vgg_contreps.m

function Y = vgg_contreps(X)

% vgg_contreps  Contraction with epsilon tensor.
%
% B = vgg_contreps(A) is tensor obtained by contraction of A with epsilon tensor.
% However, it works only if the argument and result fit to matrices, in particular:
%
% - if A is row or column 3-vector ...  B = [A]_x
% - if A is skew-symmetric 3-by-3 matrix ... B is row 3-vector such that A = [B]_x
% - if A is skew-symmetric 4-by-4 matrix ... then A can be interpreted as a 3D line Pluecker matrix
%                                               skew-symmetric 4-by-4 B as its dual Pluecker matrix.
% - if A is row 2-vector ... B = [0 1; -1 0]*A', i.e., A*B=eye(2)
% - if A is column 2-vector ... B = A'*[0 1; -1 0], i.e., B*A=eye(2)
%
% It is vgg_contreps(vgg_contreps(A)) = A.

% werner@robots.ox.ac.uk, Oct 2001

if prod(size(X)) == 3  % get [X]_\times
  Y = [0 X(3) -X(2)
      -X(3) 0 X(1)
       X(2) -X(1) 0];
elseif all(size(X) == [1 2])
  Y = [0 1; -1 0]*X';
elseif all(size(X) == [2 1])
  Y = X'*[0 1; -1 0];
elseif all(size(X) == [3 3]) % get X from [X]_\times
  Y = [X(2,3) X(3,1) X(1,2)];
elseif all(size(X) == [4 4])  % pluecker matrix dual
  Y = [0      X(3,4) X(4,2) X(2,3) 
       X(4,3) 0      X(1,4) X(3,1)
       X(2,4) X(4,1) 0      X(1,2)
       X(3,2) X(1,3) X(2,1) 0     ];
else
  error('Wrong matrix size.')
end