelliptical With Tissot's Indicatrices Of Distortion
Tobler may refer to: * Tobler (name), a surname originating in Germany (and list of people with the name) * Tom Brownlees or Tobler, an Australian rules football player * Tobler hyperelliptical projection, a family of map projections * Chocolat Tobler, a chocolate more commonly known as Toblerone * Villa Tobler Villa Tober, also known as the Theater an der Winkelwiese, is a protected building in Zürich-Hottingen that comprises the mansion built in 1853, and a public park. Location The villa (or theater) is situated in Zürich-Hottingen, between K ..., a listed building in Zürich, Switzerland See also * Tobler's crow (''Euploea tobleri''), a species of nymphalid butterfly * Tobler's first law of geography, according to Waldo Tobler {{disambiguation ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Latitude
In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north pole, with 0° at the Equator. Parallel (latitude), Lines of constant latitude, or ''parallels'', run east-west as circles parallel to the equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth. On its own, the term "latitude" normally refers to the ''geodetic latitude'' as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or ''Normal (geometry), normal'') to the ellipsoidal surface from the point, and the equatorial plane, plane of the equator. Background Two levels of abstraction are employed in the definitions of latitude and longitude. In the first step the physical surface i ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Circle Of Latitude
A circle of latitude or line of latitude on Earth is an abstract east–west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line. Circles of latitude are often called parallels because they are Parallel (geometry), parallel to each other; that is, planes that contain any of these circles never Intersection, intersect each other. A location's position along a circle of latitude is given by its longitude. Circles of latitude are unlike circles of longitude, which are all great circles with the centre of Earth in the middle, as the circles of latitude get smaller as the distance from the Equator increases. Their length can be calculated by a common sine or cosine function. For example, the 60th parallel north or 60th parallel south, south is half as long as the Equator (disregarding Earth's minor flattening by 0.335%), stemming from \cos(60^) = 0.5. On the Mercator projection or on the Gall-Peters projection, a circle of la ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |