[Proj] Geodesic distances away from the ellipsoid
ckarney at Sarnoff.com
Sun Jul 12 06:10:00 EST 2009
> > From: Charles Karney
> > 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?
> 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.
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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