[Proj] Geodesic distances away from the ellipsoid
strebe at aol.com
Mon Jul 13 21:40:52 EST 2009
I agree that a geodesic on a projected surface need not be a simple scaling of a geodesic on the original surface even if the paths of the geodesics coincide. While I wrote "the surface of constant height above an ellipsoid must also be an ellipsoid" as if "must be an ellipsoid" were a constraint for any solution, in fact I merely thought it was a constraint of the posed problem. I did not realize (or recall) that other surfaces were up for consideration. Presumably constant height is of interest for... flying? Or...? (I cannot imagine any practical use for a precise solution, but there's no reason that should shut off consideration.)
You originally stated that the geodesics should coincide because a normal section through a point on E is a normal section through the corresponding point on S. Surely that's insufficient to claim a geodesic would coincide, since all it gives is a local direction for the geodesic, not the derivative of the direction that informs the direction of the next infinitesimal section in the geodesic's path. What might be two "adjacent" normal sections on the ellipsoid along its geodesic would belong to two different geodesics on the raised surface. Not a proof by any means, but I think it's analogous to saying the slope along a two-dimensional path tells you nothing about the behavior of the path away from that point. Presumably a rigorous form of this observation is what convinced you of the error.
— daan Strebe
On Jul 12, 2009, at 4:10:00 AM, "Karney, Charles" <ckarney at Sarnoff.com> wrote:
> From: daan Stebe
> 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.
Indeed my statement is false.
But your way of showing this doesn't work because while an ellipsoid
raised by h stops being an ellipsoid, an ellipsoidal geodesic raised
by h also ceases to have the properties of an ellipsoidal geodesic.
A simple counter-example is a cylinder with a cross section which is a
stadium (two semi circles joined by straight segments). When the
cylinder is unfolded, the geodesic spiralling up such a surface is a
straight line. However if the surface is mapped into another cylinder
a distance h away, this straight line maps into a connected sequence
of straight lines which have different slopes corresponding to the
flat and round portions of the stadium. This obviously is not the
geodesic for the expanded cylinder.
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