[PROJ] Auxiliary latitudes, rhumb lines

[Charles Karney]

Two papers which may be of interest to PROJ users are available as
preprints:

   C. F. F. Karney
   On auxiliary latitudes (Dec. 2022)
   https://arxiv.org/abs/2212.05818

   C. F. F. Karney
   The area of rhumb polygons (Mar. 2023)
   https://arxiv.org/abs/2303.03219

The first is of more immediate interest in the PROJ context...  A case
could be made that all the conversions between auxiliary latitudes in
PROJ should be centralized and should support both series approximations
(fast and applicable to small flattening) and exact methods (slower and
applicable to arbitrary eccentricity).

I would single out the current treatment of authalic latitude in area
preserving projections as most in need of some loving case.  These all
end up determining the authalic latitude by taking the arc sine of the
value returned by pj_qsfn (normalized).  This results in a severe loss
of accuracy near the poles.

The second paper implements solutions to rhumb problems in terms of
auxiliary latitudes.  This could be used to provide an ellipsoid
generalization of the loximuthal projection (but I'm not sure whether
this is a pressing need).

Note that I'm careful to use the subjunctive mood here (should, would,
...).  I don't have the time to undertake the suggested changes to PROJ
myself; but I would be happy to guide someone else (perhaps these are
suitable student projects?).

-- 
Charles Karney <charles at karney.com>
702 Prospect Ave
Princeton, NJ 08540-4037