[PROJ] Ellipsoidal distances, with different heights

[Charles Karney]

Even's suggestion (compute the geodesics on a height adjusted
ellipsoid) is probably as good as you're going to get for now.  There
has been some work on the restricted version of this problem, computing
geodesics on a surface of constant height above the ellipsoid.  See

  Richard J. Mathar
  Geodetic Line at Constant Altitude above the Ellipsoid
  https://arxiv.org/abs/0711.0642

I toyed with the idea of implementing a bullet-proof solution of thie
problem, but I have other irons in the fire at present.

On Wed, Aug 28, 2024 at 6:04 AM Even Rouault via PROJ
<proj at lists.osgeo.org> wrote:
>
> Or even simpler, compute geodesic distance on ellipsoid (a + h_mean, b +
> h_mean) where h_mean is the mean of h_start and h_end.  If h_start and
> h_end are small compared to a, I would expect whatever mean formula used
> to lead to similar results. At least this method is guaranteed to give
> the correct result when h_start = h_end = 0 ...
>
> Le 28/08/2024 à 01:36, Even Rouault via PROJ a écrit :
> > Nyall,
> >
> > I'm not sure there's a real definition to what you want to accomplish.
> >
> > I guess I would :
> >
> > - use the geod_position() of geodesic.h to compute a sufficient number
> > of intermediate positions
> >
> > - linearly interpolate the ellipsoidal height
> >
> > - convert the resulting (lon, lat, h) to geocentric (X, Y, Z) using
> > +proj=cart
> >
> > - use 3D Cartesian distance to compute each intermediate segment
> >
> > - sum them up
> >
> > A cheaper alternative might be to compute the geodesic distance
> > between the start and end points both on the ellipsoid (a + h_start, b
> > + h_start) and on the one (a + h_end, b + h_end), and compute some
> > sort of mean (arithmetic, geometry, ... ?) on those 2 distances.
> >
> > Even
> >
> >
> > Le 28/08/2024 à 01:06, Nyall Dawson via PROJ a écrit :
> >> Hi list,
> >>
> >> Let's say I have two points on an ellipsoid, with each point having a
> >> different height above the ellipsoid. I want to calculate a kind of
> >> "geodesic" between these points, where there's an assumption that the
> >> gradient of the height-above-ellipsoid for the "geodesic" is constant.
> >>
> >> Is this mathematically solvable? Or, more to the point, is it possible
> >> to calculate this using any of the methods exposed via geodesic.h?
> >>
> >> Nyall