knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
For most transformation, we assume that we can compute only the translation coefficients ($a_i$). The only exception are Euclidean transformation around a single axis of rotation that allow to compute a single scaling and a single rotation coefficient. In all other cases, values of computed coefficients would depend on the assumed order of individual transformation, making them no more than a potentially misleading guesses.
Number of parameters: 2
$$ \begin{bmatrix} 1 & 0 & a_1 \ 0 & 1 & a_2 \ 0 & 0 & 1 \end{bmatrix} $$
Number of parameters: 4
$$ \begin{bmatrix} b_1 & b_2 & a_1 \ -b_2 & b_1 & a_2 \ 0 & 0 & 1 \end{bmatrix} $$
The Euclidean transformation is a special case, where we can compute rotation ($\theta$) and the single scaling ($\phi$) coefficients, as follows: $$ \phi = \sqrt{b_1^2 + b_2^2}\ \theta = tan^{-1}(\frac{b_2}{b_1}) $$
Number of parameters: 6
$$ \begin{bmatrix} b_1 & b_2 & a_1 \ b_3 & b_4 & a_2 \ 0 & 0 & 1 \end{bmatrix} $$
Number of parameters: 8
$$ \begin{bmatrix} b_1 & b_2 & a_1 \ b_3 & b_4 & a_2 \ b_5 & b_6 & 1 \end{bmatrix} $$
Number of parameters: 3
$$ \begin{bmatrix} 1 & 0 & 0 & a_1 \ 0 & 1 & 0 & a_2 \ 0 & 0 & 1 & a_3 \ 0 & 0 & 0 & 1 \end{bmatrix} $$
Number of parameters: 5
For all Euclidean rotations, we opted to use coefficient $b_3$ to code scaling ($\phi$), whereas $b_2 = sin(\theta)$ and $b_1=\phi~ cos(\theta)$. The coefficients are computed as follows: $$ \phi = \sqrt{b_1^2 + b_2^2}\ \theta = tan^{-1}(\frac{b_2}{b_1}) $$
Note that during fitting $\phi$ is computed from $b_1$ and $b_2$ on the fly. $$ \begin{bmatrix} \phi & 0 & 0 & a_1 \ 0 & b_1 &-b_2 & a_2 \ 0 & b_2 & b_1 & a_3 \ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ \begin{bmatrix} b_1 & 0 & b_2 & a_1 \ 0 & \phi & 0 & a_2 \ -b_2 & 0 & b_1 & a_3 \ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ \begin{bmatrix} b_1 &-b_2 & 0 & a_1 \ b_2 & b_1 & 0 & a_2 \ 0 & 0 & \phi & a_3 \ 0 & 0 & 0 & 1 \end{bmatrix} $$
Number of parameters: 12
$$ \begin{bmatrix} b_1 & b_2 & b_3 & a_1 \ b_4 & b_5 & b_6 & a_2 \ b_7 & b_8 & b_9 & a_3 \ 0 & 0 & 0 & 1 \end{bmatrix} $$
Number of parameters: 15
$$ \begin{bmatrix} b_1 & b_2 & b_3 & a_1 \ b_4 & b_5 & b_6 & a_2 \ b_7 & b_8 & b_9 & a_3 \ b_{10} & b_{11} & b_{12} & 1 \end{bmatrix} $$
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