function [lat, lon, azi, rk] = gnomonic_inv(lat0, lon0, x, y, ellipsoid)
%GNOMONIC_INV Inverse ellipsoidal gnomonic projection
% [LAT, LON] = GNOMONIC_INV(LAT0, LON0, X, Y)
% [LAT, LON, AZI, RK] = GNOMONIC_INV(LAT0, LON0, X, Y, ELLIPSOID)
% performs the inverse ellipsoidal gnomonic projection of points (X,Y) to
% (LAT,LON) using (LAT0,LON0) as the center of projection. These input
% arguments can be scalars or arrays of equal size. 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. GEODPROJ
% defines the projection and gives the restrictions on the allowed ranges
% of the arguments. The forward projection is given by GNOMONIC_FWD.
% AZI and RK give metric properties of the projection at (LAT,LON); AZI
% is the azimuth of the geodesic from the center of projection and RK is
% the reciprocal of the azimuthal scale. The scale in the radial
% direction is 1/RK^2.
% In principle, all finite X and Y are allowed. However, it's possible
% that the inverse projection fails for very large X and Y (when the
% geographic position is close to the "horizon"). In that case, NaNs are
% returned for the corresponding output variables.
% LAT0, LON0, LAT, LON, AZI are in degrees. The projected coordinates X,
% Y are in meters (more precisely the units used for the equatorial
% radius). RK is dimensionless.
% The ellipsoidal gnomonic projection is an azimuthal projection about a
% center point. All geodesics through the center point are projected
% into straight lines with the correct azimuth relative to the center
% point. In addition all geodesics are pass close to the center point
% are very nearly straight. The projection is derived in Section 8 of
% C. F. F. Karney, Algorithms for geodesics,
% J. Geodesy 87, 43-55 (2013);
% Addenda: http://geographiclib.sf.net/geod-addenda.html
% which also includes methods for solving the "intersection" and
% "interception" problems using the gnomonic projection.
% This routine depends on the MATLAB File Exchange package "Geodesics on
% an ellipsoid of revolution":
% See also GEODPROJ, GNOMONIC_FWD, GEODRECKON, DEFAULTELLIPSOID.
% Copyright (c) Charles Karney (2012) <email@example.com>.
% This file was distributed with GeographicLib 1.29.
if nargin < 4, error('Too few input arguments'), end
if nargin < 5, ellipsoid = defaultellipsoid; end
Z = lat0 + lon0 + x + y;
Z = zeros(size(Z));
error('lat0, lon0, x, y have incompatible sizes')
if length(ellipsoid(:)) ~= 2
error('ellipsoid must be a vector of size 2')
a = ellipsoid(1);
numit = 10;
eps1 = a * 0.01 * sqrt(eps);
lat0 = lat0 + Z; lon0 = lon0 + Z; x = x + Z; y = y + Z;
azi0 = atan2(x, y) / (pi/180);
rho = hypot(x, y);
s = a * atan(rho / a);
little = rho <= a;
rho(~little) = 1 ./ rho(~little);
g = Z == 0; trip = ~g;
lat = Z; lon = Z; azi = Z; m = Z; M = Z; ds = Z;
for k = 1 : numit
[lat(g), lon(g), azi(g), ~, m(g), M(g)] = ...
geodreckon(lat0(g), lon0(g), s(g), azi0(g), ellipsoid);
g = g & ~trip;
if ~any(g), break, end
c = little & g;
ds(c) = (m(c) ./ M(c) - rho(c)) .* M(c).^2;
c = ~little & g;
ds(c) = (rho(c) - M(c) ./ m(c)) .* m(c).^2;
s(g) = s(g) - ds(g);
trip(g) = ~(abs(ds(g)) >= eps1);
c = ~trip;
lat(c) = NaN;
lon(c) = NaN;
azi(c) = NaN;
M(c) = NaN;
rk = M;