Add projection scale factor due to elevation

[jjimenezshaw]

Reading this paper from Michael Dennis (thanks Michael for the link ;) you can see that to get a "complete" computation of the linear distortion the elevation should be taken into consideration.

https://www.fig.net/resources/proceedings/fig_proceedings/fig2023/papers/cinema03/CINEMA03_dennis_12044.pdf

In the formula 𝛿 = 𝑘 (𝑅𝐺 / (𝑅𝐺 + ℎ)) − 1 PROJ already can compute 𝑘 with proj CLI or proj_factors C method. However the scale factor component due to the elevation is not computed.

Should we include that functionality in proj_factors? (I am happy to make a PR. Before that I would like to agree on some points). RG is the Gaussian radius, easy to compute. h is the ellipsoidal height. It should be provided in the variable "PJ_COORD lp" (in meters)

a) Should I add a new parameter to PJ_FACTORS ? (I do not know how big the impact of such a change is) b) How should I name it? it is a horizontal "scale" due to elevation (see that the elevation is not scaled). c) Should I add a "combined scale factor"?

If a) is no, should I implement it in a different way?

The "combined scale factor" could be sqrt(parallel_scale * meridional_scale) * horizontal_due_to_elevation_scale. This combined or global scale factor could be a strange thing in a non-conformal projection. But most of the projections used in surveying are conformal or are really close to the projection center so parallel_scale and meridional_scale are very similar.

Any comments?

Thanks.

[jjimenezshaw]

I did this during holidays, so probably it passed unnoticed. https://github.com/OSGeo/PROJ/pull/3949 reminds me its existence.

[rouault]

For API and ABI backward compatibility reasons, we likely need a new function( pj_factors_3D ?) and a new structure (FACTORS3D ?), likely extending the existing FACTORS one.

I've not enough background to give hints on the naming of the new parameter and combined scale factor. Perhaps @busstoptaktik or @cffk have some opinions on the subject.

[cffk]

I've just given Dennis's paper a quick read. I don't have much to contribute here. There should be some acknowledgement that if a conformal projection on the reference ellipsoid is lifted up onto a surface of constant elevation, then it is no longer conformal. So I would go along with the suggestion of @jjimenezshaw to adopt a "combined scale factor" nomenclature.