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Conformal Map Projection
In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, then their images on a map with a conformal projection cross at a 39° angle. Properties A conformal projection can be defined as one that is locally conformal at every point on the Earth. Thus, every small figure on the earth is nearly similar to its image on the map. The projection preserves the ratio of two lengths in the small domain. All Tissot's indicatrices of the projections are circles. Conformal projections preserve only small figures. Large figures are distorted by even conformal projections. In a conformal projection, any small figure is similar to the image, but the ratio of similarity (scale (map), scale) varies by location, which explains the distor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can be modeled in ways that communicate spatial information effectively. The fundamental objectives of traditional cartography are to: * Set the map's agenda and select traits of the object to be mapped. This is the concern of map editing. Traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. * Represent the terrain of the mapped object on flat media. This is the concern of map projections. * Eliminate characteristics of the mapped object that are not relevant to the map's purpose. This is the concern of generalization. * Reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization. * Orchestrate the elements of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Conformal Conic Projection
A Lambert conformal conic projection (LCC) 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'' (Notes and Comments on the Composition of Terrestrial and Celestial Maps). 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. That parallel is called the ''reference parallel'' or ''standard parallel''. By scaling the resulting 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 regi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lee Conformal Projection
The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection that projects the globe onto a tetrahedron using Dixon elliptic functions. It is conformal everywhere except for the four singularities at the vertices of the polyhedron. Because of the nature of polyhedra, this map projection can be tessellated infinitely in the plane. It was developed by L. P. Lee in 1965. Coordinates from a spherical datum can be transformed into Lee conformal projection coordinates with the following formulas, where is the longitude and the latitude: : 2 \operatornamew\,\operatornamew = 2^\exp(i\lambda) \tan\bigl(\tfrac14\pi - \tfrac12\phi\bigr) where : w = x + y i and sm and cm are Dixon elliptic functions. Since there is no elementary expression for these functions, Lee suggests using the 28th degree MacLaurin series. See also * List of map projections * AuthaGraph projection, another tetrahedral projection, 1999 * Dymaxion map, 1943 * Peirce quincuncial proj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce Quincuncial Projection
The Peirce quincuncial projection is the conformal map projection from the sphere to an unfolded square dihedron, developed by Charles Sanders Peirce in 1879. Each octant projects onto an isosceles right triangle, and these are arranged into a square. The name ''quincuncial'' refers to this arrangement: the north pole at the center and quarters of the south pole in the corners form a quincunx pattern like the pips on the ''five'' face of a traditional die. The projection has the distinctive property that it forms a seamless square tiling of the plane, conformal except at four singular points along the equator. Typically the projection is square and oriented such that the north pole lies at the center, but an oblique aspect in a rectangle was proposed by Émile Guyou in 1887, and a transverse aspect was proposed by Oscar Adams in 1925. The projection has seen use in digital photography for portraying spherical panoramas. History The maturation of complex analysis led to gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and modular forms. Definition A meromorphic function is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore also called ''doubly periodic''. Period lattice and fundamental domain Iff is an elliptic function with periods \omega_1,\omega_2 it also holds that : f(z+\gamma)=f(z) for every line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' − ''bc'' ≠ 0. Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics. Möbius tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Littrow Projection
The Littrow projection is a map projection developed by Joseph Johann von Littrow in 1833. It is the only conformal, retroazimuthal map projection. As a retroazimuthal projection, the Littrow shows directions, or azimuths, correctly from any point to the center of the map. Patrick Weir of the British Merchant Navy The Merchant Navy is the maritime register of the United Kingdom and comprises the seagoing commercial interests of UK-registered ships and their crews. Merchant Navy vessels fly the Red Ensign and are regulated by the Maritime and Coastguard ... independently reinvented the projection in 1890, after which it began to see more frequent use as recognition of its retroazimuthal property spread. Maps based on the Littrow projection are sometimes referred to as Weir Azimuth diagrams. The projection transforms from latitude ''φ'' and longitude ''λ'' to map coordinates ''x'' and ''y'' via the following equations: :\begin x &= R \frac \\ y &= R \cos \left(\lambda - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GS50 Projection
GS50 is a map projection that was developed by John Parr Snyder of the USGS in 1982. The GS50 projection provides a conformal projection suitable only for maps of the 50 United States. Scale varies less than 2% throughout the area covered. Distortion is very low as well. It is not a standard projection in the sense that it uses complex polynomials (of the tenth order) rather than a trigonometric formulation, though it was developed from an oblique stereographic projection. References * * {{cite journal , last1 = Snyder , first1 = John Parr , year = 1987 , title = Map Projections: A Working Manual , url = https://pubs.er.usgs.gov/publication/pp1395 , format = PDF , publisher = United States Geological Survey The United States Geological Survey (USGS), formerly simply known as the Geological Survey, is a scientific agency of the United States government. The scientists of the USGS study the landscape of the United States, its nat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a subcontinent of Eurasia and it is located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. Comprising the westernmost peninsulas of Eurasia, it shares the continental landmass of Afro-Eurasia with both Africa and Asia. It is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, the Mediterranean Sea to the south and Asia to the east. Europe is commonly considered to be separated from Asia by the watershed of the Ural Mountains, the Ural River, the Caspian Sea, the Greater Caucasus, the Black Sea and the waterways of the Turkish Straits. "Europe" (pp. 68–69); "Asia" (pp. 90–91): "A commonly accepted division between Asia and Europe ... is formed by the Ural Mountains, Ural River, Caspian Sea, Caucasus Mountains, and the Black Sea wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Africa
Africa is the world's second-largest and second-most populous continent, after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area and 20% of its land area.Sayre, April Pulley (1999), ''Africa'', Twenty-First Century Books. . With billion people as of , it accounts for about of the world's human population. Africa's population is the youngest amongst all the continents; the median age in 2012 was 19.7, when the worldwide median age was 30.4. Despite a wide range of natural resources, Africa is the least wealthy continent per capita and second-least wealthy by total wealth, behind Oceania. Scholars have attributed this to different factors including geography, climate, tribalism, colonialism, the Cold War, neocolonialism, lack of democracy, and corruption. Despite this low concentration of wealth, recent economic expansion and the large and young population make Afric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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