How to find the reflection of the plane $ax+by+cz+d=0$ in the plane $a'x+b'y+c'z+d'=0$?
I can't really think of a method of for doing so. I do know how to reflect a line on a plane though.That could be done just by finding the projection of any two points on the line onto the plane.And then probably mid-point theorem.
But how do I reflect a plane on a plane?
Write the plane in point-normal form, then reflect the point and the normal.
Specifically, suppose we are given the plane $ax+by+cz+d=0$. Let $\mathbf{N} = (a,b,c)$, let $k = \mathbf{N} \cdot \mathbf{N} = a^2 + b^2 + c^2$, and let $\mathbf{P} = \frac{-d\mathbf{N}}{k}$. Then $\mathbf{N} \cdot \mathbf{P} = -d$, and so
\begin{align} &ax+by+cz+d=0 \\ \Leftrightarrow \quad &\mathbf{N} \cdot \mathbf{X} + d =0 \\ \Leftrightarrow \quad &\mathbf{N} \cdot (\mathbf{X} - \mathbf{P}) = 0 \\ \end{align} So the plane passes through the point $\mathbf{P}$ and has normal vector $\mathbf{N}$. Reflect $\mathbf{P}$ and $\mathbf{N}$ through the other plane to get a reflected point $\mathbf{P}'$ and a reflected normal vector $\mathbf{N}'$. The equation of the reflected plane is then $\mathbf{N'} \cdot (\mathbf{X} - \mathbf{P'}) = 0$.