# [Proj] Geodesic distances away from the ellipsoid

Karney, Charles 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
>
> No.
>
> 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
```