Rotation transformation between two frames

Question

My goal is to express the transformation between the black frame F1 and the other one F0 (Green, Red, Violet):

All what I know is the position X,Y Z of four points A, B, C, D wrt F0. My question is how can I know the transformation (3d rotation+ 3d translation) between the two frames?

Answer 1

Translation + rotation can be expressed like this

$$M=\left[\begin{array}{cc}T&0\\0&1\end{array}\right]\left[\begin{array}{cc}R&0\\0&1\end{array}\right]\left[\begin{array}{cc}T^{-1}&x\\0&1\end{array}\right]$$

Where $T$ first is transform that takes us to coordinate system where we rotate. and $R$ is rotation matrix, for example

$$R=\left[\begin{array}{ccc}\cos(\alpha)&\sin(\alpha)&0\\-\sin(\alpha)&\cos(\alpha)&0\\0&0&1\end{array}\right]$$ $$T=\left[\begin{array}{ccc}v_1&v_2&v_P\end{array}\right]$$ Where $v_P$ is vector along axis or rotation and $\{v_1,v_2\}$ is a basis for plane of rotation.

So what you have is some equations $Mw_1 = w_2$ where vectors in $w_2$ are coordinates for frame 2 and $w_1$ are same points in first frame. $w$ should be filled like this $w=[c_x,c_y,c_z,1]^T$ coordinate $x,y,z$ and don't forget the 1 at the end of vector.

The sin:s and cos:s make this slightly nonlinear, but there still exist quite okay methods to solve it. I think a few iterations of refining linear solver with some kind of correction should work.

Answer 2

Suppose you know that

• $(0,0,0)$ in the original $F_0$ frame is mapped to $(x_0,y_0,z_0)$ in the transformed $F_1$ frame
• $(1,0,0)$ in $F_0$ maps to $(x_1,y_1,z_1)+(x_0,y_0,z_0)$ (the first vector can be found from the raw coordinates of $(1,0,0)$'s image by subtraction)
• $(0,1,0)$ maps to $(x_2,y_2,z_2)+(x_0,y_0,z_0)$, $(0,0,1)$ maps to $(x_3,y_3,z_3)+(x_0,y_0,z_0)$

Then the transformation that takes a point expressed in $F_1$'s coordinates and returns it in $F_0$'s, or transforms $F_0$ to $F_1$, is $$\begin{bmatrix}x_1&x_2&x_2&x_0\\ y_1&y_2&y_3&y_0\\ z_1&z_2&z_3&z_0\\ 0&0&0&1\end{bmatrix}$$ using the homogenous coordinates trick of adding a fourth coordinate of 1 to each point to be transformed. To get the reverse transformation, simply invert this matrix.