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## Copyright (C) 2022 The Octave Project Developers
##
## This program is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {} {@var{phi} =} auth2geo (@var{xi})
## @deftypefnx {} {@var{phi} =} auth2geo (@var{xi}, @var{spheroid})
## @deftypefnx {} {@var{phi} =} auth2geo (@var{xi}, @var{spheroid}, @var{angleUnit})
## Returns the geodetic latitude given authalic latitude @var{xi}.
##
## Input
## @itemize
## @item
## @var{xi}: the authalic latitude(s); scalar, vector or nD array.
##
## @item
## (optional) @var{spheroid}: referenceEllipsoid name, EPSG code, or parameter
## struct: the default is wgs84.
##
## @item
## (optional) @var{angleUnit}: string for angular units ('degrees' or 'radians',
## case-insensitive, just the first charachter will do). Default is 'degrees'.
## @end itemize
##
## Output
## @itemize
## @var{phi}: the geodetic latitude(s), shape matching that of @var{xi}.
## @item
## @end itemize
##
## Example
## @example
## auth2geo(44.872)
## ans = 45.000
## @end example
## @seealso{geo2auth, geo2con, geo2iso, geo2rect}
## @end deftypefn
function [phi] = auth2geo (xi, spheroid = "", angleUnit = "degrees")
if (nargin < 1 || nargin > 3)
print_usage();
endif
if (! isnumeric (xi) || ! isreal (xi))
error ("auth2geo: numeric input expected");
endif
if (isnumeric (spheroid))
spheroid = num2str (spheroid);
endif
E = sph_chk (spheroid);
if (! ischar (angleUnit))
error ("auth2geo: character value expected for 'angleUnit'");
elseif (strncmpi (angleUnit, "degrees", min (length (angleUnit), 7)))
## Latitude must be within [-90 ... 90]
if (any (abs (xi) > 90))
error ("auth2geo: azimuth value(s) out of acceptable range [-90, 90]")
endif
xi = deg2rad (xi);
elseif (strncmpi (angleUnit, "radians", min (length (angleUnit), 7)))
## Latitude must be within [-pi/2 ... pi/2]
if (any (abs (phi) > pi / 2))
error("auth2geo: azimuth value(s) out of acceptable range (-pi/2, pi/2)")
endif
else
error ("auth2geo: illegal input for 'angleUnit'");
endif
ecc = E.Eccentricity;
## From Snyder's "Map Projections-A Working Manual" [pg 16].
e2 = ecc ^ 2;
e4 = ecc ^ 4;
e6 = ecc ^ 6;
phi = xi + ...
(((e2 / 3) + (31 / 180) * e4 + (517 /5040) * e6) * sin(2 * xi)) + ...
(((23 / 360) * e4 + (251 / 3780) * e6) * sin(4 * xi)) + ...
(((761 / 45360) * e6) * sin(6 * xi));
if (strncmpi (angleUnit, "degrees", length (angleUnit)))
phi = rad2deg (phi);
endif
endfunction
%!test
%! xi = [0:15:90];
%! phi = auth2geo (xi);
%! Z = degrees2dm(phi - xi);
%! check = [0
%! 3.8575173
%! 6.6751024
%! 7.6978157
%! 6.6579355
%! 3.8403504
%! 0];
%! assert (Z(:,2), check, 1e-6)
%!error <numeric> auth2geo ("s")
%!error <numeric> auth2geo (5i)
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