aff_trafo_glob2loc.m

function [C, G] = aff_trafo_glob2loc(x0, y0)
%function [C, G] = aff_trafo_glob2loc(x0, y0)
% function giving the coefficients for the affine transformation
% from original/global triangle to the reference/local one.
%
% In detail, this implements a transformation of the type
% ``T_{i,aff}(x;\mu) = C_{i,aff}(\mu) + \sum_{j=1,2} G_{ij}^k(\mu) x_j \qquad      i=1,2``
%
% @verbatim
% triangle:                     /| (x0(3),y0(3))              (0,1)  |\
%                              / |                ---T--->           | \
%               (x0(1),y0(1)) /__| (x0(2),y0(2))              (0,0)  |__\ (1,0)
% @endverbatim
%
% Parameters:
%  x0: vector of size '3 x 1' holding x values of the original/global triangle
%  y0: vector of size '3 x 1' holding y values of the original/global triangle
%
% Return values:
%  C: matrix of size '2 x 1' with entries 'C=[c1; c2]'
%  G: matrix of size '2 x 2' with entries 'G=[g11, g12; g21, g22]'
%
% See also aff_trafo_loc2glob() which gives the transformation in the other
% direction (local to global)

% Oliver Zeeb, 01.02.11

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

ref_coord = [x1; y1; x2; y2; x3; y3];

B_aff =  [1, 0, x0(1), y0(1), 0, 0; ...
          0, 1, 0, 0, x0(1), y0(1); ...
          1, 0, x0(2), y0(2), 0, 0; ...
          0, 1, 0, 0, x0(2), y0(2); ...
          1, 0, x0(3), y0(3), 0, 0; ...
          0, 1, 0, 0, x0(3), y0(3)];
      
coef_vec = B_aff \ ref_coord;

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);