function [L,L_u,L_k] = pix2PluckerRay(k,u)

% PIX2PLUCKERRAY  Plucker ray from optical center through pixel.
%   PIX2PLUCKERRAY(K,U) is a Plucker line passing over pixel U and the
%   optical center, in a pin-Hole camera of intrinsic vector K.
%
%   [R, R_k, R_u] = PIX2PLUCKERRAY(...) returns the Jacobians wrt K and U.

%   Copyright 2008-2009 Joan Sola @ LAAS-CNRS.

if nargout == 1

    n = [0;0;0];
    v = invPinHole(u,1,k);

    L = [n;v];

else

    n = [0;0;0];
    [v,V_u,~,V_k] = invPinHole(u,1,k);

    L = [n;v];
    L_u = [zeros(3,2);V_u];
    L_k = [zeros(3,4);V_k];

end

return

%%
syms u1 u2 u0 v0 au av real
k = [u0;v0;au;av];
u = [u1;u2];

[L,L_u,L_k] = pix2PluckerRay(k,u);

simplify(L_u - jacobian(L,u))
simplify(L_k - jacobian(L,k))



% ========== End of function - Start GPL license ==========


%   # START GPL LICENSE

%---------------------------------------------------------------------
%
%   This file is part of SLAMTB, a SLAM toolbox for Matlab.
%
%   SLAMTB is free software: you can redistribute it and/or modify
%   it under the terms of the GNU General Public License as published by
%   the Free Software Foundation, either version 3 of the License, or
%   (at your option) any later version.
%
%   SLAMTB is distributed in the hope that it will be useful,
%   but WITHOUT ANY WARRANTY; without even the implied warranty of
%   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%   GNU General Public License for more details.
%
%   You should have received a copy of the GNU General Public License
%   along with SLAMTB.  If not, see <http://www.gnu.org/licenses/>.
%
%---------------------------------------------------------------------

%   SLAMTB is Copyright:
%   Copyright (c) 2008-2010, Joan Sola @ LAAS-CNRS,
%   Copyright (c) 2010-2013, Joan Sola,
%   Copyright (c) 2014-2015, Joan Sola @ IRI-UPC-CSIC,
%   SLAMTB is Copyright 2009 
%   by Joan Sola, Teresa Vidal-Calleja, David Marquez and Jean Marie Codol
%   @ LAAS-CNRS.
%   See on top of this file for its particular copyright.

%   # END GPL LICENSE

