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@inra.fr
0035 % Created: 2010-12-03,    using Matlab 7.9.0.529 (R2009b)
0036 % Copyright 2010 INRA - Cepia Software Platform.
0037 
0038 % HISTORY
0039 % 2013-07-03 added support for multiple inputs (Sven Holcombe)
0040 % 2017-10-16 rewrite
0041 
0042 % size of input arguments
0043 srcSz = size(source, 1);
0044 tgtSz = size(target, 1);
0045 maxSz = max(srcSz, tgtSz);
0046 
0047 % check case of multiple inputs
0048 if maxSz > 1
0049     [t1, t2] = deal( bsxfun(@times, eye(4), ones(1,1,maxSz)) );
0050     if srcSz > 1
0051         source = permute(source, [3 2 1]);
0052     end
0053     if tgtSz > 1
0054         target = permute(target, [3 2 1]);
0055     end
0056 else
0057     [t1, t2] = deal(eye(4));
0058 end
0059 
0060 % Place source and target planes into t1 and t2 t-form matrices. If either
0061 % input is non-numeric it is assumed to mean 'global', or identity t-form.
0062 if isnumeric(source)
0063     if maxSz > 1 && srcSz == 1
0064         source = bsxfun(@times, source, ones(1,1,maxSz));
0065     end
0066     t1(1:3, 1, :) = source(1, 4:6, :);
0067     t1(1:3, 2, :) = source(1, 7:9, :);
0068     t1(1:3, 3, :) = crossProduct3d(source(1,4:6,:), source(1,7:9,:));
0069     t1(1:3, 4, :) = source(1, 1:3, :);
0070 end
0071 if isnumeric(target)
0072     if maxSz > 1 && tgtSz == 1
0073         target = bsxfun(@times, target, ones(1,1,maxSz));
0074     end
0075     t2(1:3, 1, :) = target(1, 4:6, :);
0076     t2(1:3, 2, :) = target(1, 7:9, :);
0077     t2(1:3, 3, :) = crossProduct3d(target(1,4:6,:), target(1,7:9,:));
0078     t2(1:3, 4, :) = target(1, 1:3, :);
0079 end
0080 
0081 
0082 % compute transform matrix
0083 transfo = zeros(4, 4, maxSz);
0084 for i = 1:maxSz
0085     % coordinate of four reference points in source basis
0086     po = t1(1:3, 4, i)';
0087     px = po + t1(1:3, 1, i)';
0088     py = po + t1(1:3, 2, i)';
0089     pz = po + t1(1:3, 3, i)';
0090     
0091     % express coordinates of reference points in the new basis
0092     t2i = inv(t2(:,:,i));
0093     pot = transformPoint3d(po, t2i);
0094     pxt = transformPoint3d(px, t2i);
0095     pyt = transformPoint3d(py, t2i);
0096     pzt = transformPoint3d(pz, t2i);
0097     
0098     % compute direction vectors in new basis
0099     vx = pxt - pot;
0100     vy = pyt - pot;
0101     vz = pzt - pot;
0102 
0103     % concatenate result in a 4-by-4 affine transform matrix
0104     transfo(:,:,i) = [vx' vy' vz' pot' ; 0 0 0 1];
0105 end