1. Help with a Transformation Matrix between 2 cartesian coordinate systems

I have two 3D Cartesian coordinate systems with different orientations, call them A and M. In 'A' I have 8 defined points with 3D coordinates. In 'M', I have 6 of the points in 'A' defined with 3D coordinates as oriented in 'M', 4 of which are on the same plane.

I need help trying to figure out the steps to get a transformation matrix such that I can get the unknown two points in 'M' using the known points in 'A'.

Thanks

Also, any good links for transformation matrices or help to solve a problem like this would be appreciated.

2. Re: Help with a Transformation Matrix between 2 cartesian coordinate systems

A linear transformation in 3D can be represented by a 3 by 3 matrix. If you know, for example, that the point (x, y, z) in one coordinate system is mapped into (p, q, r) in the other, you know that
$\begin{bmatrix}a & b & c \\ d & e & f\\ g & h & i\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}ax+by+ cz \\ dx+ ey+ fz \\ gx+ hy+ iz\end{bmatrix}= \begin{bmatrix}p \\ q\\ r\end{bmatrix}$

so you have the three linear equations, ax+ by+ cz= p, dx+ ey+ fz= q, and gx+ hy+ iz= r. Two more points would give 6 more equations so you need only three points to get 9 linear equations in the 9 components, a, b, c, d, e, f, g, h, and i.

3. Re: Help with Transformation Matrices

Ok, great. Thanks for the help. I just want to get this straight:

So you're saying that "Known Point 1" from 'A' = (a,b,c); "Known Point 2" from 'A' = (d,e,f); and "Known Point 3" from 'A' = (g,h,i) where a, d and g are the x-components of the known coordinate points; b, e, and h are the known y- components etc... And the matrix represented by [X, Y, Z] are basically unit vectors? Or are those a placeholder for the "Unknown Point 4" from 'M' coordinate components? And is [P, Q, R] the "Known Point 4" from 'A' components from the other coordinate system?

Then using the product matrix of that multiplication (the middle matrix in the figure) I solve for x, y, and z using p,q and r? And then I would have found one of the two missing points in 'M' and completed more of the coordinate set?

Sorry, I know that's a lot of questions, but I really appreciate your help!