nofootnotes date November 2009 one source date November 2009 The Hayford ellipsoid is a geodesy geodetic reference ellipsoid , named after the US geodesist John Fillmore Hayford 1868 1925 , which was introduced in 1910. The Hayford ellipsoid was also referred to as the International ellipsoid 1924 after it had been adopted by the International Union of Geodesy and Geophysics IUGG in 1924, and was recommended for use all over the world. Many countries retained their previous ellipsoids. The Hayford ellipsoid is defined by its semi major axis a 6,378,388.000 and its flattening f 1 297.00 . Unlike its predecessors, such as the Bessel ellipsoid a 6,377,397 m, f 1 299,15 , which was a European ellipsoid, the Hayford ellipsoid also included measurements from North America , as well as other continents to a lesser extent . It also included isostatic measurements to reduce plumbline divergences. It has since been replaced as the International ellipsoid by the newer Lucerne ellipsoid 1967 and GRS 80 1980 . Sources Defense Mapping Agency Geodesy for the layman , 1983, p. 8 http www.ngs.noaa.gov PUBS LIB Geodesy4Layman geo4lay.pdf Category Geodesy Category Geophysics de Hayford Ellipsoid ru ... more details
Unreferenced date December 2009 In optics , an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive index refractive indices in a crystal . The equation for the ellipsoid is constructed using the electric displacement field electric displacement vector D and the dielectric constant s. Defining the field energy W as math 8 pi W frac D 2 1 varepsilon 1 frac D 2 2 varepsilon 2 frac D 2 3 varepsilon 3 . math and the reduced displacement as math R i frac D i sqrt 8 pi W , math then the index ellipsoid is defined by the equation math frac R 1 2 varepsilon 1 frac R 2 2 varepsilon 2 frac R 3 2 varepsilon 3 1. math The semiaxes of this ellipsoid are dielectric constants of the crystal. This ellipsoid can be used to determine the Polarization waves polarization of an incoming wave with wave vector math vec s math by taking the intersection of the plane math vec R cdot vec s 0 math with the index ellipsoid. The axes of the resulting ellipse are the resulting polarization directions. Indicatrix An important special case of the index ellipsoid occurs when the ellipsoid is an ellipsoid of revolution, i.e. constructed by rotating an ellipse around either the minor or major axis, when two axes are equal and a third is different. In this case, there is only one optical axis, the axis of rotation, and the material is said to be uniaxial . When all axes of the index ellipsoid are equal, the material is isotropic . In all other cases, in which the ellipsoid has three distinct axes, the material is called biaxial . DEFAULTSORT Index Ellipsoid Category Optics Category Surfaces de Indexellipsoid nl Indicatrix sl Opti na indikatrisa ... more details
The Bessel ellipsoid or Bessel 1841 is an important reference ellipsoid of geodesy . It is currently used by several countries for their national geodetic surveys, in Europe and on other continents, but will be replaced in the next decades by modern ellipsoids of satellite geodesy . The Bessel ellipsoid was derived 1841 by Friedrich Wilhelm Bessel , based on several meridian arc s and other data of continental geodetic network s of Europe , Russia and the British Survey of India . It is based on 10 meridional arcs and 38 precise measurements of the astro geographic latitude and longitude see also Astro geodetic astro geodesy . The dimensions of the ellipsoid axes were defined by logarithm s in keeping with former calculation methods. The Bessel and GPS ellipsoids The Bessel ellipsoid fits especially ... that of the mean Earth ellipsoid derived by satellites. Below the two axes a , b and the flattening ... system. Bessel ellipsoid 1841 defined by log a and a 6,377,397.155 m 1 299.152815 3513233 0.003342 773154 0.000005 b 6,356,078.963 m . Earth ellipsoid WGS84 defined directly by a and a 6,378,137.0 m 1 298.257223563 b 6,356,752.30 m . Usage The ellipsoid data published by Bessel 1841 were ... all national surveys just some surveys in Asia switching to the Clarke ellipsoid of 1880. After ... ellipsoid of 1910 which was adopted in 1924 by the IAG as the International ellipsoid 1924 ..., rock density and the distribution of network data. Every reference ellipsoid deviates from the worldwide ... on the Bessel ellipsoid. In the following decades the Americas American states switched mainly to the Hayford ellipsoid 1908 internat. Ell. 1924 which was also used for the European unification project ... to use the Krassowski ellipsoid of about 1940. As of 2010 the Bessel ellipsoid is the geodetic system ... Geodesy de Bessel Ellipsoid fr Ellipso de de Bessel nl Ellipso de van Bessel ru sv Bessels ellipsoid ... more details
ellipsoid, ranging from meridian arc s up to modern satellite geodesy or the analysis and interconnection ... s are several of special importance the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and for GPS positioning the WGS84 ellipsoid. Historical Earth ellipsoids The following ... 6,356,256.913224 299.3249646 Bessel ellipsoid Bessel 1841 6,377,397.155 6,356,078.963 299.1528128 Alexander ... close to WGS84 298.30 International Ellipsoid International 1924 6,378,388.0 6,356,911.9 297.0 ... infty math where does this one go?? Mean Earth ellipsoid and reference ellipsoids A data set which describes the global average of the Earth s surface curvature is called the mean Earth Ellipsoid. It refers ... . The latter is close to the mean sea level , and therefore an ideal Earth ellipsoid has the same volume as the geoid. While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a so called reference ellipsoid may be the better choice. When geodetic measurements ... ellipsoid Hayford or the Bessel ellipsoid , despite of the fact that their main axes deviate ... adopts the axes of the Earth ellipsoid to the best available data. See also Reference ellipsoid History of geodesy datum geodesy geodetic datum DEFAULTSORT Earth Ellipsoid Category Geodesy Category ... more details
In robotics , the manipulability ellipsoid is the geometric interpretation of the scaled eigenvector s resulting from the singular value decomposition of the jacobian that describes a robot s motion. Category Robot control Category Geometry robotics stub ... more details
Ellipsoid Hill coor dm 77 48 S 163 49 E is a rounded, partly ice covered summit 1,130 m to the north of Blue Glacier , between Geoid Glacier and Spheroid Hill , in Victoria Land . The name is one of a group in the area associated with surveying applied in 1993 by New Zealand Geographic Board NZGB . Named from ellipsoid, in geodesy a mathematical figure formed by revolving an ellipse about its minor axis. usgs gazetteer Category Hills of Victoria Land Category Scott Coast ScottCoast geo stub ... more details
File Chloryl hexafluoroantimonate xtal 2008 Cl coordination CM 3D ellipsoids.png thumb right 200px A thermal ellipsoid model of the coordination environment of the chlorine atom in the crystal structure of chloryl chloryl hexafluoroantimonate , ClO sub 2 sub SbF sub 6 sub , showing F Cl interactions. Thermal ellipsoids , more formally termed atomic displacement parameters , are ellipsoid s used in crystallography to indicate the magnitudes and directions of the thermal vibration of atom s in crystal structure s. Since the vibrations are usually anisotropic different magnitudes in different directions in space , an ellipsoid is a convenient way of visualising the vibration and therefore the time averaged position of an atom in a crystal. Thermal ellipsoids can be defined by a tensor , a mathematical object which allows the definition of magnitude and orientation of vibration with respect to Cartesian coordinate system three mutually perpendicular axes . The three principal axes of the thermal vibration of an atom are denoted math U 1 math , math U 2 math , and math U 3 math , and the corresponding thermal ellipsoid is based on these axes. The size of the ellipsoid is scaled so that it occupies the space in which there is a particular probability of finding the electron density of the atom. The particular probability is usually 50 . ref Massa2nd pages 35 37 ref See also Debye Waller factor References reflist Category Crystallography ... more details
File Ellipsoid 2.png thumb 400px An iteration of the ellipsoid method In mathematical optimization , the ellipsoid ... specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm , which finds an optimal solution in a finite number of steps. The ellipsoid method ... a minimizer of a convex function . History The ellipsoid method has a long history. As an iterative ... Judin . As an algorithm for solving linear programming problems with rational data, the ellipsoid ... polynomial time solvability of linear programs. Following Khachiyan s work, the ellipsoid method was the only ... point method and variants of the simplex algorithm are much faster than the ellipsoid method ... ellipsoid math mathcal E 0 subset mathbb R n math defined as math mathcal E 0 left z in mathbb ... of the algorithm, we have a point math x k math at the center of an ellipsoid math mathcal E k left ... that math x in mathcal E k cap z g k 1 T z x k leq 0 . math We set math mathcal E k 1 math to be the ellipsoid of minimal volume containing the half ellipsoid described above and compute math x k 1 ... g k leq epsilon Rightarrow f x k f x leq epsilon. math border 1 Sample sequence of iterates File Ellipsoid 1.png thumb center 250px math k 0 math File Ellipsoid 2.png thumb center 250px math k 1 math File Ellipsoid 3.png thumb center 250px math k 2 math File Ellipsoid 4.png thumb center 250px math k 3 math File Ellipsoid 5.png thumb center 250px math k 4 math File Ellipsoid 6.png thumb center 250px ... for constrained minimization, we have a point math x k math at the center of an ellipsoid math mathcal ... math th constraint, update the ellipsoid with a feasibility cut. Our feasibility cut may be a subgradient ... feasible math z math . Application to Linear Programming Performance The ellipsoid method is used on low ..., the ellipsoid method is an important theoretical technique in combinatorial optimization . In computational complexity theory , the ellipsoid algorithm is attractive because its complexity depends ... more details
In mathematics , the John ellipsoid E K associated to a convex body K in n dimension al Euclidean space R sup n sup is the ellipsoid of maximal n dimensional volume contained within K . The John ellipsoid is named after the Germany German mathematician Fritz John . The following refinement of John s original theorem, due to Ball 1992 , gives necessary and sufficient conditions for the John ellipsoid of K to be the closed unit ball B of R sup n sup The John ellipsoid E K of a convex body K R sup n sup is B if and only if B K and there exists an integer m n and, for i 1, ..., m , real number s c sub i sub > 0 and unit vector s u sub i sub S sup n &minus 1 sup K such that math sum i 1 m c i u i 0 math and, for all x R sup n sup math x sum i 1 m c i x cdot u i u i . math References cite journal last Ball first Keith M. title Ellipsoids of maximal volume in convex bodies journal Geom. Dedicata volume 41 year 1992 issue 2 pages 241&ndash 250 issn 0046 5755 doi 10.1007 BF00182424 John, Fritz. Extremum problems with inequalities as subsidiary conditions . Studies and Essays Presented to R. Courant on his 60th Birthday , January 8, 1948, 187 204. Interscience Publishers, Inc., New York, N. Y., 1948. oclc 1871554 MR 30135 cite journal last Gardner first Richard J. title The Brunn Minkowski inequality journal Bulletin of the American Mathematical Society Bull. Amer. Math. Soc. N.S. volume 39 issue 3 year 2002 pages 355&ndash 405 electronic issn 0273 0979 doi 10.1090 S0273 0979 02 00941 2 Category Convex geometry Category Multi dimensional geometry geometry stub de John Ellipsoid ... more details
Geodesy In geodesy , a reference ellipsoid is a mathematically defined surface that approximates the geoid ... and point coordinates such as latitude , longitude , and elevation are defined. Ellipsoid parameters ... self gravitating fluid body in equilibrium takes the form of an oblate ellipsoid of revolution which ... University of Chicago Press year 1993 isbn 0 226 76747 7 page 82 ref uses the word ellipsoid alone in preference to the full term term oblate ellipsoid of revolution or the older term oblate spheroid . In the rare instances some asteroid s and planet s where a more general ellipsoid shape is required as a model the term used is Triaxial ellipsoid triaxial or scalene ellipsoid. A great many ellipsoids ... center of mass of the Earth or body being modeled. The shape of an oblate ellipsoid of revolution is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated ... radius of the ellipsoid the semi minor axis of the ellipse, b , is identified with the Geographic pole polar distances from the centre . These two lengths completely specify the shape of the ellipsoid ... systems to be defined upon the same reference ellipsoid. The longitude measures the rotational ... the equatorial plane and a line that is Surface normal normal to the reference ellipsoid. Depending ... is the angle between the equatorial plane and a line from the center of the ellipsoid. For non Earth ... normal containing the point, and the height h of the point over the reference ellipsoid. If these coordinates ... argument version of arctan is recommended. Historical Earth ellipsoids Main Earth ellipsoid Historical ... ellipsoid used, and that used in the context of the Global Positioning System, is the one defined ... American Datum of 1927 Clarke 1866 6 378 206.4 6 356 583.8 294.978 698 2 Bessel ellipsoid Bessel 1841 6 377 397.155 6 356 078.965 299.152 843 4 International Ellipsoid International 1924 6 378 388 ... enough that the average polar radius is used to define its ellipsoid. The Earth s Moon is effectively ... more details
axis frame, it must lie on an ellipsoid , called inertia ellipsoid . The ellipsoid axes values are the half of the principal moments of inertia . The path traced out on this ellipsoid by the angular ... L math Hence, the normal vector to the kinetic energy ellipsoid at math boldsymbol omega math ... energy ellipsoid that is tangent to it and rolls around on it without slipping. This is Poinsot ... energy constrains math mathbf L math to lie on an ellipsoid, whereas the angular momentum constraint ... more details
Lame s stress ellipsoid Figure coming is an alternative to Mohr s circle for the graphical representation of the Stress mechanics Stress tensor stress state at a point . The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an ellipse Figure coming . Once the equations of the ellipsoid is known ... the equation of the stress ellipsoid we consider the coordinate axes math x 1, x 2, x 3 ... 2 2 2 frac T 3 sigma 3 2 2 1 , math which is the equation of an ellipsoid centered at the origin of the coordinate system, with the lengths of the semiaxes of the ellipsoid equal to the magnitudes of the principal stresses, i.e. the intercepts of the ellipsoid with the principal axes are math pm sigma ... to the sum of the principal radii of the ellipsoid. The second stress invariant math I 2 , math is directly proportional to the sum of the three principal areas of the ellipsoid. The tree principal ... proportional to the volume of the ellipsoid. If two of the three principal stresses are numerically equal the stress ellipsoid becomes an spheroid ellipsoid of revolution . ref name timoshenko1 ... stresses are equal and of the same sign, the stress ellipsoid becomes a sphere and any three perpendicular directions can be taken as principal axes. ref name timoshenko1 The stress ellipsoid ... vector of the stress ellipsoid acts on a plane oriented parallel to the tangent plane to the stress ... 0486611876 DEFAULTSORT Lame s Stress Ellipsoid Category Classical mechanics Category Materials science ... more details
Summary This is figure 5A from US Patent 5,539,310 by Peter J. Basser, James H. Mattiello, and Denis Le Bihan, granted July 23, 1996. Images appearing in a US Patent are public domain unless a specific copyright exclusion is posted, and no such exclusion appears in this patent Patents are published as part of the terms of granting the patent to the inventor. Subject to limited exceptions reflected in 37 CFR 1.71 d & e and 1.84 s , the text and drawings of a patent are typically not subject to copyright restrictions. See http www.uspto.gov main ccpubguide.htm 37 CFR 1.84 s Copyright or Mask Work Notice 1.71 d . A copyright or mask work notice may appear in the drawing, but must be placed within the sight of the drawing immediately below the figure representing the copyright or mask work material and be limited to letters having a print size of.32 cm. to .64 cm. 1 8 to 1 4 inches high. The content of the notice must be limited to only those elements provided for by law. For example, 1983 John Doe 17 U.S.C. 401 and M John Doe 17 U.S.C. 909 would be properly limited and, under current statutes, legally sufficient notices of copyright and mask work, respectively. Inclusion of a copyright or mask work notice will be permitted only if the authorization language set forth in 1.71 e is included at the beginning preferably as the first paragraph of the specification. 37 CFR 1.71 e The authorization shall read as follows A portion of the disclosure of this patent document contains material which is subject to copyright or mask work protection. The copyright or mask work owner has no objection to the facsimile reproduction by any one of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright or mask work rights whatsoever. Licensing PD US patent ... more details
wiktionary undulation Undulation may refer to Lateral undulation , the most primitive of vertebrate locomotor patterns Law of Undulation , an invention of C. S. Lewis Undulation of the geoid , the separation between the geoid and the reference ellipsoid of the Earth disambig cs Undulace ... more details
Indicatrix may refer to Dupin indicatrix , a conic section which describes the local shape of a surface Tissot s indicatrix , which describes and visualizes the distortion of a map Indicatrix, a special case of the Index ellipsoid Indicatrix index ellipsoid used in optics to describe properties of crystals Tangent indicatrix , an object in differential geometry related to a closed space curve disambiguation ... more details
The SK 42 reference system also known as the Feodosy Krasovsky Krasovsky Reference ellipsoidellipsoid , is a coordinate system established in the Soviet Union in 1942 as Systema koordinat lang ru 1942 , and provides parameters which are linked to the geocentric Cartesian coordinate system PZ 90 . It was used in geodetic calculations, notably in military mapping and determining state borders. ref slide 11, Borodko ref Citations and notes reflist References references Borodko, Alexander, Topographic and geodetic provisions of the Russian Federation frontier delimitation and demarcation , Federal Agency of Geodesy and Cartography, International Symposium on Land and River Boundaries Demarcation and Maintenance in Support of Borderland Development, Bangkok , Thailand, 6&ndash 11 November 2006 DEFAULTSORT Sk 42 Reference System Category Coordinate systems Category Geodesy Category GPS Category Navigation Category Surveying cs Krasovsk ho elipsoid de Krassowski Ellipsoid ru ... more details
wiktionary scalene Scalene may refer to A scalene triangle , one in which all sides or angles are different A scalene ellipsoid , one in which the lengths of all three semi principal axes are different Scalene muscles of the neck Scalene tubercle , a slight ridge on the first rib prolonged internally into a tubercle disambiguation ... more details
Spheroid Hill coor dm 77 47 S 163 56 E is a mostly ice free summit 1,230 m 1 nautical mile 1.9 km east of Ellipsoid Hill , on the north side of Blue Glacier in Victoria Land . The name is one of a group in the area associated with surveying applied in 1993 by New Zealand Geographic Board NZGB . Named from spheroid sometimes referred to as an ellipsoid , a mathematical figure formed by revolving an ellipse about its minor axis. usgs gazetteer Category Mountains of Victoria Land Category Scott Coast ScottCoast geo stub ... more details
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