intersectPlanes(plane1, plane2)

function line = intersectPlanes(plane1, plane2)
%INTERSECTPLANES Return intersection line between 2 planes in space
%
%   LINE = intersectPlanes(PLANE1, PLANE2)
%   Returns the straight line belonging to both planes
%   PLANE: [x0 y0 z0 dx1 dy1 dz1 dx2 dy2 dz2]
%   LINE:  [x0 y0 z0 dx dy dz]
%
%   See also:
%   planes3d, lines3d, intersectLinePlane
%
%   ---------
%   author : David Legland 
%   INRA - TPV URPOI - BIA IMASTE
%   created the 17/02/2005.
%

%   HISTORY


% plane normal
n1 = normalizeVector3d(cross(plane1(:,4:6), plane1(:, 7:9), 2));
n2 = normalizeVector3d(cross(plane2(:,4:6), plane2(:, 7:9), 2));

% test if planes are parallel
if abs(cross(n1, n2, 2))<1e-14
    line = [NaN NaN NaN NaN NaN NaN];
    return;
end

% Uses Hessian form, ie : N.p = d
% I this case, d can be found as : -N.p0, when N is normalized
d1 = dot(n1, plane1(:,1:3), 2);
d2 = dot(n2, plane2(:,1:3), 2);

% compute dot products
dot1 = dot(n1, n1, 2);
dot2 = dot(n2, n2, 2);
dot12 = dot(n1, n2, 2);

% intermediate computations
det = dot1*dot2 - dot12*dot12;
c1  = (d1*dot2 - d2*dot12)./det;
c2  = (d2*dot1 - d1*dot12)./det;

% compute line origin and direction
p0  = c1*n1 + c2*n2;
dp  = cross(n1, n2, 2);

% concatenate result to form a new line
line = [p0 dp];