TITLE: Transform a plane to the xy plane.
QUESTION [14 upvotes]: I have a plane equation in the form:
Ax + By + Cz + D = 0

Is there any possible way to find out which operations transform the plane to the xy plane?
EDIT: I am guessing you would need a translation and a rotation because sometimes the plane won't intersect the origin so you need to translate it to the origin then rotate. However, I am still confused as to how I'm supposed to get these angles and translations just from the plane equation.

REPLY [5 votes]: I tried to add this as a comment to Emilio Novati's answer but don't have enough points. The rotation matrix here requires that the vector along the rotation axis, $\vec{u}$, is a unit vector, so:
$$\hat{u}=\dfrac{\vec{v}\times\vec{k}}{|\vec{v}\times\vec{k}|}=\dfrac{1}{\sqrt{a^2+b^2}}\left(b,-a,0\right)^T=(u_1,u_2,0)^T$$
which gives the components:
$u_1 = \frac{b}{\sqrt{a^2 + b^2}}$
$u_2 = -\frac{a}{\sqrt{a^2 + b^2}}$
for the same rotation matrix:
$$\left (
\begin{array}{cccc}
\cos \theta +u_1^2 (1-\cos \theta) &u_1u_2 (1-\cos \theta)  & u_2\sin \theta \\
 u_1u_2 (1-\cos \theta)& \cos \theta+ u_2^2 (1-\cos \theta)& -u_1 \sin \theta \\
 -u_2 \sin \theta  & u_1 \sin \theta& \cos \theta
\end {array}
\right)$$