createBasisTransform3d(source, target)

function transfo = createBasisTransform3d(source, target)
%CREATEBASISTRANSFORM3D Compute matrix for transforming a basis into another basis
%
%   TRANSFO = createBasisTransform3d(SOURCE, TARGET) will create a 4-by-4
%   transformation matrix representing the transformation from SOURCE basis
%   to TARGET basis. 
%    SOURCE and TARGET are either standard 1-by-9 geom3d PLANE
%    representations of the form: [x0 y0 z0  ex1 ey1 ez1  ex2 ey2 ez2]
%     OR
%    SOURCE and TARGET may be any string such as 'global' or 'g' in which
%    case they represent the global plane [0 0 0 1 0 0 0 1 0].
%
%   The resulting TRANSFO matrix is such that a point expressed with
%   coordinates of the first basis will be represented by new coordinates
%   P2 = transformPoint3d(P1, TRANSFO) in the target basis.
%
%   Example:
%     % Calculate local plane coords. of a point given in global coords
%     Plane = [5 50 500 1 0 0 0 1 0];
%     Tform = createBasisTransform3d('global', Plane);
%     PT_WRT_PLANE = transformPoint3d([3 8 2], Tform);
%
%   See also
%   transforms3d, transformPoint3d, createPlane
%
% ------
% Author: David Legland
% e-mail: [e-mail]
% Created: 2010-12-03,    using Matlab 7.9.0.529 (R2009b)
% Copyright 2010 INRA - Cepia Software Platform.

% init basis transform to identity
t1 = eye(4);
t2 = eye(4);

% Place source and target planes into t1 and t2 t-form matrices. If either
% input is non-numeric it is assumed to mean 'global', or identity t-form.
if isnumeric(source)
    t1(1:3, 1) = source(4:6);
    t1(1:3, 2) = source(7:9);
    t1(1:3, 3) = cross(source(4:6), source(7:9));
    t1(1:3, 4) = source(1:3);
end
if isnumeric(target)
    t2(1:3, 1) = target(4:6);
    t2(1:3, 2) = target(7:9);
    t2(1:3, 3) = cross(target(4:6), target(7:9));
    t2(1:3, 4) = target(1:3);
end

% compute transfo
% same as: transfo = inv(t2)*t1;
transfo = t2 \ t1;