Search results
Results From The WOW.Com Content Network
A geodesic, on a Riemannian surface, is a curve that is locally straight at each of its points. On the Euclidean plane the geodesics are lines, and on a sphere the geodesics are great circles. The shortest path in the surface between two points is always a geodesic, but other geodesics may exist as well.
Klein quartic with 28 geodesics (marked by 7 colors and 4 patterns). In geometry, a geodesic (/ ˌ dʒ iː. ə ˈ d ɛ s ɪ k,-oʊ-,-ˈ d iː s ɪ k,-z ɪ k /) [1] [2] is a curve representing in some sense the locally [a] shortest [b] path between two points in a surface, or more generally in a Riemannian manifold.
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.. 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.
In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. In the intrinsic approach, a great circle is a geodesic; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane ...
Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of (usually ideal) particles with no proper acceleration , their motion satisfying the geodesic equations.
The unit circle with the metric inherited from the Euclidean metric of (the chordal metric) is not a path metric space. The induced intrinsic metric on S 1 {\displaystyle S^{1}} measures distances as angles in radians , and the resulting length metric space is called the Riemannian circle .
This marked a new departure from tradition because for the first time Gauss considered the intrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space.
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, non-gravitational forces is a particular type of geodesic.