[PROJ] Transverse and oblique Mercator

[Charles Karney]

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.

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.

   --Charles

On 1/14/20 8:51 AM, Pierre Abbat wrote:
> I've been working on a surveying CAD program, one of whose parts is converting
> between lat/long and the coordinate systems used in surveying. I did Lambert
> conformal conic first, since it's easier (and I live in North Carolina), then
> figured out transverse Mercator. (I also did spherical stereographic, which
> Bezitopo uses for internal purposes.) The test for projections, before I wrote
> transverse Mercator, required that, at all but a negligible fraction of points
> (i.e. a branch cut), the inverse projection of the projection of a point be
> within 1 mm of the point and the distance between two points 1 m apart in a
> random direction match the computed scale. When I added transverse Mercator, I
> had to reduce all but a negligible fraction, but I lowered it to no less than
> 80%. It is accurate to within 100 µm over 71% of the earth, which is within
> 45° of the central meridian. How does this compare to PROJ?
> 
> Lambert conic and transverse Mercator together cover all but one of the US
> state plane grids and all of UTM, but there are others. I'd like to understand
> the oblique Mercator well enough to write code for it and what the differences
> are (besides the obvious) between Alaska and Switzerland.
> 
> Pierre
>