Quaternion

Math

Quaternion

Naranjito 2024. 9. 29. 14:43

Quaternion

A set of numbers that use Complex Numbers.

It extends the complex numbers. The algebra of quaternions is often denoted by

H

or by

H

.

If we express the three-dimensional vector $v=(a,b,c)$ in terms of the basis $i,j,k$ , we obtain $ai+bj+ck$ .

Quaternions : $d+ai+bj+ck$ with the scalar value $d$ added to it.

It can be expressed as $(d,v)$ .

∴

Quaternions can be said to be a vector based on $1,i,j,k$ .

Quaternions multiplication rules

\begin{matrix} i^{2} = j^{2} = k^{2} = - 1 \\ i j = k, j k = i, k i = j \\ j i = - k, k j = - i, i k = - j \end{matrix}

∴

i j k \cdot k = (- 1) \cdot k \Rightarrow i j = k

i \cdot i j k = i \cdot (- 1) \Rightarrow j k = i

∴

j k \cdot i = i \cdot i \Rightarrow j \cdot j k i = j \cdot (- 1)

∴ k i = j

∴

j \cdot i j \cdot j = j \cdot k \cdot j \Rightarrow - j i = i j

k \cdot j k \cdot k = k \cdot i \cdot k \Rightarrow - k j = j k

i \cdot k i \cdot i = i \cdot j \cdot i \Rightarrow - i k = k i

Quaternions operation

1. Addition and Subtraction

If

\hat{p} = (s_{p}, v_{p}), \hat{q} = (s_{q}, v_{q})

,

then

\hat{p} + \hat{q} = (s_{p} + s_{q}, v_{p} + v_{q})

In here,

$s_{p},s_{q}$ : scalar, $v_{p},v_{q}$ : vector

\hat{p} = (a_{p}, b_{p}, c_{p}, d_{p})

\hat{q} = (a_{q}, b_{q}, c_{q}, d_{q})

then

\hat{p} + \hat{q} = (a_{p} + a_{q}, b_{p} + b_{q}, c_{p} + c_{q}, d_{p} + d_{q})

\hat{p} - \hat{q} = (a_{p} - a_{q}, b_{p} - b_{q}, c_{p} - c_{q}, d_{p} - d_{q})

∴

\hat{p} + \hat{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 $\widehat{p},\widehat{q}.$ .

2. Multiplication

λ \hat{p} = (λ s_{p}, λ v_{p}) = (λ a_{p}, λ b_{p}, λ c_{p}, λ d_{p})

\hat{p} = d_{p} + a_{p} i + b_{p} j + c_{p} k

\hat{q} = d_{q} + a_{q} i + b_{q} j + c_{q} k

Then,

\hat{p} \hat{q} = (d_{p} + a_{p} i + b_{p} j + c_{p} k) (d_{q} + a_{q} i + b_{q} j + c_{q} k)

In here,

①

d_{p} d_{q} + d_{p} + d_{p} a_{q} i + d_{p} b_{q} j + d_{p} c_{q} k + a_{p} i d_{q} + a_{p} i a_{q} i + a_{p} i b_{q} j + a_{p} i c_{q} k + b_{p} j d_{q} + b_{p} j a_{q} i + b_{p} j b_{q} j + b_{p} j c_{q} k + c_{p} k d_{q} + c_{p} k a_{q} i + c_{p} k b_{q} j + c_{p} k c_{q} k

②

Align by $i,j,k$

d_{p} d_{q} + d_{p} a_{q} i + d_{p} b_{q} j + d_{p} c_{q} k + a_{p} d_{q} i + a_{p} a_{q} i^{2} + a_{p} b_{q} i j + a_{p} c_{q} i k + b_{p} d_{q} j + b_{p} a_{q} j i + b_{p} b_{q} j^{2} + b_{p} c_{q} j k + c_{p} d_{q} k + c_{p} a_{q} k i + c_{p} b_{q} k j + c_{p} c_{q} k^{2}

③

Express the product of two quaternion units as one quaternion unit (ex. $ij=k$ )

d_{p} d_{q} + d_{p} a_{q} i + d_{p} b_{q} j + d_{p} c_{q} k + a_{p} d_{q} i - a_{p} a_{q} + a_{p} b_{q} k - a_{p} c_{q} j + b_{p} d_{q} j - b_{p} a_{q} k - b_{p} b_{q} + b_{p} c_{q} i + c_{p} d_{q} k + c_{p} a_{q} j - c_{p} b_{q} i - c_{p} c_{q}

④

Bind into $i,j,k$

d_{p} d_{q} - a_{p} a_{q} - b_{p} b_{q} - c_{p} c_{q} + d_{p} a_{q} i + a_{p} d_{q} i + (b_{p} c_{q} - c_{p} b_{q}) i + d_{p} b_{q} j + b_{p} d_{q} j + (c_{p} a_{q} - a_{p} c_{q}) j + d_{p} c_{q} k + c_{p} d_{q} k + (a_{p} b_{q} - b_{p} a_{q}) k

⑤

Bind into $d_{p},d_{q},i,j,k$ .

d_{p} d_{q} - (a_{p} a_{q} + b_{p} b_{q} + c_{p} c_{q}) + d_{p} (a_{q} i + b_{q} j + c_{q} k) + d_{q} (a_{p} i + b_{p} j + c_{p} k) + (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

⑥

d_{p} d_{q} - (v_{p} \cdot v_{q}) + d_{p} v_{q} + d_{q} v_{p} + v_{p} \times v_{q}

In here,

{\vec{v}}_{p} \cdot {\vec{v}}_{q}

(a_{p} a_{q} + b_{p} b_{q} + c_{p} c_{q})

{\vec{v}}_{p} = (a_{p} i + b_{p} j + c_{p} k)

{\vec{v}}_{p} \times {\vec{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 .

Conclusion

∴

\hat{p} \hat{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/

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