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## Copyright (C) 2014-2022 Alfredo Foltran <alfoltran@gmail.com>
##
## 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 {Function File} {} @var{dist} = vincenty(@var{pt1}, @var{pt2})
## @deftypefnx {Function File} {} @var{dist} = vincenty(@var{pt1}, @var{pt2}, @var{ellipsoid})
## @deftypefnx {Function File} {[@var{dist}, @var{az}] = } {vincenty(@var{pt1}, @var{pt2})}
## @deftypefnx {Function File} {[@var{dist}, @var{az}] = } {vincenty(@var{pt1}, @var{pt2}, @var{ellipsoid})}
## Calculates the distance (in meters) between two (sets of) locations on
## an ellipsoid.
##
## The formula devised by Thaddeus Vincenty is used with an accurate
## ellipsoidal model of the earth (@var{ellipsoid}). The default ellipsoidal
## model is 'WGS84', which is the most globally accurate model.
##
## @var{pt1} and @var{pt2} are two-column matrices of the form [latitude longitude].
## The units for the input coordinates angles must be degrees.
## Optional argument @var{ellipsoid} defines the reference ellipsoid to use.
##
## Sample values for @var{ellipsoid} are the following:
##
## @multitable @columnfractions .7 .3
## @headitem Model @tab @var{ellipsoid}
## @item WGS 1984 (default) @tab referenceEllipsoid(7030)
## @item GRS 1980 @tab referenceEllipsoid(7019)
## @item G.B. Airy 1830 @tab referenceEllipsoid(7001)
## @item Internacional 1924 @tab referenceEllipsoid(7022)
## @item Clarke 1880 @tab referenceEllipsoid(7012)
## @item Australian Nat. @tab referenceEllipsoid(7003)
## @end multitable
##
## The sample model values are the following:
##
## @multitable @columnfractions .35 .20 .20 .25
## @headitem Model @tab Major (km) @tab Minor (km) @tab 1 / f
## @item WGS 1984 @tab 6378.137 @tab 6356.7523142 @tab 298.257223563
## @item GRS 1980 @tab 6378.137 @tab 6356.7523141 @tab 298.257222101
## @item G.B. Airy 1830 @tab 6377.563396 @tab 6356.256909 @tab 299.3249646
## @item Internacional 1924 @tab 6378.388 @tab 6356.911946 @tab 297.0
## @item Clarke 1880 @tab 6378.249145 @tab 6356.51486955 @tab 293.465
## @item Australian Nat. @tab 6378.1600 @tab 6356.774719 @tab 298.25
## @end multitable
##
## Usage:
## @example
## >> vincenty ([37, -76], [37, -9])
## ans = 5830.081
## >> vincenty ([37, -76], [67, -76], referenceEllipsoid (7019))
## ans = 3337.843
## @end example
##
## @seealso{distance, referenceEllipsoid}
## @end deftypefn
## Author: Alfredo Foltran <alfoltran@gmail.com>
##
## Octave style fixes and some re-coding to avoid sneaky errors by
## Philip Nienhuis <prnienhuis@users.sf.net>
function [dist, az] = vincenty (pt1, pt2, ellipsoid)
if (nargin < 3)
ellipsoid = referenceEllipsoid (7030);
endif
major = ellipsoid.SemimajorAxis;
minor = ellipsoid.SemiminorAxis;
f = ellipsoid.Flattening;
## Avoid confusion of length units impsed by ellipsoid, standardize on meters
if (isfield (ellipsoid, "LengthUnit") && ! isempty (ellipsoid.LengthUnit))
major *= unitsratio ("meters", ellipsoid.LengthUnit);
minor *= unitsratio ("meters", ellipsoid.LengthUnit);
endif
iter_limit = 20;
pt1 = deg2rad (pt1);
pt2 = deg2rad (pt2);
[lat1 lng1] = deal (pt1(1), pt1(2));
[lat2 lng2] = deal (pt2(1), pt2(2));
delta_lng = lng2 - lng1;
reduced_lat1 = atan ((1 - f) * tan (lat1));
reduced_lat2 = atan ((1 - f) * tan (lat2));
[sin_reduced1 cos_reduced1] = deal (sin (reduced_lat1), cos (reduced_lat1));
[sin_reduced2 cos_reduced2] = deal (sin (reduced_lat2), cos (reduced_lat2));
lambda_lng = delta_lng;
lambda_prime = 2 * pi;
i = 0;
while (abs (lambda_lng - lambda_prime) > 10e-12 && i <= iter_limit)
i++;
[sin_lambda_lng cos_lambda_lng] = deal (sin (lambda_lng), cos (lambda_lng));
sin_sigma = sqrt ((cos_reduced2 * sin_lambda_lng) ^ 2 + ...
(cos_reduced1 * sin_reduced2 - ...
sin_reduced1 * cos_reduced2 * cos_lambda_lng) ^ 2);
if (abs (sin_sigma < eps))
dist = 0;
return;
endif
cos_sigma = (sin_reduced1 * sin_reduced2 + ...
cos_reduced1 * cos_reduced2 * cos_lambda_lng);
sigma = atan2 (sin_sigma, cos_sigma);
sin_alpha = (cos_reduced1 * cos_reduced2 * sin_lambda_lng / sin_sigma);
cos_sq_alpha = 1 - sin_alpha ^ 2;
if (abs (cos_sq_alpha > eps))
cos2_sigma_m = cos_sigma - 2 * (sin_reduced1 * sin_reduced2 / cos_sq_alpha);
else
cos2_sigma_m = 0.0; ## Equatorial line
endif
C = f / 16.0 * cos_sq_alpha * (4 + f * (4 - 3 * cos_sq_alpha));
lambda_prime = lambda_lng;
lambda_lng = (delta_lng + (1 - C) * f * sin_alpha * (sigma + ...
C * sin_sigma * (cos2_sigma_m + C * cos_sigma * ...
(-1 + 2 * cos2_sigma_m ^ 2))));
endwhile
if (i > iter_limit)
error("Inverse Vincenty's formulae failed to converge!");
endif
u_sq = cos_sq_alpha * (major ^ 2 - minor ^ 2) / minor ^ 2;
A = 1 + u_sq / 16384.0 * (4096 + u_sq * (-768 + u_sq * (320 - 175 * u_sq)));
B = u_sq / 1024.0 * (256 + u_sq * (-128 + u_sq * (74 - 47 * u_sq)));
delta_sigma = (B * sin_sigma * (cos2_sigma_m + B / 4. * (cos_sigma * ...
(-1 + 2 * cos2_sigma_m ^ 2) - B / 6. * cos2_sigma_m * ...
(-3 + 4 * sin_sigma ^ 2) * (-3 + 4 * cos2_sigma_m ^ 2))));
dist = minor * A * (sigma - delta_sigma);
if (nargout() > 1)
alpha1 = atan2 (cos_reduced2 * sin_lambda_lng, ...
cos_reduced1 * sin_reduced2 - sin_reduced1 * ...
cos_reduced2 * cos_lambda_lng);
alpha2 = atan2 (cos_reduced1 * sin_lambda_lng, ...
-sin_reduced1 * cos_reduced2 + cos_reduced1 * ...
sin_reduced2 * cos_lambda_lng);
az = rad2deg ([alpha1 alpha2]);
endif
endfunction
%!test
%! assert (vincenty ([37, -76], [37, -9]), 5830081.063, 1e-2);
%!test
%! assert (vincenty ([37, -76], [67, -76], referenceEllipsoid (7019)), ...
%! 3337842.871, 1e-2)
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