[PROJ] Geodetic to Authalic latitude conversions

[Jérôme St-Louis]

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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