[PROJ] Relevant descrepancies between Sinusoidal and Homolosine

[Even Rouault]

Luí­s,

> after some more trials I can only but conclude that Daan is correct: the
> current implementation of the SInusoidal projection in PROJ does not match
> the historical equal-area projection associated with cartographers like
> Cossin, Sanson, Flamsteed or Mercator. I confirmed in Snyder's "Flattening
> the Earth" (pag. 113) the expressions for this projection, the relationship
> between latitude and northing is the identity (btw, Wikipaedia currently
> claims the same).

For the sphere, yes.
But for the ellipsoid, if I trust https://pubs.usgs.gov/pp/1395/report.pdf, 
Snyder's "MAP PROJECTIONS-A WORKING MANUAL ", at page 148, equations (30-9) 
and (3-21) show the northing is not the identity of latitude. And while I've 
not checked in depth, I believe that the M expression must hopefully match the 
pj_mlfn() function.

> Whomever implemented the  Homolosine projection must had thus to implement
> the real Sinusoidal internally, otherwise the scale would not match the
> Mollweide projection at the prescribed latitude (~ 40.75 º). Whether with
> the sphere or the ellipsoid, this implementation respects the expected
> relationship with latitude (and Pi).

I believe I found what is wrong with igh. igh implementation is only meant to 
work on the sphere, as there is a P->es = 0 in it. Mollweide implementation is 
only meant to work on the sphere. But Sinusoidal has a separate spherical and 
ellipsoidal implementation. The issue was that igh was forcing P->es = 0 too 
late, that is after the sinusoidal has been instanciated. Thus if using igh 
with an ellipsoid, you got the ellipsoidal sinusoidal and the spherical 
mollweide.
I've submitted https://github.com/OSGeo/PROJ/pull/2042, to make sure the 
spherical shape is force early so that spherical sinusoidal is used, which I 
believe is how igh should be fixed.

Even

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