---
title: "Transformation matrices"
author: "Alexander (Sasha) Pastukhov"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Transformation matrices}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
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.

## Bidimensional regression

### Translation

Number of parameters: 2

* translation: $a_1$, $a_2$

$$
\begin{bmatrix}
 1 & 0 & a_1 \\
 0 & 1 & a_2 \\
 0 & 0 & 1
\end{bmatrix}
$$

### Euclidean

Number of parameters: 4

* translation: $a_1$, $a_2$
* scaling: $\phi$
* rotation: $\theta$

$$
\begin{bmatrix}
 b_1 & b_2 & a_1 \\
-b_2 & b_1 & a_2 \\
 0   & 0   & 1
\end{bmatrix}
$$

The Euclidean transformation is a special case, where we can compute rotation ($\theta$) and the single scaling ($\phi$) coefficients, as follows:
$$
\phi = \sqrt{b_1^2 + b_2^2}\\
\theta = tan^{-1}(\frac{b_2}{b_1})
$$

### Affine

Number of parameters: 6

* translation: $a_1$, $a_2$
* scaling · rotation · sheer: $b_1$, $b_2$, $b_3$, $b_4$

$$
\begin{bmatrix}
 b_1 & b_2 & a_1 \\
 b_3 & b_4 & a_2 \\
 0   & 0   & 1
\end{bmatrix}
$$

### Projective

Number of parameters: 8

* translation: $a_1$, $a_2$
* scaling · rotation · sheer · projection: $b_1$...$b_6$

$$
\begin{bmatrix}
 b_1 & b_2 & a_1 \\
 b_3 & b_4 & a_2 \\
 b_5 & b_6 & 1
\end{bmatrix}
$$

## Tridimensional regression

### Translation

Number of parameters: 3

* translation: $a_1$, $a_2$, $a_3$

$$
\begin{bmatrix}
 1 & 0 & 0 & a_1 \\
 0 & 1 & 0 & a_2 \\
 0 & 0 & 1 & a_3 \\
 0 & 0 & 0 & 1
\end{bmatrix}
$$

### Euclidean

Number of parameters: 5

* translation: $a_1$, $a_2$, $a_3$
* scaling: $\phi$
* rotation: $\theta$

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:
$$
\phi = \sqrt{b_1^2 + b_2^2}\\
\theta = tan^{-1}(\frac{b_2}{b_1})
$$

#### Euclidean, rotation about x axis

Note that during fitting $\phi$ is computed from $b_1$ and $b_2$ on the fly.
$$
\begin{bmatrix}
 \phi & 0   & 0   & a_1 \\
 0    & b_1 &-b_2 & a_2 \\
 0    & b_2 & b_1 & a_3 \\
 0    & 0   & 0   & 1
\end{bmatrix}
$$


#### Euclidean, rotation about y axis

$$
\begin{bmatrix}
 b_1 & 0    & b_2 & a_1 \\
 0   & \phi & 0   & a_2 \\
-b_2 & 0    & b_1 & a_3 \\
 0   & 0    &  0   & 1
\end{bmatrix}
$$

#### Euclidean, rotation about z axis

$$
\begin{bmatrix}
 b_1 &-b_2 & 0    & a_1 \\
 b_2 & b_1 & 0    & a_2 \\
 0   & 0   & \phi & a_3 \\
 0   & 0   &  0   & 1
\end{bmatrix}
$$

### Affine

Number of parameters: 12

* translation: $a_1$, $a_2$,  $a_3$
* scaling · rotation · sheer: $b_1$...$b_9$


$$
\begin{bmatrix}
 b_1 & b_2 & b_3 & a_1 \\
 b_4 & b_5 & b_6 & a_2 \\
 b_7 & b_8 & b_9 & a_3 \\
 0   & 0   &  0   & 1
\end{bmatrix}
$$

### Projective

Number of parameters: 15

* translation: $a_1$, $a_2$,  $a_3$
* scaling · rotation · sheer · projection: $b_1$...$b_12$


$$
\begin{bmatrix}
 b_1    & b_2    & b_3    & a_1 \\
 b_4    & b_5    & b_6    & a_2 \\
 b_7    & b_8    & b_9    & a_3 \\
 b_{10} & b_{11} & b_{12} & 1
\end{bmatrix}
$$