I have 3 known points in the Cartesian, let's say, World coordinate system:
$P^W_1= (10, 10, 10)$,
$P^W_2= (10, 12, 10)$,
$P^W_3= (5, 10, 12)$.
There is a point $L$ in the World system which "sees" these 3 points with the coordinates:
$P^L_1= (3, -2, 1)$,
$P^L_2= (3, -4, 1)$,
$P^L_3= (8, -2, 3)$.
I would like find and expression $T$ that can be used to calculate any arbitrary point's $W$ coordinate from the $L$ system, or for that matter, from any coordinate system given that we know the coordinates of that 3 points.
Coordinate transformation based on three points gave me some ideas about the solution to this problem, but the answer there assumes the same origin of the two coordinate systems.
Coordinate Transformation on Local coordinate system has pretty much the same problem but it's answer is rather confusing to me.
I tried finding the coordinates of $L$ and then finding a transformation $T$ so that $T(L)=(0,0,0)$, but to no avail.
This method, I presume, could be done using a system of linear equations $Ax=B$, where $A$ is the points' coordinate in $L$ and $B$ in $W$, however, I do not really know where to start.
If you could direct me to a relevant thread I might have missed or give some hints, I would greatly appreciate it!