[PROJ] Geodetic to Authalic latitude conversions

[Jérôme St-Louis]

So it seems like we already have an implementation of Horner and 
Clenshaw in:

https://github.com/OSGeo/PROJ/blob/master/src/mlfn.cpp

called /polyval()/ and /clenshaw()/ just like in GeographicLib ( 
polyval() 
<https://github.com/geographiclib/geographiclib/blob/main/include/GeographicLib/Math.hpp#L280> 
, Clenshaw() 
<https://github.com/geographiclib/geographiclib/blob/main/src/AuxLatitude.cpp#L1319>).

It seems like Charles wrote or at least updated that :)

That is using the Cµφ (C[mu phi]) (A5) and Cφµ (C[phi mu]) (A6) from 
page  12 of the paper, where µ is called the "rectifying latitude".
I imagine that this is directly related to the "meridional distance" ?

Perhaps we could re-organize this a bit to share this /polyval()/ and 
/clenshaw()/ (they are currently static functions local to this 
/mlfn.cpp/) for use in /auth.cpp/ ?

Thanks!

Kind regards,

-Jerome

On 9/11/24 3:33 PM, Jérôme St-Louis wrote:
>
> Thanks a lot for the input Charles and Thomas,
>
> I am not familiar with either Horner 
> <https://en.wikipedia.org/wiki/Horner%27s_method> or Clenshaw 
> <https://en.wikipedia.org/wiki/Clenshaw_algorithm>, but I do see the 
> mentions now on /Section 6 - Evaluating the series/ pages 6 and 7 of 
> the papers.
> I implemented the simpler basic approach from section 3 / page 3, 
> which also happened to more easily correspond to the existing PROJ 
> implementation.
>
> I can definitely try to understand all this, with the help of this 
> Rust Geodesy code and the GeographicLib code, and have a go at 
> updating my proposed implementation for improved accuracy and performance.
>
> Kind regards,
>
> -Jerome
>
> On 9/11/24 12:18 PM, Thomas Knudsen wrote:
>> I totally agree with Charles regarding using Horner for polynomial
>> evaluation and Clenshaw for the trig series - for accuracy and speed.
>>
>> I implemented all the material from Charles' preprint
>> https://arxiv.org/pdf/2212.05818  for Rust Geodesy, when the preprint
>> appeared about 1½ years ago.
>>
>> And although (being an experiment) my handling of the raw coefficients
>> is rather clumsy, at least it gave me a reason to revise my PROJ horner
>> and clenshaw implementations (which in turn were based on material from
>> Poder & Engsager: "Some Conformal Mappings...").
>>
>> So Jérôme, perhaps take a look at the functions "taylor" and "fourier"
>> over athttps://github.com/busstoptaktik/geodesy/blob/main/src/math/series.rs
>>
>> While written in Rust, translating to C++ should be rather trivial,
>> and they may be easier to follow than my decade-old versions already
>> in the PROJ code base.
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