[PROJ] Transverse and oblique Mercator

[Melita Kennedy]

-----Original Message-----

>Date: Tue, 14 Jan 2020 21:53:28 -0500
>From: Pierre Abbat <phma at bezitopo.org>
>To: proj at lists.osgeo.org
>Subject: Re: [PROJ] Transverse and oblique Mercator
>
>On Tuesday, January 14, 2020 9:14:33 AM EST Charles Karney wrote:
>> Transverse Mercator as implemented in PROJ is accurate to about 5
>> nanometers within 3900 km of the central meridian.  See Fig. 2 on
>> 
>> https://geographiclib.sourceforge.io/html/transversemercator.html#tmfigures
>> 
>> GeographicLib implements an "exact" version of transverse Mercator (in
>> terms of elliptic functions) which is accurate to 9 nanometers over
>> the whole ellipsoid.  However, this isn't included in PROJ.
>
>I'm thinking of using TMcoords.dat in Bezitopo to test the accuracy of the 
>projection. What would be a good way to graph the results? I'd post the graphs 
>that transmer makes, but the file is 5 MB, including several graphs for each 
>ellipsoid.
>
>Bezitopo computes the projection as follows:
>1. Conformally project the ellipsoid to a sphere of equal volume.
>2. Project the sphere transversely to a plane, giving Gauss-Schreiber.
>3. Pass the resulting point as a complex number to a Fourier series and add it 
>to the point.
>This seems to be what Krüger was doing before he computed the n-series, but I 
>have trouble understanding the hundred-year-old math paper in German with 
>archaic notation,
>
>> Regarding oblique Mercator...  I understand how this is defined for a
>> sphere.  But, I'm not sure there's a well accepted definition of how
>> it is defined on an ellipsoid.  In the case of Mercator (resp.
>> transverse Mercator) the equator (resp. central meridian) projects to
>> a straight line at equal scale.  The equivalent line would need to be
>> specified for oblique Mercator.
>
>However it's defined in Alaska, Switzerland, and anywhere else that uses the 
>projection, that's how I have to do it. There may be more than one way of 
>specifying an instance of the projection (for conformal conic, you can specify 
>two parallels, or one parallel and a scale), but if it's the same projection, 
>it needs only to be implemented once.

John Snyder discusses 4 different oblique Mercator implementations on page 162 
of his book Flattening the Earth. Rosenmund's was first and is used in 
Switzerland. It's not quite the same as Hotine's which is used in Alaska, but 
it's close enough that Esri, at least, is using Hotine's version for Switzerland. 
Madagascar uses Laborde's version. It converts to a conformal sphere, then uses 
a spherical transverse Mercator, and finally a 3rd-order complex-algebra 
transformation to make it oblique. It cannot be emulated with Hotine with a 
reasonable accuracy. Snyder also lists one designed by Cole for Egypt, but it's 
not in common use.

Disclosure: I work for Esri.

Melita