Question

How do you exactly compute the distance traveled between two points at different altitudes on a spherical body? If the two points are at the same altitude it’s a simple great-circle calculation. But what is the additional term to account for a steady climb or descent precisely? Say we’re talking about a spaceplane that steadily climbs up to a great height over a great distance after taking off.

Illustration:
https://i.stack.imgur.com/3n7jL.png

Answer 1

The National Geodetic Survey (NGS) (a division of NOAA) has some information on this, and even sample Fortran code and working programs on their website and for a PC.

See:
http://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html

The program that you want is INVERS3D:
http://www.ngs.noaa.gov/PC_PROD/Inv_Fwd/

You will need to look through their code for specifics, but they calculate “ellipsoidal distance, the mark-to-mark distance, and the ellipsoid height difference” using lat/long/altitude.

From their website:

INVERS3D

Program INVERS3D is the three dimensional version of program INVERSE,
and is the tool for computing not just the geodetic azimuth and
ellipsoidal distance, but also the mark-to-mark distance, the
ellipsoid height difference, the DX, DY, DZ (differential X, Y, Z used
to express GPS vectors), and the DN, DE, DU (differential North, East,
Up using the FROM station as the origin of the NEU-coordinate system).
The program requires geodetic coordinates as input, expressed as
either: 1) latitude and longitude in degrees, minutes, and seconds or
decimal degrees along with the ellipsoid heights for both stations, or
2) rectangular coordinates (X, Y, Z in the Conventional Terrestrial
Reference System) for each station. The program works exclusively on
the GRS80 ellipsoid and the units are meters. Both types of
coordinates may be used in the same computation. The program reads
input geodetic positions with the default hemispheres for latitude and
longitude set at North and West.