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Geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a " straight line". The noun '' geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of havi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoidal Geodesic
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an '' oblate ellipsoid'', a slightly flattened sphere. A ''geodesic'' is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry . If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the equator and the meridians are the only simple closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometry, isometric embedding in Euclidean space. Surfaces naturally arise as Graph of a function, graphs of Function (mathematics), functions of a pair of Variable (mathematics), variables, and sometimes appear in parametric form or as Locus (mathematics), loci associated to Curve#Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic Curvature
In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold \bar, the geodesic curvature is just the usual curvature of \gamma (see below). However, when the curve \gamma is restricted to lie on a submanifold M of \bar (e.g. for curves on surfaces), geodesic curvature refers to the curvature of \gamma in M and it is different in general from the curvature of \gamma in the ambient manifold \bar. The (ambient) curvature k of \gamma depends on two factors: the curvature of the submanifold M in the direction of \gamma (the normal curvature k_n), which depends only on the direction of the curve, and the curvature of \gamma seen in M (the geodesic curvature k_g), which is a second order quantity. The relation between these is k = \sqrt. In particular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (graph Theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance between two vertices and is defined as the length of a shortest directed path from to consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, does not necessarily coincide with —so it is just a quasi-metric, and it might be the case that one is defined while the other is not. Related concepts A metric space defined over a set of points in terms of distances in a graph defined ov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great-circle Distance
The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the sphere. (By comparison, the shortest path passing through the sphere's interior is the chord between the points.) On a curved surface, the concept of straight lines is replaced by a more general concept of geodesics, curves which are locally straight with respect to the surface. Geodesics on the sphere are great circles, circles whose center coincides with the center of the sphere. Any two distinct points on a sphere that are not antipodal (diametrically opposite) both lie on a unique great circle, which the points separate into two arcs; the length of the shorter arc is the great-circle distance between the points. This arc length is proportional to the central angle between the points, which if measured in radians can be scaled u ... [...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] |
Geodetic
Geodesy or geodetics is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D. It is called planetary geodesy when studying other astronomical bodies, such as planets or circumplanetary systems. Geodynamical phenomena, including crustal motion, tides, and polar motion, can be studied by designing global and national control networks, applying space geodesy and terrestrial geodetic techniques, and relying on datums and coordinate systems. Geodetic job titles include geodesist and geodetic surveyor. History Geodesy began in pre-scientific antiquity, so the very word geodesy comes from the Ancient Greek word or ''geodaisia'' (literally, "division of Earth"). Early ideas about the figure of the Earth held the Earth to be flat and the heavens a physical dome spanning over it. Two early arguments for a spherical Earth were that lunar eclipses appear to an observer as circular shadows and that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray (optics), ray of light. Lines are space (mathematics), spaces of dimension one, which may be Embedding, embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two Point (geometry), points (its ''endpoints''). Euclid's Elements, Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as Non-Euclidean geometry, non-Euclidean, Project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the ''minor arc'', and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Transport
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay '' parallel'' with respect to the connection. The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of ''connecting'' the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one way of connecting up the geometries of points on a curve is tantamount to providing a ''connection''. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, ''vice versa'', parallel transport is the local realization of a connection. As parallel transport suppl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
