createBasisTransform3d

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.

   Either (or both) SOURCE or TARGET may be an N-by-9 set of N planes. In
   that case, TRANSFO will be a 4-by-4-by-N array of N transformation
   matrices.

   Example:
     % Calculate local plane coords. of a point given in global coords.
     plane = [10 10 10  1 0 0  0 1 0];
     transfo = createBasisTransform3d('global', plane);
     PT_IN_PLANE = transformPoint3d([3 8 2], transfo)
     PT_IN_PLANE =
         13  18  12

   See also
     transforms3d, transformPoint3d, planePosition, createBasisTransform

0001 function transfo = createBasisTransform3d(source, target)
0002 %CREATEBASISTRANSFORM3D Compute matrix for transforming a basis into another basis.
0003 %
0004 %   TRANSFO = createBasisTransform3d(SOURCE, TARGET) will create a 4-by-4
0005 %   transformation matrix representing the transformation from SOURCE basis
0006 %   to TARGET basis.
0007 %    SOURCE and TARGET are either standard 1-by-9 geom3d PLANE
0008 %    representations of the form: [x0 y0 z0  ex1 ey1 ez1  ex2 ey2 ez2]
0009 %     OR
0010 %    SOURCE and TARGET may be any string such as 'global' or 'g' in which
0011 %    case they represent the global plane [0 0 0 1 0 0 0 1 0].
0012 %
0013 %   The resulting TRANSFO matrix is such that a point expressed with
0014 %   coordinates of the first basis will be represented by new coordinates
0015 %   P2 = transformPoint3d(P1, TRANSFO) in the target basis.
0016 %
0017 %   Either (or both) SOURCE or TARGET may be an N-by-9 set of N planes. In
0018 %   that case, TRANSFO will be a 4-by-4-by-N array of N transformation
0019 %   matrices.
0020 %
0021 %   Example:
0022 %     % Calculate local plane coords. of a point given in global coords.
0023 %     plane = [10 10 10  1 0 0  0 1 0];
0024 %     transfo = createBasisTransform3d('global', plane);
0025 %     PT_IN_PLANE = transformPoint3d([3 8 2], transfo)
0026 %     PT_IN_PLANE =
0027 %         13  18  12
0028 %
0029 %   See also
0030 %     transforms3d, transformPoint3d, planePosition, createBasisTransform
0031 
0032 % ------
0033 % Author: David Legland
0034 % E-mail: david.legland@inrae.fr
0035 % Created: 2010-12-03, using Matlab 7.9.0.529 (R2009b)
0036 % Copyright 2010-2024 INRA - Cepia Software Platform
0037 
0038 % size of input arguments
0039 srcSz = size(source, 1);
0040 tgtSz = size(target, 1);
0041 maxSz = max(srcSz, tgtSz);
0042 
0043 % check case of multiple inputs
0044 if maxSz > 1
0045     [t1, t2] = deal( bsxfun(@times, eye(4), ones(1,1,maxSz)) );
0046     if srcSz > 1
0047         source = permute(source, [3 2 1]);
0048     end
0049     if tgtSz > 1
0050         target = permute(target, [3 2 1]);
0051     end
0052 else
0053     [t1, t2] = deal(eye(4));
0054 end
0055 
0056 % Place source and target planes into t1 and t2 t-form matrices. If either
0057 % input is non-numeric it is assumed to mean 'global', or identity t-form.
0058 if isnumeric(source)
0059     if maxSz > 1 && srcSz == 1
0060         source = bsxfun(@times, source, ones(1,1,maxSz));
0061     end
0062     t1(1:3, 1, :) = source(1, 4:6, :);
0063     t1(1:3, 2, :) = source(1, 7:9, :);
0064     t1(1:3, 3, :) = crossProduct3d(source(1,4:6,:), source(1,7:9,:));
0065     t1(1:3, 4, :) = source(1, 1:3, :);
0066 end
0067 if isnumeric(target)
0068     if maxSz > 1 && tgtSz == 1
0069         target = bsxfun(@times, target, ones(1,1,maxSz));
0070     end
0071     t2(1:3, 1, :) = target(1, 4:6, :);
0072     t2(1:3, 2, :) = target(1, 7:9, :);
0073     t2(1:3, 3, :) = crossProduct3d(target(1,4:6,:), target(1,7:9,:));
0074     t2(1:3, 4, :) = target(1, 1:3, :);
0075 end
0076 
0077 
0078 % compute transform matrix
0079 transfo = zeros(4, 4, maxSz);
0080 for i = 1:maxSz
0081     % coordinate of four reference points in source basis
0082     po = t1(1:3, 4, i)';
0083     px = po + t1(1:3, 1, i)';
0084     py = po + t1(1:3, 2, i)';
0085     pz = po + t1(1:3, 3, i)';
0086     
0087     % express coordinates of reference points in the new basis
0088     t2i = inv(t2(:,:,i));
0089     pot = transformPoint3d(po, t2i);
0090     pxt = transformPoint3d(px, t2i);
0091     pyt = transformPoint3d(py, t2i);
0092     pzt = transformPoint3d(pz, t2i);
0093     
0094     % compute direction vectors in new basis
0095     vx = pxt - pot;
0096     vy = pyt - pot;
0097     vz = pzt - pot;
0098 
0099     % concatenate result in a 4-by-4 affine transform matrix
0100     transfo(:,:,i) = [vx' vy' vz' pot' ; 0 0 0 1];
0101 end