20220621 Dual Quaternion

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Pair even

One 、 Yes, what is even number ？

https://zhuanlan.zhihu.com/p/358146509

It is a special self consistent operation for even numbers , Similar to the common plural basic unit $i$ （$i^2=-1$）, Here mathematicians define even basic units as $ϵ$ （${ϵ}^2=0$ And $ϵ\ne 0$）.

Similar to the definition of the plural $a+ib$, The definition of an even number is $a+ϵb$.

Based on the above definition , Define two pairs of even numbers $d_1=a_1+ϵb_1$ and $d_2=a_2+ϵb_2$ , According to the general algorithm, we can get the operation of even numbers ：
$$\begin{array}{rl}k{d}_1& =(k{a}_1)+ϵ(k{b}_1)\\ d_1+d_2& =(a_1+a_2)+ϵ(b_1+b_2)\\ d_1{d}_2& =a_1{a}_2+ϵ(a_1{b}_2+a_2{b}_1)+{ϵ}^2{b}_1{b}_2=a_1{a}_2+ϵ(a_1{b}_2+a_2{b}_1)\\ d_1^{\ast }& =a_1-ϵb_1\\ |d_1|& =\sqrt{d_1{d}_1^{\ast }}=|a_1|\\ d_1^{-1}& =\frac{1}{a_1+ϵb_1}=\frac{a_1-ϵb_1}{a_1^2}=\frac{1}{a_1}-ϵ\frac{b_1}{a_1^2}\text{if}{a}_1\ne 0\\ \frac{d_2}{d_1}& =\frac{a_2+ϵb_2}{a_1+ϵb_1}=\frac{(a_2+ϵb_2)(a_1-ϵb_1)}{(a_1+ϵb_1)(a_1-ϵb_1)}=\frac{a_1{a}_2+ϵ(a_1{b}_2-a_2{b}_1)}{a_1^2}=\frac{a_2}{a_1}+ϵ(\frac{b_2}{a_1}-\frac{a_2{b}_1}{a_1^2})\text{if}{a}_1\ne 0\end{array}$$

The best thing about even numbers is Taylor expansion , The superfluous items immediately disappear , namely
$$f(a+ϵb)=f(a)+ϵb{f}^{\prime }(a)+{ϵ}^2(\dots )=f(a)+ϵb{f}^{\prime }(a)$$ therefore , Further, we can get $$\begin{array}{rl}{e}^{a+ϵb}& =e^a+ϵb{e}^a\\ \ln (a+ϵb)& =\ln (a)+ϵb\frac{1}{a}\text{if}a\ne 0\\ \sin (a+ϵb)& =\sin (a)+ϵb\cos (a)\\ \cos (a+ϵb)& =\cos (a)-ϵb\sin (a)\end{array}$$

further , You can find $$f^{\prime }(a)=\frac{f(a+ϵb)-f(a)}{ϵb}$$ Note that this is direct linearization , Instead of approximate linearization , This can be used to calculate the derivative .

Two 、 Dual vector

Zhaihongmin . A dual quaternion based approach for spacecraft position and attitude synchronization planning and control . Harbin Institute of Technology , 2021

Dual vector $\mathbf{d}=\mathbf{a}+ϵ\mathbf{b}$ ,$\mathbf{a},\mathbf{b}\in {\mathbb{R}}^n$, And then there is
$$\mathbf{d}=\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right]+ϵ\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]=\left[\begin{array}{c}{a}_1+ϵb_1\\ a_2+ϵb_n\\ ⋮\\ a_n+ϵb_n\end{array}\right]$$ in other words , It can be seen as both the real part 、 The dual parts are all pairs of even numbers composed of vectors , At the same time, it is regarded as a vector with pairs of even numbers as elements .

Any line in space can be described by unit dual vector . As shown in the figure below , A straight line $\mathbf{R}$ It can be expressed as $\mathbf{l}+ϵ\mathbf{m}$,$\mathbf{l}$ Displays the parallel direction of the line ,$\mathbf{m}$ Displays the vertical distance between the line and the origin . The unit dual vector here is usually called pruck （Plücker） A straight line .

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3、 ... and 、 Dual four elements

Dual quaternions can be regarded as Elements are quaternions of even numbers , It can also be regarded as a pair of even numbers whose elements are quaternions , namely
$$\begin{array}{rl}\hat{\mathbf{q}}& =[\hat{\eta },\hat{\mathbf{\xi }}]=\mathbf{q}+ϵ{\mathbf{q}}^{\prime }\\ & =\eta +{\xi }_1\mathbf{i}+{\xi }_2\mathbf{j}+{\xi }_3\mathbf{k}+ϵ\left({\eta }^{\prime }+{\xi }_1^{\prime }\mathbf{i}+{\xi }_2^{\prime }\mathbf{j}+{\xi }_3^{\prime }\mathbf{k}\right)\end{array}$$

The conjugation here is different from the one above , Because we think of it as Elements are quaternions of even numbers . Why not take it as the conjugation of even numbers whose elements are quaternions , See https://blog.csdn.net/weixin_44382195/article/details/125400408?spm=1001.2014.3001.5501.
$${\hat{\mathbf{q}}}^{\ast }={\mathbf{q}}^{\ast }+ϵ{\mathbf{q}}^{\prime \ast }$$

Two 、 Use steps

1. Import and stock in

The code is as follows （ Example ）：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import warnings
warnings.filterwarnings('ignore')
import  ssl
ssl._create_default_https_context = ssl._create_unverified_context

2. Read in the data

The code is as follows （ Example ）：

data = pd.read_csv(
    'https://labfile.oss.aliyuncs.com/courses/1283/adult.data.csv')
print(data.head())

It's used here url Data requested by the network .

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summary

Tips ： Here is a summary of the article ：
for example ： That's what we're going to talk about today , This article only briefly introduces pandas Use , and pandas Provides a large number of functions and methods that enable us to process data quickly and conveniently .