[Proj] Geodesic distances away from the ellipsoid

strebe strebe at aol.com
Sat Jul 11 15:14:00 EST 2009

Charles Karney asks:
>(1) is this observation true?

In order for it to be true, the surface of constant height above an ellipsoid must also be an ellipsoid, and must be the same ellipsoid but scaled by a constant. That this is not true can be seen by the two-dimensional case: If you scale an ellipse, the new ellipse's major axis and minor axis are scaled by the same amount, but since the major and minor axes are different lengths, scaling by the same amount cannot result in the same differential added to both axes. If the differentials are not the same, the height of the new ellipse over the old one cannot be constant. This generalizes to three dimensions.

The exception is the sphere.

— daan Strebe

On Jul 11, 2009, at 9:03:17 AM, "Karney, Charles" <ckarney at Sarnoff.com> wrote:
Let E be the ellipsoid and S a surface a constant height h above it.
A normal section through a point on E is a normal section through the
corresponding point on S.  Thus mapping a geodesic on E to S by
elevating it by h results in a geodesic on S.  This directly gives you
the course of the geodesic.  A little extra work gives you the azimuth
and length.

A couple of questions suggest themselves:

(1) is this observation true?
(2) is it new?

Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300

Tel: +1 609 734 2312
Fax: +1 609 734 2662

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