[Proj] triaxial ellipsoid

[Charles Karney]

On 2012-10-17 11:25, Aleksander Yanovskiy wrote:
> Greetings!
>
> Could you please suggest me a software library that can calculate distances on the surface of the triaxial Earth ellipsoid ?
>
> Sincerely,
> Aleksander

The quick answer is "no", I know of no such library.

It might be worth digging a little into the issue, however...

For a geodesist, the issue has limited interest.  A triaxial model of
the earth has major/minor equatorial axes of 6378137 +/- 35 meters (see
my E-mail "Analyzing the bumps in the EGM2008 geoid" to proj.4 on
2011-07-04).  The EGM2008 geoid differs from the triaxial model by up to
+/- 70 m.  Thus while an oblate ellipsoid is a lot better at
approximating the geoid than a sphere (cutting the errors from 2 km to
100m), the triaxial ellipsoid offers only a modest increase in fidelity
(100m to 70m) at a considerable cost (see next...).

For a mathematician, a triaxial ellipsoid is of considerable interest.
Geodesics on an oblate ellipsoid are "easy" (in mathematical terms)
because the rotational symmetry of the system allows you to reduce the
problem to quadrature.  With triaxial ellipsoids the problem become a
*lot* harder (and therefore more interesting).  There was considerable
excitement when Jacobi solved this problem with a "remarkable"
substitution, see

   http://books.google.com/books?id=RbwGAAAAYAAJ&pg=PA309
   http://books.google.com/books?id=Rh8GAAAAYAAJ&pg=RA1-PA267

If you google this problem you will see that this is still an active
area.  Finally, I will note that Weierstrass treats the triaxial problem
in the geodetic context in

   http://books.google.com/books?id=9O4GAAAAYAAJ&pg=PA257

But perhaps, that's just because his funding source was tied to military
objectives (and he was basically conning his sponsors)?

-- 
Charles Karney <charles.karney at sri.com>
SRI International, Princeton, NJ 08543-5300

Tel: +1 609 734 2312
Fax: +1 609 734 2662