[Proj] Rhumb lines and Mercator on a triaxial ellipsoid

Charles Karney charles.karney at sri.com
Sun Dec 21 08:47:54 EST 2014

The longitude is undefined.  You may end up retracing your path.

On 12/21/2014 08:44 AM, Mikael Rittri wrote:
> 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
>>    only little had to be added, finds no mention."
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