aff_trafo_loc2glob.m

function [C, G] = aff_trafo_loc2glob(x0, y0)
%function [C, G] = aff_trafo_loc2glob(x0, y0)
%
% function giving the coefficients for the affine transformation
% from reference/local triangle to the original/global one,
% i.e. T_i_aff(x;mu) = C_aff_i(mu) + sum_j=1_2 G_ij_k(mu) x_j;      j=1,2, i=1,2
% transformation is calculated for the standard triangle:
%
%             (0,1)  |\                                   /| (x0(3),y0(3))
%                    | \        ---T--->                 / |
%             (0,0)  |__\ (1,0)           (x0(1),y0(1)) /__| (x0(2),y0(2))
%
% function giving c1, c2, g11, g12, g21, g22
% so: C=[c1; c2] and G=[g11, g12; g21, g22]
%
% input: 
% x0: 3-by-1-list of the x values of the original/global triangle
% y0: 3-by-1-list of the y values of the original/global triangle
%
% output:
% affine transformation can be written as T(x) = C + G*x
% C: 2-by-1-vector
% G: 2-by-2-matrix
%
% See also aff_trafo_glob2loc which gives the transformation in the other
% direction (global to local)
%
% Oliver Zeeb, 01.02.11


%standard triangle
x1 = 0;
y1 = 0;
x2 = 1;
y2 = 0;
x3 = 0;
y3 = 1;

    

%matrices for transformation:
B_aff =  [1, 0, x1, y1, 0, 0; ...
          0, 1, 0, 0, x1, y1; ...
          1, 0, x2, y2, 0, 0; ...
          0, 1, 0, 0, x2, y2; ...
          1, 0, x3, y3, 0, 0; ...
          0, 1, 0, 0, x3, y3];
     
V_aff = [x0(1); y0(1); x0(2); y0(2); x0(3); y0(3)];

coef_vec = B_aff \ V_aff;

C = [coef_vec(1); coef_vec(2)];
G = [coef_vec(3), coef_vec(4); coef_vec(5), coef_vec(6)];
% c1  = coef_vec(1);
% c2  = coef_vec(2);
% g11 = coef_vec(3);
% g12 = coef_vec(4);
% g21 = coef_vec(5);
% g22 = coef_vec(6);