How I can check if planes are parallel to each other in 3d?
Dear all kindly ask how I can check if any of these six planes of cube are parallel to each other, and which planes are adjacent to each other How I can check if planes are parallel to each other. I have cube 3*3*3 dimensions and there are planes. Thank you so much in advance for your help and advice My final goal is to check if one of my any point A(X0,Y0,Z0) lies in one of the plane and check if second point B(X1,Y1,Z1) lies in any of these planes. And at the end check if these two planes are parallel or adjusting to each other planes(:,:,1) = [0 3 3; 0 0 3; 0 3 0; 0 0 0; 0 0 0]; planes(:,:,2) = [0 0 3; 3 0 3; 0 0 0; 3 0 0; 0 0 0]; planes(:,:,3) = [3 0 3; 3 3 3; 3 0 0; 3 3 0; 3 0 0]; planes(:,:,4) = [3 3 3; 0 3 3; 3 3 0; 0 3 0; 0 3 3]; planes(:,:,5) = [0 3 0; 3 3 0; 0 0 0; 3 0 0; 0 0 0]; planes(:,:,6) = [0 3 3; 3 3 3; 0 0 3; 3 0 3; 0 0 3]; location_plane = 6; for j=1:6 % j is number of plane j plane = planes(:,:,j); p0 = plane(1,:); %p0 is top left point of plane p1 = plane(2,:); %p1 is top right point of plane p2 = plane(3,:); %p2 is bottom left point of plane p3 = plane(4,:); %p3 is bottom right point of plane V0 = plane(5,:); %point on the plane
Prashant Kumar answered .
2025-11-20
planes(:,:,1) = [0 3 3; 0 0 3; 0 3 0; 0 0 0; 0 0 0]; planes(:,:,2) = [0 0 3; 3 0 3; 0 0 0; 3 0 0; 0 0 0]; planes(:,:,3) = [3 0 3; 3 3 3; 3 0 0; 3 3 0; 3 0 0]; planes(:,:,4) = [3 3 3; 0 3 3; 3 3 0; 0 3 0; 0 3 3]; planes(:,:,5) = [0 3 0; 3 3 0; 0 0 0; 3 0 0; 0 0 0]; planes(:,:,6) = [0 3 3; 3 3 3; 0 0 3; 3 0 3; 0 0 3];
6 different planes, represented by points in each plane. First, I'll verify that each set of 5 points you designated do indeed lie in a plane in R^3. (Hey you might have made a mistake.)
for i = 1:6 rank(planes(:,:,i) - planes(1,:,i)) end
ans = 2 ans = 2 ans = 2 ans = 2 ans = 2 ans = 2
Two planes are parallel if they have the same normal vector (Though you could multiply by -1.) So just compute the normal vectors to each plane.
Nullvecs = zeros(6,3); for i = 1:6 nullvecs(:,i) = null(planes(:,:,i) - planes(1,:,i)); end nullvecs
nullvecs = 3×6
1 0 1 0 0 0
0 -1 0 1 0 0
0 0 0 0 1 1
Now, look at those vectors. See that planes 1 and 3 have the same normal vectors, The signs of the normal vectors are the same. Planes 2 and 4 have a sign flip on the normal vectors. And planes 5 and 6 are also parallel. In the last case again, the normal vectors had the same signs. Could we identify those pairs automatically? Yes, of course. We can compute a corrrelation matrix, then look for elements of the correlation matrix that are exactly either 1 or -1. I'll put a small tolerance on that result.
C = abs(abs(corr(nullvecs) - eye(6)) - 1)<2*eps
C = 6×6 logical array 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
Note that I subtracted the identity matrix so we don't identify each vector as the same as itself.
[I,J] = find(C); unique(sort([I,J],2),'rows')
ans = 3×2
1 3
2 4
5 6
So there were 3 pairs of parallel planes identified.