How to describe a position of a point w.r.t the position and orientation of 3 other points Is it possible to describe a position of a point $A$ (in 3D) w.r.t. the position and orientation of 3 other points? If so, how?

FYI, the 3 other points lie on a plane, whereas the point $A$ is not on the plane.

Just to elaborate, initially I have the info for all four points. My goal is to find where point $A$ is when the other three points changing position and orientation.
FYI, the 3 other points lie on a plane


Of course they do. Any set of three points defines a plane (except in the degenerate case joojaa mentions, where they lie on many planes). If the points are $p$, $q$, and $r$, then $(q - p) \wedge (r - p)$ is the normal to the plane; call it $n$.

This plane defines a co-ordinate system. If you make it the XY plane, then $z = a \dot\ n$. You can also project $a$ onto the plane to get the other two co-ordinates: $x = (a - p) \dot\ (q - p)$ and $y = (a - p) \dot\ (r - p)$.

Note that the co-ordinate system I've described here is pretty rubbish: it isn't even orthogonal. It's good enough to uniquely identify a point, as long as you're careful about all the signs.

But if you want an orthonormal basis, you can construct it from this one. First, instead of $q - p$ and $r - p$, normalize those vectors. Then $n$ will also have length 1, and the $x$ axis will also be normal. The last problem is the $y$ axis, which is not orthogonal to the $x$ axis. You can get a better $y$ axis by crossing the other two and normalizing, and while you're doing that you might as well also choose the order to ensure the basis is right-handed. Your three basis vectors are as follows:

$$
\begin{eqnarray}
\hat{z} &amp; = &amp; \frac{n}{|n|} \\
\hat{x} &amp; = &amp; \frac{q - p}{|q - p|} \\
\hat{y} &amp; = &amp; \hat{z} \wedge \hat{x}
\end{eqnarray}
$$