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skew3d

PURPOSE ^

SKEW3D Convert between a vector and the corresponding skew-symmetric matrix

SYNOPSIS ^

function y=skew3d(x,m)

DESCRIPTION ^

SKEW3D Convert between a vector and the corresponding skew-symmetric matrix

 Inputs:   x   input vector or matrix
                size(x) must equal [3 1], [3 3], [6 1] or [4 4]
           m   m string:
               'n'  normalize the vector to have unit magnitude
               'z'  orthoganlize the vector so that x'Jy=0

 Outputs:  y   output matrix or vector
                size(y) = [3 3], [3 1], [4 4] or [6 1] respectively
                Note that skew3d() is its own inverse: skew3d(skew3d(x)) =  x

 3D Euclidean space
 ------------------
    If A and B are 3x1 vectors then the vector cross product is given by
    skew3d(A)*B = cross(A,B) = A x B. This relationship is widely used
    in computer vision.

 3D Projective space
 -------------------
 In 3D projective space, a line has 4 degrees of freedom and may be
 represented by its homogeneous 6x1 Plucker vector, A, or 4x4 Plucker
 matrix B=skew3d(A).
 The 6x1 Plucker vector loses one degree of freedom because it is
 homogeneous (i.e. independent of a non-zero scale factor) and another
 because it must satisfy A'*flipud(A)=0. Setting the 'n' and 'z' options
 in the second input parameter will remove these redundancies by forcing
 A'*A=1 and A'*flipud(A)=0.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function y=skew3d(x,m)
0002 %SKEW3D Convert between a vector and the corresponding skew-symmetric matrix
0003 %
0004 % Inputs:   x   input vector or matrix
0005 %                size(x) must equal [3 1], [3 3], [6 1] or [4 4]
0006 %           m   m string:
0007 %               'n'  normalize the vector to have unit magnitude
0008 %               'z'  orthoganlize the vector so that x'Jy=0
0009 %
0010 % Outputs:  y   output matrix or vector
0011 %                size(y) = [3 3], [3 1], [4 4] or [6 1] respectively
0012 %                Note that skew3d() is its own inverse: skew3d(skew3d(x)) =  x
0013 %
0014 % 3D Euclidean space
0015 % ------------------
0016 %    If A and B are 3x1 vectors then the vector cross product is given by
0017 %    skew3d(A)*B = cross(A,B) = A x B. This relationship is widely used
0018 %    in computer vision.
0019 %
0020 % 3D Projective space
0021 % -------------------
0022 % In 3D projective space, a line has 4 degrees of freedom and may be
0023 % represented by its homogeneous 6x1 Plucker vector, A, or 4x4 Plucker
0024 % matrix B=skew3d(A).
0025 % The 6x1 Plucker vector loses one degree of freedom because it is
0026 % homogeneous (i.e. independent of a non-zero scale factor) and another
0027 % because it must satisfy A'*flipud(A)=0. Setting the 'n' and 'z' options
0028 % in the second input parameter will remove these redundancies by forcing
0029 % A'*A=1 and A'*flipud(A)=0.
0030 
0031 %      Copyright (C) Mike Brookes 1998-2012
0032 %      Version: $Id: skew3d.m 2152 2012-07-06 15:41:30Z dmb $
0033 %
0034 %   VOICEBOX is a MATLAB toolbox for speech processing.
0035 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
0036 %
0037 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0038 %   This program is free software; you can redistribute it and/or modify
0039 %   it under the terms of the GNU General Public License as published by
0040 %   the Free Software Foundation; either version 2 of the License, or
0041 %   (at your option) any later version.
0042 %
0043 %   This program is distributed in the hope that it will be useful,
0044 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0045 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0046 %   GNU General Public License for more details.
0047 %
0048 %   You can obtain a copy of the GNU General Public License from
0049 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0050 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
0051 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0052 
0053 [j,k]=size(x);
0054 mn=nargin>1 && any(m=='n');  % normalize
0055 mz=nargin>1 && any(m=='z');  % orthoganalize
0056 if j==3
0057     if k==1
0058         if mn && x'*x>0
0059             x=x/sqrt(x'*x);
0060         end
0061         y=zeros(3,3);
0062         y([6 7 2])=x(:)';
0063         y([8 3 4])=-x(:)';
0064     elseif k==3
0065         y=x([6 7 2]');
0066         if mn && y'*y>0
0067             y=y/sqrt(y'*y);
0068         end
0069     else
0070         error('size(x) must be [3 1], [3 3], [6 1] or [4 4]');
0071     end
0072 elseif j==6 && k==1
0073     x=x(:);
0074     u=x(1:3);
0075     v=x(6:-1:4);
0076     if mz && u'*u>0 && v'*v>0  % orthoganalize
0077         v = v - (u'*v)/(2*u'*u)*u;
0078         x = [u-(v'*u)/(v'*v)*v; v([3 2 1])];
0079     end
0080     if mn && x'*x>0
0081         x=x/sqrt(x'*x);
0082     end
0083     y=zeros(4,4);
0084     y([5 9 13 10 8 15])=x(:)';
0085     y([2 3 4 7 14 12])=-x(:)';
0086 elseif j==4 && k==4
0087     u=x([5 9 13]');
0088     v=x([15 8 10]');
0089     if mz && u'*u>0 && v'*v>0  % orthoganalize
0090         v = v - (u'*v)/(2*u'*u)*u;
0091         y = [u-(v'*u)/(v'*v)*v; v([3 2 1])];
0092     else
0093         y = [u; v([3 2 1])];
0094     end
0095     if mn && y'*y>0
0096         y=y/sqrt(y'*y);
0097     end
0098 else
0099     error('size(x) must be [3 1], [3 3], [6 1] or [4 4]');
0100 end

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