[Proj] CEA from an ellipse and +R_A

[Gerald I. Evenden]

On Friday 11 July 2008 5:57:49 am David James Brockley wrote:
> Hello
>
> I'm trying to convert geodetic lon/lat into CEA xy using the proj API
> initialised with
>
> +proj=cea +long_0=0 +ellps=WGS84 +R_A
>
> What I expect to happen is
>
> y = a * q / 2
>
> where a is the authalic radius calculated from the semi-major of WGS84.
>
> q/2 seems to be a common approximation to q/q_p which is sin
> ( authalic latitude)
>
>
> What actually happens is that proj uses the equation for a sphere
> rather than an ellipse.

That is exactly what is supposed to happen as implied with the capital R in 
option's acronym,  The R in this case is based upon a sphere of surface area 
equal to the ellipse selected.  *All* of rhe R_* option series create radii 
to be used with the spherical earth projection formulas.

Quite frankly, I do not understand what you are trying to do and your 
operations do not make sense to me.  Authalic projections are based upon the 
math of the transformation and not on simple manipulation of the elliptic 
elements.

> y = a sin( phi )
>
> I can get proj to do what I expect by initialising with
>
> +proj=cea +long_0=0 +a=6371007 +e=0.0818191
>
> but that seems somewhat back to front.
>
>
> Can anyone tell me the correct way to configure proj for what I am
> trying to do?
>
> I'm fairly sure that my expected function for y is correct, since
> I've both copied it from the literature and derived it by hand from
> the spherical case by substituting authalic radius for radius and
> authalic latitude for latitude - I've also realised that the /2 is
> due to the maximum possible value of q_p being 2 when e=0, and that
> it doesn't affect the properties of the transform (still equal area).
>
> Regards, Brock
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