---
title: "幾何代數 (Geometric Algebra) I"
path: "幾何代數 Geometric Algebra I"
---

{%hackmd @RintarouTW/About %}

# 幾何代數 (Geometric Algebra) I

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$\require{cancel}$

又稱外代數 (exterior algebra)，合併 Dot Product 與 Wedge Product 形成與四元數類似的應用。

## Context

- $G\ (\mathbb{R^2})$
- $u, v \in G$
- 2 base vectors: $e_1, e_2$ $, where\ e_1\cdot e_2 = 0$

## Dot product

### Definition

$$
\begin{aligned}
u\cdot u &= |u|^2\\
u\cdot v &= v\cdot u = |u||v|\cos{\theta}\\
\end{aligned}
$$

$$
\begin{cases}
u = ae_1 + be_2\\
v = ce_1 + de_2
\end{cases}\\
\begin{aligned}
u\cdot v &= ac + bd\\
u\perp v \implies u\cdot v &= 0\\
\end{aligned}
$$

## Wedge Product

works for any dimention, better than cross product.
- anti-communitive (order matters)
- bivectors : $(e_1e_2)$ also be notated as $I$

### Definition

$$
\begin{aligned}
u\wedge u &= 0\\
u\wedge v &= - v\wedge u\\
&= |u||v|\sin{\theta}(e_1e_2)
\end{aligned}
$$

### Induction

$$
\begin{cases}
u = ae_1 + be_2\\
v = ce_1 + de_2
\end{cases}\\
\begin{aligned}
u\wedge v &= \bcancel{ac(e_1\wedge e_1)} + ad(e_1\wedge e_2)\\
&+ bc(e_2\wedge e_1) + \bcancel{bd(e_2\wedge e_2)}\\
&= ad(e_1\wedge e_2) + bc(e_2\wedge e_1)\\
&= ad(e_1\wedge e_2) - bc(e_1\wedge e_2)\\
&= (ad-bc)(e_1\wedge e_2)\\
\end{aligned}
$$

## Geometric Product

### Definition

$$
uv = u\cdot v + u\wedge v
$$

$$
\begin{cases}
u = ae_1+be_2\\
v = ce_1+de_2
\end{cases}\\
\begin{aligned}
uv &= u\cdot v + u\wedge v\\
&= (ac+bd) + (ad-bc)(e_1\wedge e_2)
\end{aligned}
$$

### Induction

$$
\begin{aligned}
uu = u^2 &= u\cdot u + \bcancel{u\wedge u} = |u|^2\\
\therefore \frac{u}{|u|^2}u &=1
\implies u^{-1} = \frac{u}{|u|^2}
\end{aligned}
$$

:::info
$$
\begin{equation}
u^{-1} =\frac{u}{|u|^2}\tag{1}\label{eq:eq1}
\end{equation}
$$
:::

$$
\begin{aligned}
u\perp v \implies uv &= \bcancel{u\cdot v} + u\wedge v\\
&= u\wedge v = -v\wedge u = -vu\\[2ex]
u\parallel v \implies uv &= u\cdot v + \bcancel{u\wedge v}\\
&= u\cdot v = v\cdot u = vu\\
\end{aligned}
$$

:::info
$$
\begin{equation}
u\perp v \implies uv = u\wedge v = -v\wedge u = -vu\\
u\parallel v \implies uv = u\cdot v = v\cdot u = vu\tag{2}\label{eq:eq2}
\end{equation}
$$
:::

$$
\begin{aligned}
e_1e_1 &= e_1\cdot e_1 + \bcancel{e_1\wedge e_1}\\
&= |e_1|^2\\
&= 1\\
\therefore e_1^2&=1\\
e_2^2&=1\quad (for\ the\ same\ reason.)
\end{aligned}
$$

:::info
$$
\begin{equation}
e_1^2 = e_2^2 = 1 \tag{3}\label{eq:eq3}
\end{equation}
$$
:::

$$
\begin{aligned}
e_1e_2 &= \bcancel{e_1\cdot e_2} + e_1\wedge e_2\\
&=e_1\wedge e_2\\
e_2e_1 &=e_2\wedge e_1 = - e_1\wedge e_2 = -e_1e_2\\
&(anti-communitive)
\end{aligned}
$$

:::info
$$
\begin{equation}
e_1e_2 = e_1\wedge e_2 = I\\
e_1e_2 = -e_2e_1\tag{4}\label{eq:eq4}
\end{equation}
$$
:::

$$
\begin{aligned}
(e_1e_2)^2 &= e_1e_2e_1e_2\\
&= -e_1e_2e_2e_1\\
&= -e_1e_2^2e_1\\
&= -e_1e_1\\
&= -1\\
\end{aligned}
$$

:::info
$$
\begin{equation}
(e_1e_2)^2 = -1 \tag{5}\label{eq:eq5}
\end{equation}
$$
:::

## Results

$$
\begin{cases}
u^{-1} &=\frac{u}{|u|^2} &\eqref{eq:eq1}\\
u\perp v \implies & uv = u\wedge v = -v\wedge u = -vu &\eqref{eq:eq2}\\
u\parallel v \implies & uv = u\cdot v = v\cdot u = vu\\
e_1^2 = e_2^2 = 1 &&\eqref{eq:eq3}\\
e_1e_2 &= e_1\wedge e_2 = I &\eqref{eq:eq4}\\
e_1e_2 &= -e_2e_1\\
(e_1e_2)^2 &= -1 &\eqref{eq:eq5}
\end{cases}
$$

$G(\mathbb{R^2})$
- base vectors: $e_1，e_2$
- bivector: $e_1\wedge e_2 = e_1e_2 = I$ (Area with orientation)
- scalar: 1
- Dim(G) = 4 formed by $\{\ 1, e_1, e_2, I\ \}$

# Applications

### Rotate $90^\circ = uI$

:::info
$uI$ is isomorphic to $ui$.
:::

$$
\eqalign{
u &= ae_1+be_2\\
uI &= (ae_1+be_2)I = (ae_1+be_2)e_1e_2\\
&= ae_1e_1e_2+be_2e_1e_2\\
&= ae_2-be_1\\
&= -be_1 + ae_2\\
\therefore uI \implies ui\\[2ex]
}
$$

:::warning
$Iu$ is not the same with $iu$ since order matters.
:::

$$
\eqalign{
but,\ Iu &= I(ae_1+be_2) = e_1e_2(ae_1+be_2)\\
&= ae_1e_2e_1 + be_1e_2e_2\\
&= -ae_2e_1e_1 + be_1\\
&= be_1 - ae_2
}
$$

This is different from $i$ which is communative. Actually,

$$
Iu = -uI = -ui
$$

### Rotate $\theta = ue^{\theta I}$

$$
\begin{aligned}
u &= ae_1+be_2\\
u'=ue^{\theta I} &= (ae_1+be_2)(\cos{\theta}+\sin{\theta}I)\\
&=a\cos{\theta}e_1+a\sin{\theta}e_1e_1e_2\\
&+b\cos{\theta}e_2+b\sin{\theta}e_2e_1e_2\\
&=a\cos{\theta}e_1+a\sin{\theta}e_2\\
&+b\cos{\theta}e_2-b\sin{\theta}e_1\\
&=(a\cos{\theta}-b\sin{\theta})e_1+(a\sin{\theta}+b\cos{\theta})e_2
\end{aligned}
$$

:::info
$ue^{\theta I}$ still works like $e^{\theta i}u$
:::

## Vector Projection

$$
\begin{cases}
uv = u\cdot v + u\wedge v\\
vu = v\cdot u + v\wedge u = u\cdot v - u\wedge v
\end{cases}\\
uv-vu = 2(u\wedge v) \implies u\wedge v = \frac{uv-vu}{2}\\
uv+vu = 2(u\cdot v) \implies u\cdot v = \frac{uv+vu}{2}
$$

:::info
$\begin{cases}u\wedge v = \frac{uv-vu}{2}\\
u\cdot v = \frac{uv+vu}{2}\end{cases}$
:::

$$
\begin{aligned}
u &= u_\parallel + u_\perp\ ,\ (\ u_\parallel\ is\ parallel\ to\ v,\ u_\perp\ is\ perpendicular\ to\ v\ )\\
uv &= u_\parallel v - v u_\perp\\
&= u_\parallel v - v(u - u_\parallel)\\
&= u_\parallel v - vu + vu_\parallel\\
&= 2u_\parallel v - vu\\
\implies u_\parallel v &= \frac{uv+vu}{2} = u\cdot v
\end{aligned}
$$

:::info
$u_\parallel v = u\cdot v$
:::

$$
\begin{aligned}
u_\parallel vv^{-1} &= (u\cdot v)v^{-1}\\
u_\parallel &= (u\cdot v)v^{-1}\\
u_\perp &= u - (u\cdot v)v^{-1}\\
&= (u - (u\cdot v)v^{-1})(vv^{-1})\\
&= (uv - u\cdot v) v^{-1}\\
&= (u\wedge v)v^{-1}\\
\end{aligned}
$$

:::info
$\begin{cases}
u_\parallel &= (u\cdot v)v^{-1}\\
u_\perp &= (u\wedge v)v^{-1}
\end{cases}$
:::

$$
\text{Also, since } v^{-1} = \frac{v}{|v|^2} = \frac{v}{v\cdot v}\\
\implies u_\parallel = \frac{u\cdot v}{v\cdot v}v\\
\text {if }|v| = 1\ then\ v^{-1} = v\\
\implies u_\parallel = (u\cdot v)v
$$

### Use Case

$$
\begin{cases}
u = e_1+e_2\\
v = 2e_1
\end{cases}
\implies v^{-1} = \frac{2e_1}{|2^2|} = \frac{e_1}{2}\\
\begin{aligned}
u_\parallel &= (u\cdot v)v^{-1}\\
&= 2 \frac{e_1}{2}\\
&= e_1\\
u_\perp &= (u\wedge v)v^{-1}\\
&= ((e_1+e_2)\wedge(2e_1))\frac{e_1}{2}\\
&= (2e_2\wedge e_1)\frac{e_1}{2}\\
&= (2e_2e1) \frac{e_1}{2}\\
&= e_2
\end{aligned}
$$

## Vector Reflection

$$
\begin{aligned}
u &= u_\parallel + u_\perp\\
u' &= u_\parallel - u_\perp\\
vu' &= vu_\parallel - vu_\perp\\
&= u_\parallel v + u_\perp v\\
&= uv\\
\therefore v^{-1}vu' &= v^{-1}uv\\
u' &= v^{-1}uv\\
\end{aligned}
$$

:::info
$u' = v^{-1}uv$
:::

### Use Case

$$
\begin{cases}
u &= e_1+e_2\\
v &= 2e_1
\end{cases}
\implies v^{-1} = \frac{e_1}{2}\\
\begin{aligned}
u' &= \frac{e1}{2} (e_1+e_2) 2e_1\\
&= e_1+e_1e_2e_1\\
&= e_1-e_2
\end{aligned}
$$

###### tags: `geometric` `geometry` `wedge` `dot` `algebra`