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cart2utm.m
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cart2utm.m
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function [E, N, U] = cart2utm(X, Y, Z, zone)
%CART2UTM Transformation of (X,Y,Z) to (N,E,U) in UTM, zone 'zone'.
%
%[E, N, U] = cart2utm(X, Y, Z, zone);
%
% Inputs:
% X,Y,Z - Cartesian coordinates. Coordinates are referenced
% with respect to the International Terrestrial Reference
% Frame 1996 (ITRF96)
% zone - UTM zone of the given position
%
% Outputs:
% E, N, U - UTM coordinates (Easting, Northing, Uping)
%Kai Borre -11-1994
%Copyright (c) by Kai Borre
%
% CVS record:
% $Id: cart2utm.m,v 1.1.1.1.2.6 2007/01/30 09:45:12 dpl Exp $
%This implementation is based upon
%O. Andersson & K. Poder (1981) Koordinattransformationer
% ved Geod\ae{}tisk Institut. Landinspekt\oe{}ren
% Vol. 30: 552--571 and Vol. 31: 76
%
%An excellent, general reference (KW) is
%R. Koenig & K.H. Weise (1951) Mathematische Grundlagen der
% h\"oheren Geod\"asie und Kartographie.
% Erster Band, Springer Verlag
% Explanation of variables used:
% f flattening of ellipsoid
% a semi major axis in m
% m0 1 - scale at central meridian; for UTM 0.0004
% Q_n normalized meridian quadrant
% E0 Easting of central meridian
% L0 Longitude of central meridian
% bg constants for ellipsoidal geogr. to spherical geogr.
% gb constants for spherical geogr. to ellipsoidal geogr.
% gtu constants for ellipsoidal N, E to spherical N, E
% utg constants for spherical N, E to ellipoidal N, E
% tolutm tolerance for utm, 1.2E-10*meridian quadrant
% tolgeo tolerance for geographical, 0.00040 second of arc
% B, L refer to latitude and longitude. Southern latitude is negative
% International ellipsoid of 1924, valid for ED50
a = 6378388;
f = 1/297;
ex2 = (2-f)*f / ((1-f)^2);
c = a * sqrt(1+ex2);
vec = [X; Y; Z-4.5];
alpha = .756e-6;
R = [ 1 -alpha 0;
alpha 1 0;
0 0 1];
trans = [89.5; 93.8; 127.6];
scale = 0.9999988;
v = scale*R*vec + trans; % coordinate vector in ED50
L = atan2(v(2), v(1));
N1 = 6395000; % preliminary value
B = atan2(v(3)/((1-f)^2*N1), norm(v(1:2))/N1); % preliminary value
U = 0.1; oldU = 0;
iterations = 0;
while abs(U-oldU) > 1.e-4
oldU = U;
N1 = c/sqrt(1+ex2*(cos(B))^2);
B = atan2(v(3)/((1-f)^2*N1+U), norm(v(1:2))/(N1+U) );
U = norm(v(1:2))/cos(B)-N1;
iterations = iterations + 1;
if iterations > 100
fprintf('Failed to approximate U with desired precision. U-oldU: %e.\n', U-oldU);
break;
end
end
%Normalized meridian quadrant, KW p. 50 (96), p. 19 (38b), p. 5 (21)
m0 = 0.0004;
n = f / (2-f);
m = n^2 * (1/4 + n*n/64);
w = (a*(-n-m0+m*(1-m0))) / (1+n);
Q_n = a + w;
%Easting and longitude of central meridian
E0 = 500000;
L0 = (zone-30)*6 - 3;
%Check tolerance for reverse transformation
tolutm = pi/2 * 1.2e-10 * Q_n;
tolgeo = 0.000040;
%Coefficients of trigonometric series
%ellipsoidal to spherical geographical, KW p. 186--187, (51)-(52)
% bg[1] = n*(-2 + n*(2/3 + n*(4/3 + n*(-82/45))));
% bg[2] = n^2*(5/3 + n*(-16/15 + n*(-13/9)));
% bg[3] = n^3*(-26/15 + n*34/21);
% bg[4] = n^4*1237/630;
%spherical to ellipsoidal geographical, KW p. 190--191, (61)-(62)
% gb[1] = n*(2 + n*(-2/3 + n*(-2 + n*116/45)));
% gb[2] = n^2*(7/3 + n*(-8/5 + n*(-227/45)));
% gb[3] = n^3*(56/15 + n*(-136/35));
% gb[4] = n^4*4279/630;
%spherical to ellipsoidal N, E, KW p. 196, (69)
% gtu[1] = n*(1/2 + n*(-2/3 + n*(5/16 + n*41/180)));
% gtu[2] = n^2*(13/48 + n*(-3/5 + n*557/1440));
% gtu[3] = n^3*(61/240 + n*(-103/140));
% gtu[4] = n^4*49561/161280;
%ellipsoidal to spherical N, E, KW p. 194, (65)
% utg[1] = n*(-1/2 + n*(2/3 + n*(-37/96 + n*1/360)));
% utg[2] = n^2*(-1/48 + n*(-1/15 + n*437/1440));
% utg[3] = n^3*(-17/480 + n*37/840);
% utg[4] = n^4*(-4397/161280);
%With f = 1/297 we get
bg = [-3.37077907e-3;
4.73444769e-6;
-8.29914570e-9;
1.58785330e-11];
gb = [ 3.37077588e-3;
6.62769080e-6;
1.78718601e-8;
5.49266312e-11];
gtu = [ 8.41275991e-4;
7.67306686e-7;
1.21291230e-9;
2.48508228e-12];
utg = [-8.41276339e-4;
-5.95619298e-8;
-1.69485209e-10;
-2.20473896e-13];
%Ellipsoidal latitude, longitude to spherical latitude, longitude
neg_geo = 'FALSE';
if B < 0
neg_geo = 'TRUE ';
end
Bg_r = abs(B);
[res_clensin] = clsin(bg, 4, 2*Bg_r);
Bg_r = Bg_r + res_clensin;
L0 = L0*pi / 180;
Lg_r = L - L0;
%Spherical latitude, longitude to complementary spherical latitude
% i.e. spherical N, E
cos_BN = cos(Bg_r);
Np = atan2(sin(Bg_r), cos(Lg_r)*cos_BN);
Ep = atanh(sin(Lg_r) * cos_BN);
%Spherical normalized N, E to ellipsoidal N, E
Np = 2 * Np;
Ep = 2 * Ep;
[dN, dE] = clksin(gtu, 4, Np, Ep);
Np = Np/2;
Ep = Ep/2;
Np = Np + dN;
Ep = Ep + dE;
N = Q_n * Np;
E = Q_n*Ep + E0;
if neg_geo == 'TRUE '
N = -N + 20000000;
end;
%%%%%%%%%%%%%%%%%%%% end cart2utm.m %%%%%%%%%%%%%%%%%%%%