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Two-point Equidistant Projection
The two-point equidistant projection or doubly equidistant projection is a map projection first described by Hans Maurer in 1919 and Charles Close in 1921. It is a generalization of the much simpler azimuthal equidistant projection. In this two-point form, two locus points are chosen by the mapmaker to configure the projection. Distances from the two loci to any other point on the map are correct: that is, they scale to the distances of the same points on the sphere. The two-point equidistant projection maps a family of confocal spherical conics onto two families of planar ellipses and hyperbolas. The projection has been used for all maps of the Asian continent by the National Geographic Society atlases since 1959, though its purpose in that case was to reduce distortion throughout Asia rather than to measure from the two loci. The projection sometimes appears in maps of air routes. The Chamberlin trimetric projection is a logical extension of the two-point idea to three point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-point Equidistant Projection SW
Hunt seat is a style of forward seat riding commonly found in North American horse shows. Along with dressage, it is one of the two classic forms of English riding. The hunt seat is based on the tradition of fox hunting. Hunt seat competition in North America includes both flat and over fences for show hunters, which judge the horse's movement and form, and equitation classes, which judge the rider's ability both on the flat and over fences. The term ''hunt seat'' may also refer to ''any'' form of forward seat riding, including the kind seen in show jumping and eventing. Hunt seat is a popular form of riding in the United States, recognized by the USHJA (United States Hunter/Jumper Association) and the United States Equestrian Federation, and in Canada. While hunt seat showing ''per se'' is not an Olympic discipline, many show jumping competitors began by riding in hunter and equitation classes before moving into the jumper divisions. Rider position The Hunt seat is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-point Equidistant With Tissot's Indicatrices Of Distortion
Hunt seat is a style of forward seat riding commonly found in North American horse shows. Along with dressage, it is one of the two classic forms of English riding. The hunt seat is based on the tradition of fox hunting. Hunt seat competition in North America includes both flat and over fences for show hunters, which judge the horse's movement and form, and equitation classes, which judge the rider's ability both on the flat and over fences. The term ''hunt seat'' may also refer to ''any'' form of forward seat riding, including the kind seen in show jumping and eventing. Hunt seat is a popular form of riding in the United States, recognized by the USHJA (United States Hunter/Jumper Association) and the United States Equestrian Federation, and in Canada. While hunt seat showing ''per se'' is not an Olympic discipline, many show jumping competitors began by riding in hunter and equitation classes before moving into the jumper divisions. Rider position The Hunt seat is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in several fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azimuthal Equidistant Projection
The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians (lines of longitude) as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection. History While it may have been used by ancient Egyptians for star maps in some holy books,, p.29 the earliest text describing the azimuthal equidistant projection is an 11th-century work by al-Biruni. An example of this system is the world map by ‛Ali b. Ahmad al-Sharafi of Sfax in 1571. The projection appears in many Renaissance maps, and Gerardus Mercator used it for an inset of the north polar regions in sheet 13 and le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locus (mathematics)
In geometry, a locus (plural: ''loci'') (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.. In other words, the set of the points that satisfy some property is often called the ''locus of a point'' satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be ''located'' or may move. History and philosophy Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the ''locus'' of a point that is at a given dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in the planar case, a spherical conic can be defined as the locus of points the sum or difference of whose great-circle distances to two foci is constant. By taking the antipodal point to one focus, every spherical ellipse is also a spherical hyperbola, and vice versa. As a space curve, a spherical conic is a quartic, though its orthogonal projections in three principal axes are planar conics. Like planar conics, spherical conics also satisfy a “reflection property”: the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors. Many theorems about conics in the plane extend to spherical conics. For example, Graves’s theorem and Ivory’s theorem about c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Geographic Society
The National Geographic Society (NGS), headquartered in Washington, D.C., United States, is one of the largest non-profit scientific and educational organizations in the world. Founded in 1888, its interests include geography, archaeology, and natural science, the promotion of environmental and historical conservation, and the study of world culture and history. The National Geographic Society's logo is a yellow portrait frame—rectangular in shape—which appears on the margins surrounding the front covers of its magazines and as its television channel logo. Through National Geographic Partners (a joint venture with The Walt Disney Company), the Society operates the magazine, TV channels, a website, worldwide events, and other media operations. Overview The National Geographic Society was founded on 13 January 1888 "to increase and diffuse geographic knowledge". It is governed by a board of trustees whose 33 members include distinguished educators, business executives, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chamberlin Trimetric Projection
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 projection's 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 base points, the true distances to a point on the mapping area are calculated. The distances from each of the three base points are then drawn on the plane by compass circles. Unlike triangulation on a plane where thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waldo Tobler
Waldo Rudolph Tobler (November 16, 1930 – February 20, 2018) was an American-Swiss geographer and cartographer. Tobler's idea that "Everything is related to everything else, but near things are more related than distant things" is referred to as the "first law of geography." He has proposed a second law as well: "The phenomenon external to an area of interest affects what goes on inside". Tobler was an active Professor Emeritus at the University of California, Santa Barbara Department of Geography until his death. Academic background In 1961, Tobler received his Ph.D. in the Department of Geography at the University of Washington at Seattle. At Washington, he participated in geography's William Garrison-led quantitative revolution of the late 1950s. After graduating in 1961, Tobler became an Assistant Professor at the University of Michigan, where he remained until moving to the University of California, Santa Barbara in 1977. Until his retirement he held the positions of Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Mean Square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the squares) of the set. The RMS is also known as the quadratic mean (denoted M_2) and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted f_\mathrm) can be defined in terms of an integral of the squares of the instantaneous values during a cycle. For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load. In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data. Definition The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions may vary with author and context. Example: roots of polynomials The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation :ax^2+bx+c=0, is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions: :x=\frac. Similarly, solutions of cubic and quartic (third and fourth degree) equations can be expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid Of Revolution
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry. If the ellipse is rotated about its major axis, the result is a ''prolate spheroid'', elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an ''oblate spheroid'', flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid, instead o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
