Plotting path between sphere or ellipsoid points? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:22:28Z http://mathoverflow.net/feeds/question/25531 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25531/plotting-path-between-sphere-or-ellipsoid-points Plotting path between sphere or ellipsoid points? nicros 2010-05-21T21:31:52Z 2010-05-24T15:18:20Z <p>Hi, my apologies if this is not the right place to ask this- I am not a mathematician (I'm a software engineer) and Im working on some 3D applications.</p> <p>My situation is this- given an origin of 0,0,0 and any two points I need to be able to return the xyz coordinate of any point that is on the line between these two points- with the assumption being that the two points must fall on a symmetrical closed surface surrounding the origin, sphere or ellipsoid. So the line would follow the surface shortest distance between the two points.</p> <p>So for example, I would need the xyz point that is 0.1 of the line length... but I have no idea how to get it...</p> <p>Is this possible from 2 points and an origin only? I am implementing this in C# if that helps.</p> <p>Thanks!</p> http://mathoverflow.net/questions/25531/plotting-path-between-sphere-or-ellipsoid-points/25583#25583 Answer by coudy for Plotting path between sphere or ellipsoid points? coudy 2010-05-22T13:30:00Z 2010-05-22T17:17:25Z <p>I guess you are in the GPS business, aren't you ? I think that Vincenty 1975 <a href="http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf" rel="nofollow">paper</a> is what you are looking for. At least it should be a starting point for a bibliographic search.</p> <p>Let me add a few remarks. Fortunately, free motion on the ellipsoid is an integrable system. Which means (loosely) that you can explicitely solve the equations of the trajectory using just a few integrals. This was done by Jacobi (1838).</p> <p>So if you are not happy with Vincenty approach, there are two paths you can follow. Either look in a book (or click <a href="http://mathworld.wolfram.com/EllipsoidGeodesic.html" rel="nofollow">here</a>) for the differential equations satisfied by the geodesics, and do a numerical integration. Or you can start from the <a href="http://mathworld.wolfram.com/OblateSpheroidGeodesic.html" rel="nofollow">solutions</a> of these equations, which are given by elliptic functions. There are standard libraries in C for computing numerical values for these functions.</p> <p>As a reference, I recommend the book "Elliptic functions and applications" by Derek F. Lawden. As far as I recall, the problem is solved in the book (I hope my memory is not betraying me). And I should add, this is a great book for everybody interested in making the connection between elliptic functions and classical mechanics.</p> <p>By the way, if you are interested in the following question: on which manifold is the geodesic flow is integrable ? then you can have a look at a short <a href="http://www.dpmms.cam.ac.uk/~hk244/miller.cp.pdf" rel="nofollow">survey</a> by Andre Miller. And if you are interested in a clever proof of the integrability of the geodesic flow that works in any dimension, there is an <a href="http://www.math.psu.edu/tabachni/prints/ellipsoids.pdf" rel="nofollow">online</a> paper by S. Tabachnikov. </p>