[PROJ] Latitude of natural origin in Transverse Mercator

[Even Rouault]

Hi Javier,

interesting. Quick numerical experimentation seems to confirm that they 
are "equivalent", as well as looking at the drawing of how the 
projection is built.

And looking at the maths of the approximate ellipsoidal forward method 
in src/projections/tmerc.cpp, one can see:

xy.y = P->k0 * (.... - Q->ml0 + ....... )

(the false northing value is added in generic PROJ code)

where

Q->ml0 = pj_mlfn(P->phi0, sin(P->phi0), cos(P->phi0), Q->en);

where pj_mlfn() computes the meridional distance (formula in src/mlfn.cpp)

So - P->k0 * Q->ml0 should be the value of the false northing equivalent 
to the effect of phi0. (I believe :-))

Also confirmed by the formula N=N0 +y(lam,phi) - k0 * m(phi0) of 
https://en.wikipedia.org/wiki/Transverse_Mercator_projection#Coordinates,_grids,_eastings_and_northings

Even

Le 13/01/2023 à 15:28, Javier Jimenez Shaw a écrit :
> Hi
>
> I have a question about Transverse Mercator.
> What is the impact of the parameter "Latitude of natural origin", 
> compared with the "False Northing"? (apart from the fact that one is 
> in degrees and the other in meters)
> Is there any difference on using a Latitude of natural origin of, 
> let's say, 40 degrees, or compute the equivalent False Northing?
>
> Thanks.
> Javier
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