---
title: "Rodrigues’ Rotation Formula"
path: "Rodrigues’ Rotation Formula"
---

{%hackmd @RintarouTW/About %}

# Rodrigues’ Rotation Formula
不使用四元數時，以任一向量為轉軸來旋轉另一向量

:::info
參考原文 https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
但原文東拉西扯太多東西，其實真正的思路很單純證明也很短。
:::

<center>
<img src="https://i.imgur.com/nCg0tbN.png" width="50%" style="background-color:#fff !important;filter:invert(0.9)"/>

<span class="caption">(此圖來自 Wikipedia)</span>
</center>

$\def \vk {\hat{k}}
\def \vpara {\vec{v_{\parallel}}}
\def \vperp {\vec{v_{\perp}}}
\def \vv {\vec{v}}
\def \vrot {\vec{v_{rot}}}$

$\begin{cases}
\vk := 轉軸單位向量 \implies |\vk|=1 \\
\vv := 被旋轉的向量 \\
\theta :=旋轉角\\
\vrot := 旋轉後的向量\\
\end{cases}\\[2ex]
\implies
\vrot = \cos{\theta}\ \vv + (1-\cos{\theta})(\vk\cdot \vv)\ \vk\ + \sin{\theta}(\vk\times{\vv})$

## 證明

令 $\begin{cases}\vpara = (\vk\cdot\vv)\ \vk\ \\
\vperp = \vv-\vpara\end{cases}
\iff \vec{v} = \vpara + \vperp$

$\vrot = \vpara + \vec{v_{\perp rot}} \quad(\vpara\ 本身就是轉軸\ \vk\ 的縮放，自不受轉動影響)$

令
$\begin{aligned}
\vec{w} &:= \vk\times{\vperp}\ (|\vec{w}|=|\vk||\vperp|sin{90^\circ}，且\ |\vk|=1, \sin{90^\circ}=1\implies|\vec{w}| = |\vperp|)\\
&= \vk\times(\vv-\vpara) \\
&= \vk\times\vv \quad(\because\vk\times\vpara=0)\\[3ex]
\vec{v_{\perp rot}} &:= \vperp\ 以\ \vperp\ 和\ \vec{w}\ 為平面轉動\ \theta\\
&= \cos{\theta}\ \vperp + \sin{\theta}\ \vec{w}\\
&= \cos{\theta}(\vv - \vpara) + \sin{\theta}(\vk\times\vv)\\
\end{aligned}$

$\begin{aligned}
\vrot &= \vpara + \vec{v_{\perp rot}}\\
&= \vpara + \cos{\theta}(\vv - \vpara) + \sin{\theta}(\vk\times\vv)\\
&=\cos{\theta}\ \vec{v}+(1-\cos{\theta})(\vk\cdot\vv)\ \vk + \sin{\theta}(\vk\times\vv)\ \blacksquare 得證
\end{aligned}$

$若\ v,k\ 固定，則\ (\vk\cdot\vv)\ \vk\ 與\ \vk\times\vv 也是固定的，只需要\ \cos{\theta}\ 與\ \sin{\theta}\ 代入即可很快得到答案。$

## 矩陣型式 (Matrix Notation)

寫成矩陣形式，目標是將 $\vec{v}$ 從之前的結果中提出來。
將

$\begin{aligned}
\vec{v_{\perp rot}} &:= \vperp\ 以\ \vperp\ 和\ \vec{w}\ 為平面轉動\ \theta\\
&= \cos{\theta}\ \vperp + \sin{\theta}\ \vec{w}\\
&= \cos{\theta}\ \vperp + \sin{\theta}(\vk\times\vv)
\end{aligned}$
代入
$\begin{aligned}
\vrot &= \vpara + \vec{v_{\perp rot}}\\
&= \vpara + \cos{\theta}\ \vperp + \sin{\theta}(\vk\times\vv) \\
\because\vpara = \vec{v} - \vperp\\
&= (\vv - \vperp) + \cos{\theta}\ \vperp + \sin{\theta}(\vk\times\vv) \\
&= \vv + (\cos{\theta}-1)\ \vperp + \sin{\theta}(\vk\times\vv)\\
\vperp = -(\vk\times(\vk\times\vv))\\
&=\vv + (1-\cos{\theta})(\vk\times(\vk\times\vv)) + \sin{\theta}(\vk\times\vv)\\
\end{aligned}$

$$
\vk := \pmatrix{k_x\\k_y\\k_x} 為單位向量,\
\vv := \pmatrix{v_x\\v_y\\v_z}\\
\begin{aligned}
\vk\times\vec{v} &= \pmatrix{k_x\\k_y\\k_x}\times\pmatrix{v_x\\v_y\\v_z}\\[2ex]
&=\pmatrix{k_{y}v_{z}-k_{z}v_{y}\\k_{z}v_{x}-k_{x}v_{z}\\k_{x}v_{y}-k_{y}v_{x}} \\[2ex]
&=\begin{bmatrix}0 & -k_z & k_y\\k_z & 0 & -k_x\\-k_y & k_x & 0\end{bmatrix}\begin{bmatrix}v_x\\v_y\\v_z\end{bmatrix}\\
\end{aligned}
$$

$$
令 K := \begin{bmatrix}0 & -k_z & k_y\\k_z & 0 & -k_x\\-k_y & k_x & 0\end{bmatrix}\\[5ex]
K 其實就是\ \vk\ 對任意向量作外積的矩陣，因此\\
\begin{aligned}
\vk\times\vec{v} &= K\ \vv\\
\vk\times(\vk\times\vv) &= \vk\times(K\vv)\\
&= K(K\vv)\\
&= K^{2}\vv
\end{aligned}
$$

$$
\begin{aligned}
\therefore
\vec{v_{rot}} &= \vec{v} + (1-\cos{\theta})K^2\vv + \sin{\theta}(K\vec{v})\\[2ex]
&= \underbrace{(I + (1-\cos\theta)K^2 + \sin\theta K)}_\text{R := Rotation Matrix}\ \vv
\end{aligned}
$$

###### tags: `math` `Rodrigues`