[Proj] Geodesic distances away from the ellipsoid

Karney, Charles ckarney at Sarnoff.com
Sat Jul 11 11:03:17 EST 2009

> From: Jay Hollingsworth <jhollingsworth at houston.oilfield.slb.com>
> Sent: Fri 20-Mar-09 11:56
> Subject: [Proj] Geodesic distances away from the ellipsoid

> At the risk of asking a dumb question, do any of the geodesic
> algorithms allow calculation of the geodesic distance if the path is
> not on the ellipsoid? Like in an airplane or satellite whose path
> could be assumed to be a constant height above the ellipsoid?

The paper

  Richard Mathar
  Geodetic Line at Constant Altitude above the Ellipsoid

addresses this problem.  However, I believe there is a simple solution.

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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