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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] |
Mollweide Projection SW
Mollweide may refer to: *Karl Mollweide, mathematician (1774–1825). :*Mollweide projection, a pseudocylindrical map projection. :*Mollweide Glacier, a glacier the Victoria region of Antarctica. :*Mollweide's formula, a mathematical equation. {{dab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goode Homolosine Projection
The Goode homolosine projection (or interrupted Goode homolosine projection) is a pseudocylindrical, equal-area, composite map projection used for world maps. Normally it is presented with multiple interruptions, most commonly of the major oceans. Its equal-area property makes it useful for presenting spatial distribution of phenomena. Development 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 and the world’s oceans. Some variants include extensions that repeat regions in two different lobes of the interrupted 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" (meaning, ''equal-area map''), Goode fused the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hammer Projection
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, where it is extreme in the Mollweide. Development Directly inspired by the Aitoff projection, Hammer suggested the use of the equatorial form of the Lambert azimuthal equal-area projection instead of Aitoff's use of the azimuthal equidistant projection: :\begin x &= \operatorname_x\left(\frac, \varphi\right) \\ y &= \tfrac12 \operatorname_y\left(\frac, \varphi\right) \end where laea and laea are the ''x'' and ''y'' components of the equatorial Lambert azimuthal equal-area projection. Written out explicitly: :\begin x &= \frac \\ y &= \frac \end The inverse is calculated with the intermediate variable :z \equiv \sqrt The longitude and latitudes can then be calculated by :\begin \lambda &= 2 \arctan \frac \\ \varphi &= \arcsin zy \end where ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aitoff Projection
The Aitoff projection is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Based on the equatorial form of the azimuthal equidistant projection, Aitoff first halves longitudes, then projects according to the azimuthal equidistant, and then stretches the result horizontally into a 2:1 ellipse to compensate for having halved the longitudes. Expressed simply: :x = 2 \operatorname_x\left(\frac, \varphi\right), \qquad y = \operatorname_y \left(\frac\lambda 2, \varphi \right) where azeq and azeq are the ''x'' and ''y'' components of the equatorial azimuthal equidistant projection. Written out explicitly, the projection is: :x = \frac, \qquad y = \frac where :\alpha = \arccos\left(\cos\varphi\cos\frac\right)\, and sinc ''α'' is the unnormalized sinc function with the discontinuity removed. In all of these formulas, ''λ'' is the longitude from the central meridian and ''φ'' is the latitude. Three years later, Ernst Hermann Heinrich Hammer suggested ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Map Projections
This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise WP:NOTABLE, notable. Because there is no limit to the number of possible map projections, there can be no comprehensive list. Table of projections *The first known popularizer/user and not necessarily the creator. Key Type of projection surface ; Cylindrical: In normal aspect, these map regularly-spaced meridians to equally spaced vertical lines, and parallels to horizontal lines. ; Pseudocylindrical: In normal aspect, these map the central meridian and parallels as straight lines. Other meridians are curves (or possibly straight from pole to equator), regularly spaced along parallels. ; Conic: In normal aspect, conic (or conical) projections map meridians as straight lines, and parallels as arcs of circles. ; Pseudoconical: In normal aspect, pseudoconical projections represent the central meridian as a straight line, other meridians as complex curves, and parallels as ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are ''n''th root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions. The ''closed-form problem'' arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it. Example: roots of polynomials The quadratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division By Zero
In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplication, multiplied by the divisor. That is, c = \tfrac ab is equivalent to c \cdot b = a. By this definition, the quotient q = \tfrac is nonsensical, as the product q \cdot 0 is always 0 rather than some other number a. Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called field (mathematics), fields leaves division by zero undefined (mathematics), undefined, and situations where division by zero might occur m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function , its derivative , and an initial guess for a root of . If satisfies certain assumptions and the initial guess is close, then x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the x-intercept of the tangent of the graph of at : that is, the improved guess, , is the unique root of the linear approximation of at the initial guess, . The process is repeated as x_ = x_n - \frac until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, and was succeeded by Halley's method. The method can also be extended t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States Geological Survey
The United States Geological Survey (USGS), founded as the Geological Survey, is an agency of the U.S. Department of the Interior whose work spans the disciplines of biology, geography, geology, and hydrology. The agency was founded on March 3, 1879, to study the landscape of the United States, its natural resources, and the natural hazards that threaten it. The agency also makes maps of planets and moons, based on data from U.S. space probes. The sole scientific agency of the U.S. Department of the Interior, USGS is a fact-finding research organization with no regulatory responsibility. It is headquartered in Reston, Virginia, with major offices near Lakewood, Colorado; at the Denver Federal Center; and in NASA Research Park in California. In 2009, it employed about 8,670 people. The current motto of the USGS, in use since August 1997, is "science for a changing world". The agency's previous slogan, adopted on its hundredth anniversary, was "Earth Science in the Pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boggs Eumorphic Projection
The Boggs eumorphic projection is a Map projection#Pseudocylindrical, pseudocylindrical, equal-area projection, equal-area map projection used for world maps. Normally it is presented with Interrupted projection, multiple interruptions. Its equal-area property makes it useful for presenting spatial distribution of phenomena. The projection was developed in 1929 by Samuel Whittemore Boggs (1889–1954) to provide an alternative to the Mercator projection, Mercator projection for portraying global areal relationships. Boggs was geographer for the United States Department of State from 1924 until his death. The Boggs eumorphic projection has been used occasionally in textbooks and atlases. Boggs generally repeated regions in two different lobes of the Interrupted projection, interrupted map in order to show Greenland or eastern Russia undivided. He preferred his interrupted version, and named it "eumorphic”, meaning "goodly shaped" (in Boggs's own words). The projection's mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
