# [Proj] triaxial ellipsoid

Charles Karney charles.karney at sri.com
Wed Jan 9 15:57:24 EST 2013

```On 2013-01-09 15:06, Aleksander Yanovskiy wrote:
> Thank you both Charles and Jean-Marc for your work!
> The visualizations at http://geographiclib.sourceforge.net/1.29/triaxial.html are really helpfull for understanding the material. Could you please advise me some free software to generate the figures with another initial conditions ?
>
> Aleksander

I can tell you what I did to make the plots:

(1) Compute the path of geodesic using Matlab.  The code for transpolar
and circumpolar geodesics is straightforward: tabulate the two integrals
over a period and then use interp1q to interpolate and reverse
interpolate these integrals, e.g., for circumpolar geodesics:

sigma -> integral -> omega

Umbilical geodesics are trickier because the rules for traversing the
umbilical points need to be worked out and coded up.  In addition the
line isn't well parameterized by either beta or omega when it lies near
the middle ellipse, so I ended up combining both parameterizations to
get a smooth result.

(2) To plot, I converted beta,omega to cartesian coordinates, projected
these to a 2d orthographic plane, replaced the back portions of the
geodesics by NaN, and used Matlab's 2d plot function.  (I suppose I
could have used the 3d graphics capabilities of Matlab; but I liked the
control that I had over the output doing the 2d conversion myself.)

You can equally well use octave for these tasks if you want a free
solution.

I might end up coding these routines for public consumption.  However
that's on a back burner for me.  First, I need to write up my algorithm
for exact sampling from a normal distribution.  See

http://randomlib.sf.net/html/classRandomLib_1_1ExactNormal.html
```