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Flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-major and semi-minor axes, semi-axes a and b of the resulting ellipse or ellipsoid is : f =\frac . The ''compression factor'' is b/a in each case; for the ellipse, this is also its aspect ratio. Definitions There are three variants: the flattening f, sometimes called the ''first flattening'', as well as two other "flattenings" f' and n, each sometimes called the ''second flattening'', sometimes only given a symbol, or sometimes called the ''second flattening'' and ''third flattening'', respectively. In the following, a is the larger dimension (e.g. semimajor axis), whereas b is the smaller (semiminor axis). All flattenings are zero for a circle (). :: Identities The flattenings can be related t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earth Flattening
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the figure of the Earth, Earth's form, used as a frame of reference, reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations. It is a spheroid (an ellipsoid of Surface of revolution, revolution) whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the ''equatorial axis'' () and the ''polar axis'' (); their radial difference is slightly more than 21 km, or 0.335% of (which is not quite 6,400 km). Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reference Ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations. It is a spheroid (an ellipsoid of revolution) whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the ''equatorial axis'' () and the ''polar axis'' (); their radial difference is slightly more than 21 km, or 0.335% of (which is not quite 6,400 km). Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayfo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, 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 ball (gridiron football), 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's, M&M. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and rotation of the Earth, rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattening, flattened in the direction of its axis of rotation. For that reason, in cartography and geode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equatorial Bulge
An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On Earth The planet Earth has a rather slight equatorial bulge; its equatorial diameter is about greater than its polar diameter, with a difference of about of the equatorial diameter. If Earth were scaled down to a globe with an equatorial diameter of , that difference would be only . While too small to notice visually, that difference is still more than twice the largest deviations of the actual surface from the ellipsoid, including the tallest mountains and deepest oceanic trenches. Earth's rotation also affects the sea level, the imaginary surface used as a reference frame from which to measure altitudes. This surface coincides with the mean water surface level in oceans, and is extrapolated over land by taking into account the loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planetary Flattening
A planetary coordinate system (also referred to as ''planetographic'', ''planetodetic'', or ''planetocentric'') is a generalization of the geographic, geodetic, and the geocentric coordinate systems for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the ''selenographic coordinates'' for the Moon. The coordinate systems for almost all of the solid bodies in the Solar System were established by Merton E. Davies of the Rand Corporation, including Mercury, Venus, Mars, the four Galilean moons of Jupiter, and Triton, the largest moon of Neptune. A planetary datum is a generalization of geodetic datums for other planetary bodies, such as the Mars datum; it requires the specification of physical reference points or surfaces with fixed coordinates, such as a specific crater for the reference meridian or the best-fitting equigeopotential as zero-level surface. Longitude The longitude systems of most of those bodies wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesy
Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional space, 3D. It is called planetary geodesy when studying other astronomical body, astronomical bodies, such as planets or Natural satellite, circumplanetary systems. Geodynamics, Geodynamical phenomena, including crust (geology), crustal motion, tides, and polar motion, can be studied by designing global and national Geodetic control network, control networks, applying space geodesy and terrestrial geodetic techniques, and relying on Geodetic datum, datums and coordinate systems. Geodetic job titles include geodesist and geodetic surveyor. History Geodesy began in pre-scientific Classical antiquity, antiquity, so the very word geodesy comes from the Ancient Greek word or ''geodaisia'' (literally, "division of Earth"). Early ideas about t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eccentricity (mathematics)
In mathematics, the eccentricity of a Conic section#Eccentricity, conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is 0. * The eccentricity of a non-circular ellipse is between 0 and 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of Line (geometry), lines is \infty. Two conic sections with the same eccentricity are similarity (geometry), similar. Definitions Any conic section can be defined as the Locus (mathematics), locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the ''eccentricity'', commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a Cone (geometry), double-napped cone associated with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity (mathematics), eccentricity e, a number ranging from e = 0 (the Limiting case (mathematics), limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but for Perimeter of an ellipse, its perimeter (also known as circumference), Integral, integration is required to obtain an exact solution. The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted and . An ellipse has four extreme points: two ''Vertex (geometry), vertices'' at the endpoints of the major axis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ovality
In telecommunications and fiber optics, ovality or noncircularity is the degree of deviation from perfect circularity of the cross section of the core or cladding of the fiber. The cross-sections of the core and cladding are assumed to a first approximation to be elliptical, and ovality is defined to be twice the third flattening of the ellipse, 2\frac , where ''a'' is the length of the major axis and ''b'' is the length of the minor axis. This dimensionless quantity is between and , and may be multiplied by to express ovality as a percentage. Alternatively, ovality of the core or cladding may be specified by a tolerance field consisting of two concentric circles, within which the cross section boundaries must lie. In measurements, ovality is the amount of out-of-roundness of a hole or cylindrical part in the typical form of an oval. In chemistry In computational chemistry, especially in QSAR studies, ovality refers to, a measure of how the shape of a molecule approaches a sphe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Celestial Mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data. History Modern analytic celestial mechanics started with Isaac Newton's ''Principia'' (1687). The name celestial mechanics is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term ''celestial mechanics''. Prior to Kepler, there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of the physical causes of the planets' motion. Laws of planetary motion Johannes Kepler was the first to closely integrate the predictive geometrical a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roundness (object)
Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in Plane (mathematics), two dimensions, such as the cross section (geometry), cross sectional circles along a cylinder, cylindrical object such as a shaft (mechanical engineering), shaft or a rolling-element bearing#Cylindrical roller, cylindrical roller for a bearing. In geometric dimensioning and tolerancing, control of a cylinder can also include its fidelity to the longitudinal axis, yielding cylindricity. The analogue of roundness in three-dimensional space, three dimensions (that is, for spheres) is ''sphericity''. Roundness is dominated by the shape's gross features rather than the definition of its edges and corners, or the surface roughness of a manufactured object. A smooth ellipse can have low roundness, if its eccentricity (mathematics), eccentricity is large. Regular polygons increase their roundness with increasing numbers of sides, ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
