I'm a computer scientist and I'm not very good with mathematical stuff...
I have got 3 points A, B, C that doesn't create an orthogonal space. I have the coordinates of those 3 points in two 3D orthogonal spaces: M (the global space) and T (the local space). In Pic.1, we can see the coordinates of the 3 points in M space. The point Ot is the center of the circumcircle of the triangle ABC. That point Ot is also the origin of the T space but we don't know the coordinates in M space.
In Pic.2, we can see the coordinates of the 3 points in T space.
In this example, I know there is just a translation to pass to one space to another but I don't know how to calculate it.
I saw on the Internet that to change basis, you need to have a matrix that "link" the two spaces:
$ M = P.T $ where P is this "link" matrix.
I can understand that formula but I don't know how to calculate P or even M and T... Do I need to defined 3 vectors that defined the space? How can I do that from 3 random points?
PS: sorry for my english...
Now that I understand my problem better, I'm going to reformulate it.
I have three arbitrary points $A$, $B$ and $C$ in two systems that represents a local system $T$ and a world system $M$.
You can see an example in Pic.3 and Pic.4, $A$, $B$ and $C$ are defined in both systems but the origin $O_T$ of the local system is just defined in the local system and I have no possibility to get it.
The pictures that I show you are just an example but in most cases, we're practically always going to have a rotation and a translation.
We also know that the transformations are always perform to the whole local system and the world system is fixed.
In this example, the transformations between the local system and the world system are a rotation (along the Z-axis with an angle of $-\pi/2$) and a translation (with the vector $(-4, 0, 0)$). In real case, I do not know the transformation applied to the local system.
My problem is to calculate the transition matrix between the local system and the world system.