Find scale, rotation, and translation that maps three points to three points in 3D Given three source points $\mathbf{P}_1 = [x_1\ y_1\ z_1]^T,$
$\mathbf{P}_2 = [x_2\ y_2\ z_2]^T,$ $\mathbf{P}_3 = [x_3\ y_3\ z_3]^T,$ and
three destination points $\mathbf{P}'_1 = [x'_1\ y'_1\ z'_1]^T,$
$\mathbf{P}'_2 = [x'_2\ y'_2\ z'_2]^T,$ $\mathbf{P'}_3 = [x'_3\ y'_3\ z'_3]^T,$
how do we find the scale $S,$ rotation $R,$ and translation $T$ that transforms
the source points into the destination points as in the following equation?
$$
\left[\begin{array}{ccc}
s_x &     & \\
    & s_y & \\
    &     & s_z \\
\end{array}\right]
\left[\begin{array}{cccc}
r_{11} & r_{12} & r_{13} & t_x \\
r_{21} & r_{22} & r_{23} & t_y \\
r_{31} & r_{32} & r_{33} & t_z 
\end{array}\right]
\left[\begin{array}{ccc}
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3 \\
z_1 & z_2 & z_3 \\
1 & 1 & 1
\end{array}\right]
=
\left[\begin{array}{ccc}
x'_1 & x'_2 & x'_3 \\
y'_1 & y'_2 & y'_3 \\
z'_1 & z'_2 & z'_3
\end{array}\right]
$$
Since the $3 \times 3$ rotation $R$ is orthonormal and can be parameterized by three rotation
angles (e.g. yaw, pitch, roll) there are 9 equations and 9 unknowns. Ideally the scale would be uniform $(s_x = s_y = s_z)$ resulting in a similarity transformation, but then the system would be over constrained. As long as the source and destination points are not co-linear, this should have a unique solution.
One could solve for a general affine transformation $A$ and decompose this, but this would require 4 non-coplanar source and destination points and may include shears.
EDIT: A possible approach is to solve for the general affine case
$X A = B$
$$
\left[\begin{array}{cccc}
x_1 & y_1 & z_1 & 1\\
x_2 & y_2 & z_2 & 1\\
x_3 & y_3 & z_3 & 1
\end{array}\right]
\left[\begin{array}{ccc}
a_{11} & a_{21} & a_{31} \\
a_{12} & a_{22} & a_{32} \\
a_{13} & a_{23} & a_{33} \\
a_{14} & a_{24} & a_{34} \\
\end{array}\right]
=
\left[\begin{array}{ccc}
x'_1 & y'_1 & z'_1 \\
x'_2 & y'_2 & z'_2 \\
x'_3 & y'_3 & z'_3
\end{array}\right]
$$
which is under-constrained since there are 9 equations and 12 unknowns which means there are infinite solutions.
Therefore, following the recipe here, we will find the solution $A'$ that has the minimum norm $\|A\|.$
The normal equation for an underdetermined system is
$$
A^* = X^T (X X^T)^{-1} B
$$
from which we can get the optimal value via SVD.
The SVD for $X$ is
$$
\begin{eqnarray}
X &=& U \Sigma V^T \\
  &=&
\left[\begin{array}{ccc}
\cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot 
\end{array}\right]
\left[\begin{array}{cccc}
  \sigma_1 & & & \cdot \\
     & \sigma_2 & & \cdot \\
     & & \sigma_3 & \cdot
\end{array}\right]
\left[\begin{array}{cccc}
\cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot\\
\cdot & \cdot & \cdot & \cdot\\
\cdot & \cdot & \cdot & \cdot
\end{array}\right]^T
\end{eqnarray}
$$
where $\sigma_1 \geq \sigma_2 \geq \sigma_3 \geq 0$ are the
singular values (if $\sigma_3 = 0$ then the source points are
co-linear, i.e., $X$'s rank is less than 3) and $U$ and $V$ are
orthonormal matrices.
Plugging this into our normal equation we have
$$
\begin{eqnarray}
A^* &=& V \Sigma^T (\Sigma \Sigma^T )^{-1} U^T B \\
  &=&
V
\left[\begin{array}{ccc}
\frac{1}{\sigma_1} & & \\
 & \frac{1}{\sigma_2} & & \\
 & & \frac{1}{\sigma_3} \\
 \cdot & \cdot & \cdot
\end{array}\right]
U^T B
\end{eqnarray}
$$
where the matrix above containing the reciprocals of the singular values
is the pseudo-inverse of $\Sigma.$
The hope is the solution $A^*$ which the minimal norm from the set of all solutions will be a similar transformation. Yet to be verified.
 A: The assumption is that this can be accomplished with a similarity transformation.
We use the following sequence of transformations to compose the desired transformation:

*

*Translate $\mathbf{P}_0$ to the origin;

*Orient $\triangle \mathbf{P}_0 \mathbf{P}_1 \mathbf{P}_2$
to align with $\triangle \mathbf{P}'_0 \mathbf{P}'_1 \mathbf{P}'_2$ via rotation $R;$

*Uniform scale by
$s = \frac{\|\mathbf{P}'_1 - \mathbf{P}'_0\|}{\|\mathbf{P}_1 - \mathbf{P}_0\|}$;

*Translate to $\mathbf{P'}_0$
$$
\begin{eqnarray}
S &=& 
\left[\begin{array}{cccc}
1 & 0 & 0 & x_0' \\
0 & 1 & 0 & y_0' \\
0 & 0 & 1 & z_0' \\
0 & 0 & 0 & 1
\end{array}\right]
\left[\begin{array}{cccc}
s & 0 & 0 & 0 \\
0 & s & 0 & 0 \\
0 & 0 & s & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\left[\begin{array}{cccc}
r_{11} & r_{12} & r_{13} & 0 \\
r_{21} & r_{32} & r_{23} & 0 \\
r_{31} & r_{32} & r_{33} & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\left[\begin{array}{cccc}
1 & 0 & 0 & -x_0 \\
0 & 1 & 0 & -y_0 \\
0 & 0 & 1 & -z_0 \\
0 & 0 & 0 & 1
\end{array}\right] \\
&=&
%\left[sI\ | \mathbf{P}'_0 \right] \left[R\ | -R \mathbf{P}_0 \right] 
\left[\begin{array}{cccc}
s & 0 & 0 & x_0' \\
0 & s & 0 & y_0' \\
0 & 0 & s & z_0' \\
0 & 0 & 0 & 1
\end{array}\right]
\left[\begin{array}{cccc}
r_{11} & r_{12} & r_{13} & -x''_0 \\
r_{21} & r_{32} & r_{23} & -y''_0 \\
r_{31} & r_{32} & r_{33} & -z''_0 \\
0 & 0 & 0 & 1
\end{array}\right]\ \ \ \ \mbox{(let $\mathbf{P}''_0 = R \mathbf{P}_0)$} \\
&=&
\left[\begin{array}{cccc}
s r_{11} & s r_{12} & s r_{13} & x'_0 - s x''_0 \\
s r_{21} & s r_{32} & s r_{23} & y'_0 - s y''_0 \\
s r_{31} & s r_{32} & s r_{33} & z'_0 - s z''_0 \\
0 & 0 & 0 & 1
\end{array}\right]
\end{eqnarray}
$$
To find the rotation $R$ we align the edge $\overline{\mathbf{P}_0 \mathbf{P}_1}$ with the edge $\overline{\mathbf{P}'_0 \mathbf{P}'_1}$ and align the plane of
$\triangle \mathbf{P}_0 \mathbf{P}_1 \mathbf{P}_2$  with the plane of
$\triangle \mathbf{P}'_0 \mathbf{P}'_1 \mathbf{P}'_2.$
We define the source orthonormal coordinate axes as
$$
\begin{eqnarray}
\mathbf{X} &=& \frac{\mathbf{P}_1 - \mathbf{P}_0}{\|\mathbf{P}_1 - \mathbf{P}_0 \|}\\
\mathbf{Z} &=& \frac{(\mathbf{P}_2 - \mathbf{P}_0) \times \mathbf{X}}{\| (\mathbf{P}_2 - \mathbf{P}_0) \times \mathbf{X}\|} \\
\mathbf{Y} &=& \mathbf{X} \times \mathbf{Z}
\end{eqnarray}
$$
and the target coordinate axes as
$$
\begin{eqnarray}
\mathbf{X}' &=& \frac{\mathbf{P}'_1 - \mathbf{P}'_0}{\|\mathbf{P}'_1 - \mathbf{P}'_0 \|}\\
\mathbf{Z}' &=& \frac{(\mathbf{P}'_2 - \mathbf{P}'_0) \times \mathbf{X}'}{\| (\mathbf{P}'_2 - \mathbf{P}'_0) \times \mathbf{X}'\|} \\
\mathbf{Y}' &=& \mathbf{X}' \times \mathbf{Z}'
\end{eqnarray}
$$
We are looking for the rotation $R$ that maps the source coordinate axes
into the target axes:
$$
\begin{eqnarray}
R \left[ \mathbf{X}\ \mathbf{Y}\ \mathbf{Z} \right]
&=&
\left[ \mathbf{X}'\ \mathbf{Y}'\ \mathbf{Z}' \right] \\
R &=& \left[ \mathbf{X}'\ \mathbf{Y}'\ \mathbf{Z}' \right] 
\left[ \mathbf{X}\ \mathbf{Y}\ \mathbf{Z} \right]^T %\\
%   &=& \left[ \mathbf{X}'\ \mathbf{Y}'\ \mathbf{Z}' \right]
%    \left[ \begin{array}{c}
%        \mathbf{X}^T \\
%        \mathbf{Y}^T \\
%        \mathbf{Z}^T
%    \end{array}
%   \right]
\end{eqnarray}
$$
So we have a uniform scale $s,$ rotation $R,$ and translation
$\mathbf{P}'_0 - s R \mathbf{P}_0.$
