Timeline for Determining geodesics between two points in curved space [closed]
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 4, 2015 at 11:37 | history | closed |
Deane Yang Willie Wong Benoît Kloeckner Marco Golla Anton Petrunin |
Not suitable for this site | |
| Nov 3, 2015 at 19:18 | vote | accept | imranal | ||
| Nov 3, 2015 at 14:23 | answer | added | Vladimir S Matveev | timeline score: 6 | |
| Nov 3, 2015 at 13:23 | answer | added | Narasimham | timeline score: 5 | |
| Nov 2, 2015 at 22:48 | comment | added | imranal | Let us continue this discussion in chat. | |
| Nov 2, 2015 at 21:12 | comment | added | imranal | These equations can most often only be solved numerically. That is also my purpose. I will check out those methods, thanks! | |
| Nov 2, 2015 at 21:09 | comment | added | imranal | The boundary value problem seems superfluous, from a practical point of view (the shortest distance in Euclidean space is a straight line that connects the two points - why is it not possible to restate this for curved space?). What is the practical point if the problem can not be restated as a way of determining the geodesics that connect two points (i.e any method which can determine u' and v' initial values for two points - provided the two points (u0,v0) and (u1,v1) are valid - unlike the case Igor stated). | |
| Nov 2, 2015 at 20:57 | comment | added | Willie Wong | I ask because if you want to solve "numerically" then you may want to look into geometric flows like the curve-shortening flow with fixed end points, or the harmonic map heat flow with fixed end points. | |
| Nov 2, 2015 at 20:54 | comment | added | Willie Wong | (1) You don't need to determine the initial directions to solve the geodesic equation, which it is soluble. (2) You can get the initial directions by first solving the geodesic equation and then taking derivative at initial time. (3) You seem to be very interested in "solving" the geodesic equation. What concept of a solution are you using? | |
| Nov 2, 2015 at 20:40 | comment | added | imranal | @Willie-Wong Phrased in a different manner : how can I determine the initial directions u',v', if I know at prior, two points that exist upon the geodesic : (u0,v0) and (u1,v1). For example on a sphere I can determine the directions geometrically, since I know that the geodesic curve connects the two points along the great circle. I thought that the entire purpose of determining the geodesics, originally, was to establish the shortest distance between two points (underline the two). | |
| Nov 2, 2015 at 20:12 | comment | added | Willie Wong | You need to look up "boundary value problem for ODEs". Short summary: there is an established theory for solving second order ODEs by specifying the values at the left and right end points, instead of specifying initial position and velocity. On geodesically complete manifolds the existence of a solution is guaranteed. In general this is an optimization problem and is studied in calculus of variations. | |
| Nov 2, 2015 at 19:34 | review | Close votes | |||
| Nov 2, 2015 at 21:42 | |||||
| Nov 2, 2015 at 19:33 | answer | added | Igor Rivin | timeline score: 8 | |
| Nov 2, 2015 at 19:04 | history | asked | imranal | CC BY-SA 3.0 |