I need to find the transformation matrix (homogeneous coordinates) that flips an object about a plane whose normal is directed towards (-2 2 0), and intersects the y-axis at y=2.

I know how to do that when I have 3 points on the plane (A,B,C): translate to origin, find the orthonormal basis of the plane (u,v,w axes) and then rotate the plane to match XYZ coordinate system (with the axes I found), perform the reflection, rotate back and translate back.

However in this case I have a normal and another data which I don't know how to use.

Any ideas?

This is the same problem as the rotation about an axis. What you need is a matrix (any matrix) that satisfies the following corner constraint: one axis must point toward $(-2, 2, 0)$ in the case of a rotation this matrix must be a rotation matrix, in the case of a planar mirror it just has to span the 3D space. But since we know how to make an arbitrary rotation matrix we use the same approach.
• I'd expect something like: $w=\dfrac{n}{|n|}$, $v=\dfrac{n\times (0,0,1)^T}{|n\times (0,0,1)^T|}$, $u=v\times w$ Aug 28 '16 at 21:46