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.
 A: 3 points not collinear in 3d space always give the possibility to assign to them coordinate frame. 


*

*Note these points as $D,E,F$ and vectors describing them in the base
frame (denote this frame as $0$) as $r_d,r_e,r_f$.

*From these vectors you can calculate vectors $r_{de}=r_e-r_d$ and
$r_{df}=r_f-r_d$.

*Now made orthogonalization and normalization of the frame based on
$r_{de}$ and $r_{df}$ obtaining orthogonal unit vectors $i$ and $j$. Assign origin of the frame (denoted as $1$) to the point  $D$.   

*The third vector of coordinate frame is $k=i \times j$

*Now you have full coordinate frame $1$ described by the homogeneous matrix $^0H_1 = \begin {bmatrix} i & j & k & r_d \\ 0 & 0 & 0 & 1 \end{bmatrix}_{4 \times 4}$ in the base frame $0$.

*Now calculate vector $^0h_a$ describing point $A$ in the base frame $0$ as a homogeneous vector $ {^0h}_{a} =\begin {bmatrix}   r_a \\  1 \end{bmatrix}_{1\times 4}$

*The following equation with the use of homogeneous coordinates is satisfied
$^0h_{a}={^0H}_1(^1h_{a})$. It allows to transform coordinates of point $A$ from $1$ frame to $0$ frame.

*From the equation above calculate  $^1h_{a}=(^0H_1)^{-1} (^0 h_{a})$ what is position of the $A$ point in the frame 1 (described with homogeneous coordinates).
A: Yes. It is in fact quite easy to describe. The easier way is to say that the distance to each of the point is a specific one. 
You have $AD=d$, $AC=c$ and $AB=b$, where $b,c,d$ are set numbers. You have two solutions if $A,B,C,D$ are distincts (edited thanks to serg_1 comment). 
