Rotate normal vector onto axis plane

Question

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

Answer 1

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.

• Let's call your plane's normal vector the new z axis.
• You can generate the new y axis by crossing the old x axis with the new z axis (your plane's normal).
• Generate the new x axis by crossing the new z with the new y.
• Make all your new axis vectors into unit vectors (length 1).
• For every point you have, create a vector that's from your new origin to the point (vector subtraction of point - plane_origin). Just dot with the new x and new y unit vectors and you get a pair (x,y) you can plot!

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:

• Pay attention to which directions your vectors are pointing. If they point the wrong way, negate the resultant vector or change the order of the cross product.
• You have trouble if your plane's normal is exactly the same as your original x axis.