For a plane, I have the normal $n$, and also a point $P$ that lies on the plane.
Now, how am I going to find extra arbitrary two points ($P_1$ and $P_2$) for the plane so that these three points $P$, $P_1$ and $P_2$ completely define the plane?
The solution here suggests that one assumes a certain $x$ and $y$ to substitute into the plane equation and find the remaining $z$. But this method is only suitable for hand calculation; it breaks down for plane $z=0$. As such, it is not suitable for computer implementation.
I would need an algorithm that is robust and can handle all the cases, any idea how to construct it?
There is a similar question here, and the answer suggests me to use Gram-Schmidt, but I don't see how it can be applied in my case here.