V_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 v_skew3d() is its own inverse: v_skew3d(v_skew3d(x)) = x
3D Euclidean space
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If A and B are 3x1 vectors then the vector cross product is given by
v_skew3d(A)*B = cross(A,B) = A x B. This relationship is widely used
in computer vision.
3D Projective space
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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=v_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)=0.0001 function y=v_skew3d(x,m) 0002 %V_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 v_skew3d() is its own inverse: v_skew3d(v_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 % v_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=v_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: v_skew3d.m 10865 2018-09-21 17:22:45Z 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