Quaternion

• Quaternion

 

A set of numbers that use Complex Numbers.

It extends the complex numbers. The algebra of quaternions is often denoted by $\mathbf{H}$ or by $\mathbb{H}$.

If we express the three-dimensional vector in terms of the basis , we obtain .

Quaternions : with the scalar value added to it.

It can be expressed as .

∴

Quaternions can be said to be a vector based on .

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• Quaternions multiplication rules 

 

$$\begin{array}{c}{i}^2=j^2=k^2=-1\\ ij=k,jk=i,ki=j\\ ji=-k,kj=-i,ik=-j\end{array}$$

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∴

$ijk\cdot k=(-1)\cdot k\Rightarrow ij=k$

$i\cdot ijk=i\cdot (-1)\Rightarrow jk=i$

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∴

$jk\cdot i=i\cdot i\Rightarrow j\cdot jki=j\cdot (-1)$

$\therefore ki=j$

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∴

$j\cdot ij\cdot j=j\cdot k\cdot j\Rightarrow -ji=ij$

$k\cdot jk\cdot k=k\cdot i\cdot k\Rightarrow -kj=jk$

$i\cdot ki\cdot i=i\cdot j\cdot i\Rightarrow -ik=ki$

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• Quaternions operation

 

1. Addition and Subtraction

If

$\widehat{p}=(s_p,v_p),\widehat{q}=(s_q,v_q)$,

then

$\widehat{p}+\widehat{q}=(s_p+s_q,v_p+v_q)$

In here,

: scalar, : vector

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If

$\widehat{p}=(a_p,b_p,c_p,d_p)$, $\widehat{q}=(a_q,b_q,c_q,d_q)$

then

$\widehat{p}+\widehat{q}=(a_p+a_q,b_p+b_q,c_p+c_q,d_p+d_q)$

$\widehat{p}-\widehat{q}=(a_p-a_q,b_p-b_q,c_p-c_q,d_p-d_q)$

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∴

$\widehat{p}+\widehat{q}=(s_p-s_q,v_p-v_q)$

In here,

the operation that looks like addition is actually component-wise subtraction. Both the scalar and vector parts are subtracted, reflecting a "difference" operation between .

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2. Multiplication

$\lambda \widehat{p}=(\lambda s_p,\lambda v_p)=(\lambda a_p,\lambda b_p,\lambda c_p,\lambda d_p)$

If

$\widehat{p}=d_p+a_{p}i+b_{p}j+c_{p}k$, $\widehat{q}=d_q+a_{q}i+b_{q}j+c_{q}k$

Then,

$\widehat{p}\widehat{q}=(d_p+a_{p}i+b_{p}j+c_{p}k)(d_q+a_{q}i+b_{q}j+c_{q}k)$

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In here,

 

①

$$\begin{gathered} d_p{d}_q+d_p+d_p{a}_q\mathbf{i}+d_p{b}_q\mathbf{j}+d_p{c}_q\mathbf{k}+ \\ {a}_p\mathbf{i}{d}_q+a_p\mathbf{i}{a}_q\mathbf{i}+a_p\mathbf{i}{b}_q\mathbf{j}+a_p\mathbf{i}{c}_q\mathbf{k}+ \\ {b}_p\mathbf{j}{d}_q+b_p\mathbf{j}{a}_q\mathbf{i}+b_p\mathbf{j}{b}_q\mathbf{j}+b_p\mathbf{j}{c}_q\mathbf{k}+ \\ {c}_p\mathbf{k}{d}_q+c_p\mathbf{k}{a}_q\mathbf{i}+c_p\mathbf{k}{b}_q\mathbf{j}+c_p\mathbf{k}{c}_q\mathbf{k} \end{gathered}$$

②

Align by

$$\begin{gathered} d_p{d}_q+d_p{a}_q\mathbf{i}+d_p{b}_q\mathbf{j}+d_p{c}_q\mathbf{k}+ \\ {a}_p{d}_q\mathbf{i}+a_p{a}_q{\mathbf{i}}^2+a_p{b}_q\mathbf{i}\mathbf{j}+a_p{c}_q\mathbf{i}\mathbf{k}+ \\ {b}_p{d}_q\mathbf{j}+b_p{a}_q\mathbf{j}\mathbf{i}+b_p{b}_q{\mathbf{j}}^2+b_p{c}_q\mathbf{j}\mathbf{k}+ \\ {c}_p{d}_q\mathbf{k}+c_p{a}_q\mathbf{k}\mathbf{i}+c_p{b}_q\mathbf{k}\mathbf{j}+c_p{c}_q{\mathbf{k}}^2 \end{gathered}$$

③

Express the product of two quaternion units as one quaternion unit (ex. )

$$\begin{gathered} d_p{d}_q+d_p{a}_q\mathbf{i}+d_p{b}_q\mathbf{j}+d_p{c}_q\mathbf{k}+ \\ {a}_p{d}_q\mathbf{i}-a_p{a}_q+a_p{b}_q\mathbf{k}-a_p{c}_q\mathbf{j}+ \\ {b}_p{d}_q\mathbf{j}-b_p{a}_q\mathbf{k}-b_p{b}_q+b_p{c}_q\mathbf{i}+ \\ {c}_p{d}_q\mathbf{k}+c_p{a}_q\mathbf{j}-c_p{b}_q\mathbf{i}-c_p{c}_q \end{gathered}$$

④

Bind into

$$\begin{gathered} d_p{d}_q-a_p{a}_q-b_p{b}_q-c_p{c}_q \\ +d_p{a}_q\mathbf{i}+a_p{d}_q\mathbf{i}+(b_p{c}_q-c_p{b}_q)\mathbf{i} \\ +d_p{b}_q\mathbf{j}+b_p{d}_q\mathbf{j}+(c_p{a}_q-a_p{c}_q)\mathbf{j} \\ +d_p{c}_q\mathbf{k}+c_p{d}_q\mathbf{k}+(a_p{b}_q-b_p{a}_q)\mathbf{k} \end{gathered}$$

⑤

Bind into .

$$\begin{gathered} d_p{d}_q-(a_p{a}_q+b_p{b}_q+c_p{c}_q) \\ +d_p(a_q\mathbf{i}+b_q\mathbf{j}+c_q\mathbf{k}) \\ +d_q(a_p\mathbf{i}+b_p\mathbf{j}+c_p\mathbf{k}) \\ +(b_p{c}_q-c_p{b}_q)\mathbf{i}+(c_p{a}_q-a_p{c}_q)\mathbf{j}+(a_p{b}_q-b_p{a}_q)\mathbf{k} \end{gathered}$$

⑥

$$\begin{gathered} d_p{d}_q-({\mathbf{v}}_p\cdot {\mathbf{v}}_q) \\ +d_p{\mathbf{v}}_q+d_q{\mathbf{v}}_p+{\mathbf{v}}_p\times {\mathbf{v}}_q \end{gathered}$$

In here,

${\overrightarrow{v}}_p\cdot {\overrightarrow{v}}_q$ : $(a_p{a}_q+b_p{b}_q+c_p{c}_q)$,

${\overrightarrow{v}}_p=(a_{p}i+b_{p}j+c_{p}k)$,

${\overrightarrow{v}}_p\times {\overrightarrow{v}}_q:$ $(b_p{c}_q-c_p{b}_q)i+(c_p{a}_q-a_p{c}_q)j+(a_p{b}_q-b_p{a}_q)k.$

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Conclusion

 

∴

$\widehat{p}\widehat{q}=(d_p,v_p)(d_q,v_q)=(d_p{d}_q-v_p\cdot v_q,d_p{v}_q+d_q{v}_p+v_p\times v_q)$

 

- Quaternion * Quaternion -> Quaternion

 

https://gaussian37.github.io/vision-concept-quaternion/