[Proj] Rhumb lines and Mercator on a triaxial ellipsoid

Charles Karney charles.karney at sri.com
Sat Dec 20 09:33:52 EST 2014

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


The integrals that Jacobi gives can be written in terms of elliptic
integrals, see


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
   only little had to be added, finds no mention."

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