# [Proj] triaxial ellipsoid

Charles Karney charles.karney at sri.com
Fri Jan 25 18:06:56 EST 2013

On 2013-01-09 11:53, Charles Karney wrote:
> Jean-Marc Baillard has coded up Jacobi's solution for geodesics on a
> triaxial ellipsoid for an HP-41 calculator, see
>
>    http://hp41programs.yolasite.com/geod3axial.php
>
> I've also written up some notes on the problem which may be of interest
>
>    http://geographiclib.sourceforge.net/1.29/triaxial.html
>
> Disclaimer: A triaxial ellipsoid approximates the earth only slightly
> better than an ellipsoid of revolution. If you are really considering
> measuring distances on the earth using a triaxial ellipsoid, you should
> also be worrying about the shape of the geoid (which essentially makes
> the geodesic problem a hopeless mess).

In the limit, when the minor semiaxis vanishes, the problem of geodesics
on a triaxial ellipsoid reduces to the problem of a billiard ball
bouncing on an elliptical table.  Billiard problems on tables of various
shapes have been thoroughly studied since the 1970s.  Searching for
"billiard ellipse" will take you to animations.  In brief:

(1) the 3 closed orbits for the ellipsoid (the 3 principal sections)
become (a) rolling around the perimeter, (b) the major axis of the
ellipse, (c) the minor axis of the ellipse.  (a) and (c) are stable;
(b) is unstable!

(2) the umbilical points are the foci of the ellipse.

(3) a circumpolar geodesic is the ball bouncing outside the foci, where
the envelope of the motion is a confocal ellipse.

(4) a transpolar geodesic is the ball bouncing between the foci, where
the envelope of the motion is a confocal hyperbola.

(5) a umbilical geodesic is the ball intersecting the foci.

Note that the billiard table is really double sided with even (resp.
odd) numbered segments corresponding to the top (resp. bottom) of the
table or ellipsoid.

--Charles