# [Proj] Rhumb lines and Mercator on a triaxial ellipsoid

Mikael Rittri Mikael.Rittri at carmenta.com
Sun Dec 21 08:44:16 EST 2014

```So, does this mean that if I travel along a rhumb line, and reach a pole after a finite distance, and continue the same distance beyond the pole, I would end up at my original latitude but at the opposite (antipodal) meridian?
That sounds fairly intuitive, but I always thought it was undefined how to continue a rhumb line beyond a pole, due to the infinite number of encirclings around the pole.

Interesting.

Mikael Rittri
Carmenta
Sweden

20 dec 2014 kl. 16:50 skrev "Noel Zinn (cc)" <ndzinn at comcast.net>:

> Nice.  A meridian is a limiting case of a rhumb line.  It's not at all
> surprising that a meridian undergoes a N/S reversal passing through a pole.
> E/W reversal in the general case is not so intuitive, but Fermat's spiral is
> a template for that.  -Noel
>
> Noel Zinn, Principal, Hydrometronics LLC
> +1-832-539-1472 (office), +1-281-221-0051 (cell)
> noel.zinn at hydrometronics.com (email)
> http://www.hydrometronics.com (website)
>
> -----Original Message-----
> From: Charles Karney
> Sent: Saturday, December 20, 2014 8:33 AM
> To: PROJ.4 and general Projections Discussions
> Subject: [Proj] Rhumb lines and Mercator on a triaxial ellipsoid
>
> It is well known that a rhumb line arrives at a pole in a finite
> distance after encircling the pole infinitely many times.  Craig Rollins
> recently asked me what heading a rhumb line has *after* passing through
> the pole.
>
> One way of answering this is to consider a rhumb line on a triaxial
> ellipsoid and to take the limit as the two large axes approach one
> another.  The (somewhat surprising) result is that the heading of the
> rhumb line is reversed.  E.g., if the initial heading is NE, the heading
> after passing through the pole is SW.
>
> This got me to thinking about the Mercator projection on a triaxial
> ellipsoid.  This was given by Jacobi in 1843, see section 28 of
>
>   https://www.worldcat.org/oclc/440645889
>
> The integrals that Jacobi gives can be written in terms of elliptic
> integrals, see
>
>   https://dx.doi.org/10.1007/978-3-642-32618-9_17
>   http://geographiclib.sf.net/html/triaxial.html#triaxial-conformal
>
> Finally, Jacobi has an interesting take on Gauss' work on conformal
> projections (excerpted from Balagangadharan's translation):
>
>   "Among the different ways of representing a curved surface on a plane,
>   as is necessary for a map, one prefers, above all, the method of
>   projection in which infinitely small elements remain similar.  In the
>   preceding century Lambert had been concerned with various aspects of
>   this projection, of which one can learn in detail from his
>   contributions to mathematics.  Because of these, Lambert's colleague
>   at that time, Lagrange, was induced to undertake an investigation from
>   the same standpoint and gave the solution completely for all surfaces
>   of revolution.  The Copenhagen Academy which later announced a prize
>   for the solution of this problem for all curved surfaces awarded it to
>   the treatise sent in by Gauss.  In this, Lagrange's work, to which