[PROJ] Transverse and oblique Mercator

[Pierre Abbat]

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.

Pierre
-- 
gau do li'i co'e kei do