6/12/2023 0 Comments Curved space coordinate radius![]() ![]() The difference is that geodesics are only locally the shortest distance between points, and are parameterized with "constant speed". In general, geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. In such a case, any of these curves is a geodesic.Ī contiguous segment of a geodesic is again a geodesic. It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. The resulting shape of the band is a geodesic. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its width, and in so doing will minimize its energy. Equivalently, a different quantity may be used, termed the energy of the curve minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimization). It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from f( s) to f( t) along the curve equals | s− t|. This has some minor technical problems because there is an infinite-dimensional space of different ways to parameterize the shortest path. Timelike geodesics in general relativity describe the motion of free falling test particles.Ī locally shortest path between two given points in a curved space, assumed to be a Riemannian manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations. Geodesics are of particular importance in general relativity. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing geodesic curvature. 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. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). In the original sense, a geodesic was the shortest route between two points on the Earth's surface. 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. ![]() It is a generalization of the notion of a " straight line". The term also has meaning in any differentiable manifold with a connection. ə ˈ d ɛ s ɪ k, - oʊ-, - ˈ d iː s ɪ k, - z ɪ k/) is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. For other uses, see Geodesic (disambiguation). For the application on Earth, see Earth geodesic. For the study of Earth's shape, see Geodesy. For geodesics in general relativity, see Geodesic (general relativity). This article is about geodesics in general. ![]()
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