Disney World , a geodesic sphere A geodesicdome is a spherical or partial spherical thin shell structure ... to form a complete sphere, it is known as a geodesic sphere . The term dome refers to an enclosed ... found on playgrounds. Typically the design of a geodesicdome begins with an icosahedron inscribed ... of the sphere. The edges of the triangles form approximate geodesic paths over the surface of the dome ... of design and fabrication of any geodesicdome, builders have tended to standardize using a few basic ... , ref http www.physics.princeton.edu trothman domes.html First GeodesicDome Planetarium in Jena ... Fuller R. Buckminster Fuller named the domegeodesic from field experiments with artist Kenneth .... ref The geodesicdome appealed to Fuller because it was extremely strong for its weight,its omnitriangulated ... volume for the least surface area. Fuller hoped that the geodesicdome would help address the postwar ... the geodesicdome s stability, the US Air Force experimented with helicopter deliverable units ... himself lived in a geodesicdome in Carbondale, Illinois , at the corner of Forest and Cherry. ref ... sphere corresponds to the structural strut of the physical geodesicdome . The general definition ... geodesic sphere or dome is the number of parts or segments into which a side edge of the underlying ... Methods of construction File Vitra geodesicdome tubing.jpg thumb Construction details of a permanently installed tent type geodesicdome by Synergetics. Inc . Wooden domes have a hole drilled ... in the structure. It also has the advantage of being watertight. Largest geodesicdome structures ... science geodesic domes worlds 10 largest domes dead link date October 2010 ref Nagoya Dome Nagoya ... news 11880141.html title News & 124 Kansas City Southern razes geodesicdome Baton Rouge ... 2 column count 2 Geodesic airframe Dome Concrete dome Cloud nine Tensegrity sphere Domed city Fullerene s, molecules which resemble the geodesicdome structure Hoberman sphere Monolithic dome Radome Silent ... more details
Infobox nrhp name ASM Headquarters and GeodesicDome nrhp type image caption location 9639 Kinsman Rd., Russell Township, Geauga County, Ohio Russell Township, Ohio lat degrees 41 lat minutes 27 lat seconds 36 lat direction N long degrees 81 long minutes 17 long seconds 56 long direction W or decimal latitude longitude coord display inline,title coord parameters region US type landmark locmapin Ohio area convert 25 acre ref name nrhpreg built architect John Terrence Kelly , R. Buckminster Fuller architecture added October 22, 2009 ref name newlistings2009oct30 cite web url http www.nps.gov history nr listings 20091030.htm title Announcements and actions on properties for the National Register of Historic Places date October 30, 2009 accessdate December 20, 2009 work Weekly Listings publisher National Park Service ref governing body Private refnum 09000849 ref name newlistings2009oct30 The ASM Headquarters and GeodesicDome , in Russell Township, Geauga County, Ohio Russell Township , Geauga County, Ohio Geauga County , Ohio , United States , are modernist structures that were built in 1959. ref name nrhpreg The building serves as the headquarters of ASM International society ASM International , formerly the American Society of Metals. The dome is the world s largest open air geodesicdome , and is rare among Buckminster Fuller designed geodesic domes in that it was never intended to be a covered structure. ref name nrhpreg The building was listed on the U.S. National Register of Historic Places on October 22, 2009. ref name nrhpreg cite web title National Register of Historic Places Registration ASM Headquarters and GeodesicDome url http www.nps.gov history nr feature weekly features ASMHeadquarters GeodesicDome.pdf format Portable Document Format PDF date May 1, 2009 author Anke Schreiber publisher National Park Service accessdate December 20, 2009 110 pages, incl. figures ... in 1959 Category Geodesic domes Category Buildings and structures on the National Register of Historic ... more details
Differential geometry of curves Exponential map GeodesicdomeGeodesic general relativity Geodesics ...File Spherical triangle.svg thumb right 150px A geodesic triangle on the sphere. The geodesics are great circle arcs. In mathematics , a geodesic pron en d i di z k , IPA en d i d s k respell JEE ... transport transported along it. The term geodesic comes from geodesy , the science of measuring the size and shape of Earth in the original sense, a geodesic was the shortest route between ... theory , one might consider a geodesic between two vertices nodes of a graph. Geodesics are of particular ... with constant velocity , meaning that the distance from f s to f t along the geodesic is proportional ... minimizing the energy leads to the same equations for a geodesic here constant velocity is a consequence ... its energy the resulting shape of the band is a geodesic. In Riemannian geometry geodesics are not the same ... is a geodesic but not the shortest path between the points. The map t t sup 2 sup from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity ... geodesic general relativity discusses the special case of general relativity in greater detail. Examples .... Metric geometry In metric geometry , a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve I M from an interval I of the reals to the metric space M is a geodesic ... v t 1 t 2 . , math This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with Curve Length of curves natural parametrization ... equality is satisfied for all t sub 1 sub , t sub 2 sub I , the geodesic is called a minimizing geodesic or shortest path . In general, a metric space may have no geodesics, except constant curves ... of rectifiable path s, although this minimizing sequence need not converge to a geodesic. Riemannian ... symbols of the metric. This is the geodesic equation , discussed Affine geodesics below . Calculus ... more details
A geodesic airframe alternatively, geodetic is a type of construction for the airframe s of aircraft developed by United Kingdom British aeronautical engineer Barnes Wallis in the 1930s. It makes use of a space frame formed from a spirally crossing basket weave of load bearing members. ref name Buttler93 Buttler, p.93 ref The principle is that two geodesicdomegeodesic arcs can be drawn to intersect on a curving surface the fuselage in a manner that the Torsion mechanics torsional load on each cancels out that on the other. ref name Buttler94 Buttler, p.94 ref Early examples Image Constitutiondiagonalriders.gif thumb right 300px 18th century American warships used geodesic Diagonal rider in their construction. The diagonal rider structural element was used by Joshua Humphreys in the Original six frigates of the United States Navy first US Navy sail frigates in 1794. Citation needed date March 2011 Diagonal riders are viewable in the interior hull structure of the preserved USS Constitution ... hogging in the ship s hull. A more complex geodesic form can be seen in an aircraft s fuselage ... or diagonal loads an example of geodesic design is a misnomer. In a geodetic structure, the strength ..., illustrating the geodesic construction and the level of punishment it could absorb while maintaining integrity and airworthiness. Image Vickers Warwick geodesic fuselage.JPG thumb right A section of the rear fuselage from a Warwick showing the geodesic construction in duralumin. On exhibit at the Armstrong & Aviation Museum at Bamburgh Castle . The geodesic construction method was developed by the United ... geodesic wiring harnesses to hold the gasbags in his commercial airship design, the R100 . Wallis used the term geodetic to apply to the airframe and distinguish it from geodesic which is the proper ... fabric skin burning off, leaving the naked frames exposed see photo . The benefits of the geodesic ... Geodesic Airframe Category Airship technology Category Structural system Category Aviation ... more details
Unreferenced date October 2008 In differential geometry and dynamical systems , a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M . Examples On the unit sphere , every great circle is an example of a closed geodesic. On a compact hyperbolic surface , closed geodesics are in one to one correspondence with non trivial conjugacy class es of elements in the Fuchsian group of the surface. A prime geodesic is an example of a closed geodesic. Definition Geodesic Flow mathematics flow is an math mathbb R math group action action on tangent bundle T M of a manifold M defined in the following way math G t V dot gamma V t math where math t in mathbb R math , math V in T M math and math gamma V math denotes the geodesic with initial data math dot gamma V 0 V math . It defines a Hamiltonian flow on co tangent bundle with the pseudo Riemannian metric as the Hamiltonian quantum mechanics Hamiltonian . In particular it preserves the pseudo Riemannian metric math g math , i.e. math g G t V ,G t V g V,V . , math That makes possible to define geodesic flow on unit tangent bundle math UT M math of the Riemannian manifold math M math when the geodesic math gamma V math is of unit speed. See also Selberg trace formula Zoll surface geodesic References Arthur Besse Besse, A. Manifolds all of whose geodesics are closed , Ergebisse Grenzgeb. Math. , no. 93, Springer, Berlin, 1978. Category Differential geometry Category Dynamical systems ... more details
In mathematics &mdash specifically, in differential geometry &mdash a geodesic map or geodesic mapping or geodesic diffeomorphism is a Function mathematics function that preserves geodesic s . More precisely, given two pseudo Riemannian manifold pseudo Riemannian manifold s M , g and N , h , a function &phi M &rarr N is said to be a geodesic map if &phi is a diffeomorphism of M onto N and the image under &phi of any geodesic arc in M is a geodesic arc in N and the image under the inverse function &phi sup &minus 1 sup of any geodesic arc in N is a geodesic arc in M . Examples If M , g and N , h are both the n dimension al Euclidean space E sup n sup with its usual flat Riemannian metric metric , then any Euclidean isometry is a geodesic map of E sup n sup onto itself. Similarly, if M , g and N , h are both the n dimensional unit hypersphere sphere S sup n sup with its usual round metric, then any isometry of the sphere is a geodesic map of S sup n sup onto itself. If M , g is the unit sphere S sup n sup with its usual round metric and N , h is the sphere of radius 2 with its usual round metric, both thought of as subsets of the ambient coordinate space R sup n 1 sup , then the expansion map &phi R sup n 1 sup &rarr R sup n 1 sup given by &phi x 2 x induces a geodesic map of M onto N . There is no geodesic map from the Euclidean space E sup n sup onto the unit sphere S sup n sup , since they are not homeomorphism homeomorphic , let alone diffeomorphic. Let D , g be the unit disc D &sub R sup 2 sup equipped with the Euclidean metric, and let D , h be the same ... map i D &rarr D , i is not a geodesic map, since g geodesics are always ... to the plane is a geodesic map as it takes great circles to lines and its inverse takes lines ... id 1118149 External links MathWorld urlname GeodesicMapping title Geodesic mapping Category ... more details
orphan date October 2009 In mathematics , a complex geodesic is a generalization of the notion of geodesic to complex number complex spaces. Definition Let X , be a complex Banach space and let B be the open set open unit ball in X . Let denote the open unit disc in the Complex plane Other meanings of complex plane complex plane C , thought of as the Poincar disc model for 2 dimensional real 1 dimensional complex hyperbolic geometry . Let the Poincar metric on be given by math rho a, b tanh 1 frac a b 1 bar a b math and denote the corresponding Carath odory metric on B by d . Then a holomorphic function f B is said to be a complex geodesic if math d f w , f z rho w, z , math for all points w and z in . Properties and examples of complex geodesics Given u X with u 1, the map f B given by f z zu is a complex geodesic. Geodesics can be reparametrized if f is a complex geodesic and g Aut is a bi holomorphic automorphism of the disc , then f small o small g is also a complex geodesic. In fact, any complex geodesic f sub 1 sub with the same image as f i.e., f sub 1 sub f arises as such a reparametrization of f . If math d f 0 , f z rho 0, z math for some z &ne 0, then f is a complex geodesic. If math alpha f 0 , f 0 1, math where &alpha denotes the Caratheodory length of a tangent vector, then f is a complex geodesic. References cite book author Earle, Clifford J. and Harris, Lawrence A. and Hubbard, John H. and Mitra, Sudeb chapter Schwarz s lemma and the Kobayashi and Carath odory pseudometrics on complex Banach manifolds title Kleinian groups and hyperbolic 3 manifolds Warwick, 2001 editor Komori, Y., Markovic, V. and Series, C. eds series London Math. Soc. Lecture Note Ser. 299 pages 363&ndash 384 publisher Cambridge Univ. Press location Cambridge year 2003 Category Hyperbolic geometry Category Metric geometry ... more details
Unreferenced date December 2009 In mathematics , a prime geodesic on a hyperbolic geometry hyperbolic surface is a primitive closed geodesic , i.e. a geodesic which is a curve closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic analysis asymptotic distribution law similar to the prime number theorem . Technical background We briefly present some facts from hyperbolic geometry which are helpful in understanding prime geodesics. Hyperbolic isometries Consider the Poincar half plane model H of 2 dimensional hyperbolic geometry . Given a Fuchsian group , that is, a discrete subgroup of projective linear group PSL 2, R , group action acts on H via linear fractional transformation . Each element of PSL 2, R in fact defines an isometry of H , so is a group of isometries of H . There are then 3 types of transformation hyperbolic, elliptic, and parabolic. The loxodromic transformations are not present because we are working with real number s. Then an element of has 2 distinct real fixed points if and only if is hyperbolic. See Linear fractional transformation Classification Classification ... a closed geodesic of H first, by connecting the geodesic semicircle joining the fixed points of h , we get a geodesic on H called the axis of h , and by projecting this geodesic to M , we get a geodesic on H . This geodesic is closed because 2 points which are in the same orbit under the action ... and possibly false, as it fails to distinguish between geodesic loops and closed geodesics. It can ... fields. Dynamical systems and ergodic theory In dynamical systems, the closed geodesic s represent the Periodic function periodic group action orbits of the GeodesicGeodesic flow geodesic flow . Number theory In number theory, various prime geodesic theorems have been proved which are very similar ... Fuchsian model Analytic number theory Zoll surface DEFAULTSORT Prime Geodesic Category Riemann surfaces ... more details
In Riemannian geometry , the geodesic curvature math k g math of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic. For instance it applies to Curvature Curves on surfaces curves on surfaces . The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold the normal curvature math k n math and the one that comes from the curve itself. The curvature math k math of the curve is related to these two by math k sqrt k g 2 k n 2 math . In particular geodesics have no geodesic curvature they are straight , and that is their definition, so that math k k n math , which explains why they appear to be curved in ambient space whenever the submanifold is. Definition Consider a curve lying on a submanifold math M math in ambient manifold math bar M math , parametrized by arclength math s math , with unit tangent vector math T math . The geodesic curvature is the norm of the projection of the derivative math dT ds math on the tangent plane to the submanifold. Conversely the normal curvature is the norm of the projection of math dT ds math on the normal bundle to the submanifold at the point considered. Example Let math M math be the unit sphere math S 2 math in three dimensional Euclidean space. The normal curvature of math S 2 math is identically 1. Great circles have curvature math k 1 math , which implies zero geodesic curvature, thus they are geodesics. Smaller circles of radius math r math will have curvature math 1 r math and geodesic curvature math k g sqrt 1 r 2 r math . Some results involving geodesic curvature The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold math M math . It does not depend ... Surfaces isbn 0 486 63433 7 . springer id G g044070 title Geodesic curvature first Yu.S. last Slobodyan year 2001 . External links Mathworld urlname GeodesicCurvature title Geodesic curvature curvature ... more details
Disputed date April 2009 In mathematics &mdash specifically, in Riemannian geometry &mdash geodesic convexity is a natural generalization of convex set convexity for sets and convex function functions to Riemannian manifold s. It is common to drop the prefix geodesic and refer simply to convexity of a set or function. Definitions Let M , g be a Riemannian manifold. A subset C of M is said to be a geodesically convex set if, given any two points in C , there is a geodesic arc contained within C that joins those two points. Let C be a geodesically convex subset of M . A function f C R is said to be a strictly geodesically convex function if the composition math f circ gamma 0, T to mathbb R math is a strictly convex function in the usual sense for every unit speed geodesic arc &gamma 0, T &rarr M contained within C . Properties A geodesically convex subset of a Riemannian manifold is also a convex metric space with respect to the geodesic distance. Examples A subset of n dimensional Euclidean space E sup n sup with its usual flat metric is geodesically convex if and only if it is convex in the usual sense, and similarly for functions. The northern hemisphere of the 2 dimensional sphere S sup 2 sup with its usual metric is geodesically convex. However, the subset A of S sup 2 sup consisting of those points with latitude further north than 45 south is not geodesically convex, since the geodesic great circle joining two points on the southern boundary of A may well leave A e.g. in the case of two points 180 apart in longitude , in which case the geodesic arc passes over the south pole . References cite book last Rapcs k first Tam s title Smooth nonlinear optimization in R sup n sup series Nonconvex Optimization and its Applications 19 publisher Kluwer Academic Publishers location Dordrecht year 1997 pages xiv 374 isbn 0 7923 4680 7 MathSciNet id 1480415 cite book last Udriste first Constantin title Convex functions ... more details
Image Geodesic Grid ISEA3H illustrated.png Geodesic Discrete Global Grid PYXIS WorldView 400px right A geodesic grid is a technique used to model the surface of a sphere such as the Earth with a subdivided polyhedron , usually an icosahedron . Introduction A geodesic grid is a global Earth reference that uses cells or tiles to statistically represent data encoded to the area covered by the cell location. The focus of the discrete cells in a geodesic grid reference is different from that of a conventional lattice based Earth reference where the focus is on a continuity of points used for addressing location and navigation. In biodiversity science, geodesic grids are a global extension of local discrete grids that are staked out in field studies to ensure appropriate statistical sampling and larger multi use grids deployed at regional and national levels to develop an aggregated understanding of biodiversity. These grids translate environmental and ecological monitoring data from multiple spatial and temporal scales into assessments of current ecological condition and forecasts of risks to our natural resources. A geodesic grid allows local to global assimilation of ecologically significant ... into a grid in this case, over the geodesy shape of the Earth . Geodesic grids have been .... Another approach gaining favour uses geodesic sphere grids generated by the subdivision of a platonic ... the new cells onto a sphere . In this geodesic grid , each of the vertices in the resulting geodesic ... geodesic grid inherits many of the virtues of 2D hexagonal grids, and specifically avoids problems ... in video games. The quadrilateralized spherical cube is a kind of geodesic grid based on subdividing ... longitude latitude grids in computers History The earliest use of the icosahedral geodesic grid in geophysical ... equation on a spherical geodesic grid journal Tellus volume 20 pages 642 653 year 1968 doi 10.1111 ... BUGS geodesic BUGS climate model page on geodesic grids http www.sou.edu cs sahr dgg Discrete ... more details
In mathematics , a complete manifold or geodesically complete manifold is a Pseudo Riemannian manifold pseudo Riemannian manifold for which every maximal inextendible geodesic is defined on math mathbb R math . Examples All compact space compact manifolds and all homogeneous space homogeneous manifolds are geodesically complete. Euclidean space math mathbb R n math , the sphere s math mathbb S n math and the torus tori math mathbb T n math with their usual Riemannian metric s are all complete manifolds. A simple example of a non complete manifold is given by the punctured plane math M mathbb R 2 setminus 0 math with its usual metric . Geodesics going to the origin cannot be defined on the entire real line. Path connectedness, completeness and geodesic completeness It can be shown that a finite dimensional Connected space Path connectedness path connected Riemannian manifold is a complete metric space if and only if it is geodesically complete. This is the Hopf Rinow theorem . This theorem does not hold for infinite dimensional manifolds. The example of a non complete manifold the punctured plane given above fails to be geodesically complete because, although it is path connected, it is not a complete metric space any sequence in the plane converging to the origin is a non converging Cauchy sequence in the punctured plane. References Citation last1 O Neill first1 Barrett title Semi Riemannian Geometry publisher Academic Press isbn 0 12 526740 1 year 1983 . See chapter 3, pp. 68 . DEFAULTSORT Complete Manifold Category Riemannian geometry ... more details
Other uses2 Dome File StPetersDomePD.jpg thumb right 250px Dome of St. Peter s Basilica in Rome crowned by a cupola . Designed primarily by Michelangelo , the dome was not completed until 1590 A dome is a structural element of architecture that resembles the hollow upper half of a sphere . Dome structures ... pendentive dome of the 6th century church Hagia Sophia . Squinches , the technique of making a transition from a square shaped room to a circular dome, was most likely invented by the ancient ... have a carcass and an outer shell made of wood or metal. The onion dome became another distinctive .... As a domestic feature the dome is less common, tending only to be a feature of the grandest houses ... to admit light and vent air, but gives an extra dimension to the decorated interior of the dome ... arch right . A dome can be thought of as an arch which has been rotated around its central ... from the base of the dome to the top. File Pendentive and Dome.png thumb right 200px A compound dome red with pendentives yellow from a sphere of greater radius than the dome. Drums, also called tholobate s or tambour s, are cylindrical or polygonal walls supporting a dome which may contain windows. When the base of the dome does not match the plan of the supporting walls beneath it for example, a circular dome on a square bay , techniques are employed to transition between the two. The simplest ... support more weight. The invention of pendentives , triangular segments of an even larger dome filling the spaces between the circular bottom of the dome and each of the four corners of the square base ... are triangular sections of a sphere used to blend the curved surface of a dome with the flat surfaces of supporting walls. In the case of the simple dome , the pendentives are part of the same sphere as the dome itself however, such domes are rare. ref name Fletcher Fletcher, Sir Banister, and Dan ... 2267 7. ref In the case of the more common compound dome , the pendentives are part of the surface ... more details
The Dome commonly refers to Millennium Dome , a former Millennium exhibition venue in London, England, now redeveloped as The O2 entertainment venue Louisiana Superdome , home of the New Orleans Saints american football team Hubert H. Humphrey Metrodome , home of the Minnesota Vikings american football team The Dome may also refer to The Dome, Edinburgh , an 1847 built Graeco Roman style building in Edinburgh s New Town, Scotland The Dome Dubai , a planned 44 floor skyscraper in Jumeirah Lake Towers, Dubai, UAE The Dome periodical , a British arts periodical published from 1897 to 1900 The Dome Sydney , an indoor sports arena in the Sydney Olympic Park, Australia The Dome television program , a German television program and music event The Dome Center incorporating the Dome Arena, a fair and convention complex in Henrietta, New York The Dome Leisure Centre , an arena and leisure centre in Doncaster, England Dome of Discovery , a building of the 1951 Festival of Britain, demolished on closure Brighton Dome , an 1805 built arts venue housing three venues Brighton, England Carrier Dome , a stadium owned by Syracuse University, Syracuse, New York See also Dome disambiguation disambig ... more details
Infobox mountain name Dome A photo elevation m 4091 elevation ref prominence m 1639 prominence ref listing ... inline type age first ascent easiest route Dome A or Dome Argus coord 80 22 S 77 21 E type mountain ... a dome or eminence of 4,093 meters elevation above sea level. It is located in the Australian ... sovereignty over Dome A. Description Dome Argus is the summit of the massive East Antarctic ... Dome Argus is considered to be the highest point in the East Antarctica Ranges. Dome A is a plain and the elevation visually is not noticeable. Below Dome A underneath at least 2400 meters thickness of ice sheet is the Gamburtsev Mountain Range . The name Dome Argus was given by the Scott Polar ... for the research of climates in the past. ref name wo Temperatures at Dome A fall below 80 Celsius ... Expeditions CHINARE traversed 1228 km from Zhongshan Station to Dome A and located the highest ... station AWS was deployed at Dome A, and a second station was installed approximately half way between .... Station at Dome A is powered by solar cell s and diesel fuel and requires yearly service and refuelling ... title Dome A coldest place on Earth publisher Wondermondo ref The coldest air temperature recorded at Dome A since January 2005 thus far 28 September 2010 was in July 2005 82.5 C. The Extremes on Earth ... Vostok , which is almost 600 m lower in elevation than Dome A. Analysis of satellite data and atmospheric models shows that the Ridge A which is located 144 km south east from Dome A is potentially ... Observatory on the dome in January 2008. ref cite web url http www.spaceref.com news viewpr.nl.html ... PLATO Dome A robotic observatory publisher UNSW accessdate 2009 12 22 ref Various institutions from ... technology based observation system called DomeA WSN on the dome in January 2008. ref cite web url http www.chinanews.com.cn gn news 2008 01 24 1144158.shtml title Dome A accessdate ... Station in Antarctica, was set up at the dome on January 27, 2009. ref cite web url http www.chinadaily.com.cn ... more details
Unreferenced date November 2006 Orphan date February 2009 Context date October 2009 Tracking the interference amongst various layers, in a multi directional Solid Freeform Fabrication, is extremely important. The geodesic interference level is a common attribute of all the mutually interfering layers. The algorithm for multi directional deposition using geodesic interference level attribute was introduced by Rajeev Dwivedi and Radovan Kovacevic in 2005, at the Research Center for Advanced Manufacturing . DEFAULTSORT Intereference Geodesic Level Category Thin films ... more details
Expert subject Physics date February 2009 Technical date August 2009 In general relativity , the geodesic deviation equation is an equation involving the Riemann curvature tensor , which measures the change in separation of neighbouring geodesic s or, equivalently, the tidal force experienced by a rigid body moving along a geodesic. In the language of mechanics it measures the rate of relative acceleration of two particles moving forward on neighbouring geodesics. In differential geometry , the geodesic deviation equation is more commonly known as the Jacobi field Jacobi equation . Let T sup a sup be the tangent vector to a given geodesic , and X sup a sup a vector field along connecting it to an infinitesimally near geodesic the deviation vector . The relative acceleration of the infinitesimally near geodesic is defined by math a a T b nabla b T c nabla c X a. math The geodesic deviation equation asserts that math a a R bcd aX bT cT d. math To more rigorously formulate the equation, let sub s sub t be a 1 parameter variation through geodesics i.e., for each fixed s , the curve swept out by sub s sub t as t varies is a geodesic with affine parameter. The tangent vector and deviation vector are respectively defined by math begin align T & frac d dt gamma 0 t X & left. frac d ds gamma s t right s 0 . end align math In order that sub s sub be a variation through geodesics, a necessary condition is that the geodesic equation holds math frac D 2 dt 2 X R X,T T. math The geodesic deviation equation can be derived from the second variation of the point particle Lagrangian ... to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much ... appears to have a corresponding generalization of geodesic deviation . Citation needed date September ... Cosmology http vishnu.mth.uct.ac.za omei gr chap6 node11.html Geodesic Deviation DEFAULTSORT Geodesic Deviation Equation Category Differential geometry Category Riemannian geometry Category ... more details
In geometric data analysis and statistical shape analysis , principal geodesic analysis is a generalization of principal component analysis to a Euclidean geometry non Euclidean , non linear setting of manifolds suitable for use with shape descriptors such as medial representation s. References http midag.cs.unc.edu pubs papers TMI04 Fletcher PGA.pdf Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape statistics stub geometry stub Category Image processing Category Digital geometry Category Differential geometry Category Topology Category Data analysis ... more details
Expert subject Mathematics date November 2008 How to date October 2009 Solving the geodesic equations is a procedure used in mathematics , particularly Riemannian geometry , and in physics , particularly in general relativity , that results in obtaining geodesic s. Physically, these represent the paths of usually ideal particles with no four acceleration proper acceleration , their motion satisfying the geodesic equations. Because the particles are subject to no four acceleration, the geodesics generally represent the straightest path between two points in a curved spacetime . The geodesic equation Main Geodesic On an n dimensional Riemannian manifold math M math , the geodesic equation written in a coordinate chart with coordinates math x a math is math frac d 2x a ds 2 Gamma a bc frac dx b ds frac dx c ds 0 math where the coordinates x sup a sup s are regarded as the coordinates of a curve s in math M math and math Gamma a bc math are the Christoffel symbol s. The Christoffel symbols are functions of the Metric mathematics metric and are given by math Gamma a bc frac 1 2 g ad left ... , the geodesic equations are a system of math n math ordinary differential equation s for the math ... to choose one that simplifies the geodesic equations. Mathematically, this means, a coordinate chart is chosen in which the geodesic equations have a particularly tractable form. Effective potentials When the geodesic equations can be separated into terms containing only an undifferentiated ... diagram s apply, in particular the location of turning points. Solution techniques Solving the geodesic ... Definitions general solution , of the geodesic equations. Most attacks secretly employ the point symmetry group of the system of geodesic equations. This often yields a result giving a family of solutions ... of equations equivalent to the geodesic equations. This method has the advantage of bypassing a tedious ... guidelines not WP CSL, which is only a proposal . DEFAULTSORT Solving The Geodesic Equations ... more details
about the use of geodesics in general relativity the general concept in geometry geodesic In general relativity , a geodesic generalizes the notion of a straight line to curved spacetime . Importantly, the world line of a particle free from all external force is a particular type of geodesic. In other words, a freely moving particle always moves along a geodesic. In general relativity, gravity is not a force but is instead a curved spacetime geometry where the source of curvature is the stress energy tensor representing matter, for instance . Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4 D spacetime geometry around the star onto ... geodesics have a tangent vector whose norm is positive. Note that a geodesic cannot be spacelike ... to a timelike Killing vector a spacelike geodesic with its affine parameter within such a space ... expression A timelike geodesic is a worldline which parallel transport s its own tangent vector ... of the tangent is zero. The above equation can be Geodesic general relativity Proofs Proof ... r a cdot s b math . Geodesics as extremal curves A geodesic between two events could also ... d over d tau ln U nu U nu qquad qquad 8 math This is just one step away from the geodesic equation ... because math U nu U nu math is constant . Finally, we have the geodesic equation math Gamma lambda mu nu dot x mu dot x nu ddot x lambda 0 . math Geodesic incompleteness and singularities The notion of geodesic incompleteness is used in the study of gravitational singularities . Approximate geodesic motion True geodesic motion is an idealization where one assumes the existence of test particle ... true geodesic motion. In qualitative terms, the problem is solved the smaller the gravitational ... s field is tiny in comparison with the Sun s , the closer this object s motion will be geodesic ... the same for other matter distributions. See also Geodesic Geodesics as Hamiltonian flows References ... more details
TOCright The French Geodesic Mission also called the Geodesic Mission to Peru , Geodesic Mission to the Equator and the Spanish French Geodesic Mission was an 18th century expedition to what is now Ecuador carried out for the purpose of measuring the roundness of the Earth and measuring the length of a degree of longitude at the Equator . The mission was one of the first geodesic or geodetic missions carried out under modern scientific principles, and the first major international scientific expedition. Background In the 18th century, there was significant debate in the scientific community, specifically in the French Academy of Sciences Acad mie des sciences , as to whether the circumference of the Earth was greater around the Equator or around the poles. French astronomer Jacques Cassini held to the view that the polar circumference was greater. Louis XV of France Louis XV , the King of France and the Academy sent two expeditions to determine the answer one was sent to Lapland, Finland Lapland , close to the North Pole , under Swedish physicist Anders Celsius and French mathematician Pierre Louis Maupertuis Pierre Maupertuis . The other mission was sent to Ecuador, at the Equator. Previous accurate measurements had been taken in Paris by Cassini and others. Expedition Image Ec map.png thumb 250px Ecuador The equatorial mission was led by French astronomer s Charles Marie de La Condamine , Pierre Bouguer , Louis Godin and Spanish geographers Jorge Juan y Santacilia Jorge Juan and Antonio de Ulloa . They were accompanied by several assistants, including the naturalist Joseph ... by the French Geodesic Mission influenced the adoption of the name, Republic of Ecuador when the country ... of the First Geodesic Mission. This second mission was led by General Georges Perrier . Publications ... of the arrival of the First Geodesic Mission. They raised a 10 meter high monument at Mitad del ... mitaddelmundo.html Mitad del Mundo Half of the World First Geodesic Mission http www.mobot.org ... more details
Golden Dome may refer to Main Administration Building University of Notre Dame Golden Dome , the main administration building at the University of Notre Dame in Notre Dame, Indiana Dome of the Rock , a shrine of great religious significance in Jerusalem St. Michael s Golden Domed Monastery in Kiev, Ukraine Gold Dome , a geodesic shaped cultural center in Oklahoma City, Oklahoma Golden Dome Monaca , a multi purpose geodesicdome d arena in Monaca, Pennsylvania Assassins Gate Green Zone , a landmark on the International Zone in Baghdad, Iraq, known as The Golden Dome The Maharishi Patanjali Golden Dome of Pure Knowledge from men and the Bagambhrini Golden Dome for ladies are located on the Maharishi University of Management Gold Dome Georgia State Capitol , is referred to as the Gold Dome because of the gold leaf applied to the structure. See also Dome disambiguation disambig Category Domes ... more details
wikt Dome A dome is a structural element of architecture that resembles the hollow upper half of a sphere. Dome may refer to Geology Dome geology , a deformational feature consisting of symmetrically dipping anticline s Granite dome , a dome of granite , formed by exfoliation Lava dome , a mound shaped growth resulting from the eruption of high silica lava from a volcano Lunar dome , a type of shield volcano found on the surface of the Earth s moon Resurgent dome , a volcanic dome that is swelling or rising due to movement in the magma chamber Salt dome , formed when a thick bed of evaporite minerals mainly salt, or halite found at depth intrudes vertically into surrounding rock strata Architecture Geodesicdome Monolithic dome Geography Antarctica Anderson Dome Arctowski DomeDome A Bonnabeau DomeDome C Dome F Law Dome Siege Dome Titan Dome Canada Dome, Ontario People Malcolm Dome Ram Chandra Dome Other Dome band , a 1980s post punk band Dome coffeehouse , a chain of caf restaurants based in Perth, Western Australia Perth, Australia Dome constructor , a Japanese based racing car constructor Dome mathematics , a closed geometrical surface which can be obtained by sectioning off a portion of a sphere with an intersecting plane The Dome periodical The Dome periodical , a quarterly ... D me, a garagiste wine label of Bordeaux wine producer Ch teau Teyssier Le Dome Cafe Le D me Caf , historical Paris intellectual venue Dome car , a type of railway passenger car Dome GWCC Philips Arena CNN Center MARTA station , a passenger rail station in Atlanta, Georgia named after the Georgia Dome Teapot Dome scandal , Wyoming, United States Steam dome , a steam locomotive component Dome , a 1987 science fiction novel by Michael Reaves and Steve Perry author Steve Perry Dome, slang for fellatio Dome, slang for a bald head A nickname for the Hubert H. Humphrey Metrodome . disambig de Dome fr D me it D me nl Dome pl Dome pt Domo th ... more details
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