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Subject: Re: Plot line between two points on ellipsoid
Date: Mon, 17 May 2010 02:35:05 +0000 (UTC)
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"Els " <y.e.t.reeuwijk@student.utwente.nl> wrote in message <hsp41n\$28e\$1@fred.mathworks.com>...
> I have two points on the surface of my ellipsoid. I calculated the distance between them with the Vincenty formula. But now I want to plot this distance on the ellipsoid distance as a line as well. For a sphere this is easy, but I do not know how to do it for an ellipsoid.
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Correct me if I am wrong, but the impression I get is that Vincenty's formulae make the assumption that the ellipsoid in question is an oblate spheroid, meaning that two of its semi-axes are of equal length, since he was primarily interested in the earth considered as an ellipsoid and the geodesics thereupon.  See: "Vincenty's formulae" at:

http://en.wikipedia.org/wiki/Vincenty's_formulae

Els, I believe you implied you wanted to deal with a general ellipsoid with all differing semi-axes when you said, "I am using a small ellipsoid, which does not look like the earth."

On checking I found an interesting article at:

http://mathworld.wolfram.com/EllipsoidGeodesic.html

that makes no such assumption.  It states that an ellipsoidal geodesic can be expressed in the form of their eqs. (8) and (9) which together could constitute a pair of not very frightening-looking differential equations involving a two confocal ellipsoidal coordinates as explained in:

http://mathworld.wolfram.com/ConfocalEllipsoidalCoordinates.html ,

rather than more conventional coordinates.  The independent variable would be the arc length along the geodesic, so that would fall out naturally.  This seems like a fascinating possibility.  There is an arbitrary constant present which they call 'theta' which presumably corresponds to the infinitely many possible closed geodesics possible on an ellipsoid through each point.  The task would be to start at one point using that point's pair of confocal coordinates, and so adjust theta that you will eventually arrive at the other desired point with its particular pair of confocal coordinates.

Unfortunately the author of this article was very short on details so it would require quite a bit of study to see if there could be a useful algorithm lurking therein.

Roger Stafford
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