I have a set of data points in 3D space which apparently all fall onto a specific plane. I use PCA to compute the plane parameters. The 3rd component of PCA gives me the normal
Although there were other interesting responses, this is the solution we figured out while waiting for answers:
function roti = magic_cosini(n)
b = acos(n(2) / sqrt(n(1)*n(1) + n(2)*n(2)));
bwinkel = b * 360 / 2 / pi;
if (n(1) >= 0)
rotb = [cos(-b) -sin(-b) 0; sin(-b) cos(-b) 0; 0 0 1];
else
rotb = [cos(-b) sin(-b) 0; -sin(-b) cos(-b) 0; 0 0 1];
end
n2 = n * rotb;
a = acos(n2(3) / sqrt(n2(2)*n2(2) + n2(3)*n2(3)));
awinkel = a * 360 / 2 / pi;
rota = [1 0 0; 0 cos(-a) -sin(-a); 0 sin(-a) cos(-a)];
roti = rotb * rota;
(It's returning a hopefully correct double rotation matrix)
The flaw we had before and fixed here was to esp. deal with the sign of the X component, which was not covered in the cosine computations. This made us rotate in the wrong direction once (rotating with 180° - angle).
I hope I will also find time to try Nosredna's solution! It's always good to avoid trigonometry.
If you have a plane, you have a normal vector and an origin. I wouldn't do any "rotations" at all. You're just a few vector operations away from your answer.
If you have cross and dot product functions already, this is just a few lines of code. I know it works because most of the 3D videogames I wrote worked this way.
Tricks:
How about:
Decompose the normal vector into a vector in the XY-plane and a Z vector. Then apply a rotation around the Z-axis to line up the XY vector with one of the axis. Then find the dot product of the the normal with the Z-axis, and rotate along which ever of X,Y you lined up with.
The idea is to line the normal vector up with Z, and by doing that your plane is now the XY plane.