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Geodesics on an ellipsoid of revolution

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Geodesics on an ellipsoid of revolution

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20 Nov 2012 (Updated )

Solutions of the direct and inverse geodesic problems

geodarea(lats, lons, ellipsoid)
function [A, P, N] = geodarea(lats, lons, ellipsoid)
%GEODAREA  Surface area of polygon on an ellipsoid
%
%   A = GEODAREA(lats, lons)
%   [A, P, N] = GEODAREA(lats, lons, ellipsoid)
%
%   calculates the surface area A of the geodesic polygon specified by the
%   input vectors lats, lons (in degrees).  The ellipsoid vector is of the
%   form [a, e], where a is the equatorial radius in meters, e is the
%   eccentricity.  If ellipsoid is omitted, the WGS84 ellipsoid (more
%   precisely, the value returned by DEFAULTELLIPSOID) is used.  There is
%   no need to "close" the polygon by repeating the first point.  Multiple
%   polygons can be specified by separating the vertices by NaNs in the
%   vectors.  Thus a series of quadrilaterals can be specified as two 5 x K
%   arrays where the 5th row is NaN.  The output, A, is in meters^2.
%   Counter-clockwise traversal counts as a positive area.  Only simple
%   polygons (which do not intersect themselves) are supported.  Also
%   returned are the perimeters of the polygons in P (meters) and the
%   numbers of vertices in N.  GEODDOC gives the restrictions on the
%   allowed ranges of the arguments.
%
%   GEODAREA loosely duplicates the functionality of the AREAINT function
%   in the MATLAB mapping toolbox.  The major difference is that the
%   polygon edges are taken to be geodesics and the area contributed by
%   each edge is computed using a series expansion with is accurate
%   regardless of the length of the edge.  The formulas are derived in
%
%     C. F. F. Karney, Algorithms for geodesics,
%     J. Geodesy 87, 43-55 (2013);
%     http://dx.doi.org/10.1007/s00190-012-0578-z
%     Addenda: http://geographiclib.sf.net/geod-addenda.html
%
%   See also GEODDOC, GEODDISTANCE, GEODRECKON, POLYGONAREA,
%     DEFAULTELLIPSOID.

% Copyright (c) Charles Karney (2012-2013) <charles@karney.com>.
%
% This file was distributed with GeographicLib 1.31.

  if nargin < 2, error('Too few input arguments'), end
  if nargin < 3, ellipsoid = defaultellipsoid; end
  if ~isequal(size(lats), size(lons))
    error('lats, lons have incompatible sizes')
  end
  if length(ellipsoid(:)) ~= 2
    error('ellipsoid must be a vector of size 2')
  end

  lat1 = lats(:);
  lon1 = lons(:);
  M = length(lat1);
  ind = [0; find(isnan(lat1 + lon1))];
  if length(ind) == 1 || ind(end) ~= M
    ind = [ind; M + 1];
  end
  K = length(ind) - 1;
  A = zeros(K, 1); P = A; N = A;
  if M == 0, return, end

  lat2 = [lat1(2:end, 1); 0];
  lon2 = [lon1(2:end, 1); 0];
  m0 = min(M, ind(1:end-1) + 1);
  m1 = max(1, ind(2:end) - 1);
  lat2(m1) = lat1(m0); lon2(m1) = lon1(m0);

  a = ellipsoid(1);
  e2 = ellipsoid(2)^2;
  f = e2 / (1 + sqrt(1 - e2));

  b = (1 - f) * a;
  c2 = (a^2 + b^2 * atanhee(1, e2)) / 2;
  area0 = 4 * pi * c2;

  [s12, ~, ~, S12] = geoddistance(lat1, lon1, lat2, lon2, ellipsoid);
  cross = transit(lon1, lon2);

  for k = 1 : K
    N(k) = m1(k) - m0(k) + 1;
    P(k) = accumulator(s12(m0(k):m1(k)));
    [As, At] = accumulator(S12(m0(k):m1(k)));
    crossings = sum(cross(m0(k):m1(k)));
    if mod(crossings, 2) ~= 0,
      [As, At] = accumulator( ((As < 0) * 2 - 1) * area0 / 2, As, At);
    end
    As = -As; At = -At;
    if As > area0/2
      As = accumulator( -area0 / 2, As, At);
    elseif As <= -area0/2
      As = accumulator(  area0 / 2, As, At);
    end
    A(k) = As;
  end
end

function cross = transit(lon1, lon2)
%TRANSIT  Count crossings of prime meridian
%
%   CROSS = TRANSIT(LON1, LON2) return 1 or -1 if crossing prime meridian
%   in east or west direction.  Otherwise return zero.

  lon1 = AngNormalize(lon1);
  lon2 = AngNormalize(lon2);
  lon12 = AngDiff(lon1, lon2);
  cross = zeros(length(lon1), 1);
  cross(lon1 < 0 & lon2 >= 0 & lon12 > 0) = 1;
  cross(lon2 < 0 & lon1 >= 0 & lon12 < 0) = -1;

end

function [s, t] = accumulator(x, s, t)
%ACCUMULATOR  Accurately sum x
%
%   [S, T] = ACCUMULATOR(X, S, T) accumulate the sum of the elements of X
%   into [S, T] using extended precision.  S and T are scalars.

  if nargin < 3, t = 0; end
  if nargin < 2, s = 0; end

  for y = x(:)',
    % Here's Shewchuk's solution...
    [z, u] = sumx(y, t);
    [s, t] = sumx(z, s);
    if s == 0
      s = u;
    else
      t = t + u;
    end
  end
end

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