Code covered by the BSD License

# Geodesic projections for an ellipsoid

### Charles Karney (view profile)

07 Dec 2012 (Updated )

Four map projections based on geodesics

gnomonic_fwd(lat0, lon0, lat, lon, ellipsoid)
```function [x, y, azi, rk] = gnomonic_fwd(lat0, lon0, lat, lon, ellipsoid)
%GNOMONIC_FWD  Forward ellipsoidal gnomonic projection
%
%   [X, Y] = GNOMONIC_FWD(LAT0, LON0, LAT, LON)
%   [X, Y, AZI, RK] = GNOMONIC_FWD(LAT0, LON0, LAT, LON, ELLIPSOID)
%
%   performs the forward ellipsoidal gnomonic projection of points
%   (LAT,LON) to (X,Y) 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 inverse
%   projection is given by GNOMONIC_INV.
%
%   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.
%
%   If the point lies "over the horizon", i.e., if RK <= 0, then NaNs are
%   returned for X and Y (the correct values are returned for AZI and RK).
%
%   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);
%     http://dx.doi.org/10.1007/s00190-012-0578-z
%
%   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":
%
%     http://www.mathworks.com/matlabcentral/fileexchange/39108
%
%   See also GEODPROJ, GNOMONIC_INV, GEODDISTANCE, DEFAULTELLIPSOID.

% Copyright (c) Charles Karney (2012) <charles@karney.com>.
%
% This file was distributed with GeographicLib 1.29.

if nargin < 4, error('Too few input arguments'), end
if nargin < 5, ellipsoid = defaultellipsoid; end
try
[~] = lat0 + lon0 + lat + lon;
catch err
error('lat0, lon0, lat, lon have incompatible sizes')
end

[~, azi0, azi, ~, m, M] = geoddistance(lat0, lon0, lat, lon, ellipsoid);
rho = m ./ M;
azi0 = azi0 * (pi/180);
x = rho .* sin(azi0);
y = rho .* cos(azi0);
rk = M;
x(M <= 0) = NaN;
y(M <= 0) = NaN;
end
```