Transformations about a Plane 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.


Pick any vector (u) that's not parallel to the normal (n)
Take cross product of u and n for b
Take cross product of n and b for a


You now have an arbitrary rotation matrix with one axis defined using vectors n, b and a. Otherwise same as the last question.