Kuaternion

Dalam matematika, Kuaternion adalah perluasan dari bilangan-bilangan kompleks yang tidak komutatif, dan diterapkan dalam mekanika tiga dimensi. Kuaternion ditemukan oleh ahli matematika dan astronomi Inggris, William Rowan Hamilton, yang memperpanjang aritmetika kompleks nomor ke kuaternion.

Segera setelah itu penemuan Hamilton, matematikawan Jerman Hermann Grassmann mulai menyelidiki vektor. Meskipun karakter abstrak, fisikawan Amerika JW Gibbs diakui dalam aljabar vektor sistem utilitas besar bagi fisikawan, seperti Hamilton mengakui kegunaan kuaternion. Pengaruh luas dari pendekatan abstrak yang dipimpin George Boole untuk menulis Hukum Thought (1854), perawatan aljabar dasar logika.

Definisi

Sebagai himpunan, kuaternion, berlambang H, sama dengan R⁴ yang merupakan ruang vektor bilangan riil empat dimensi. H memiliki tiga macam operasi: pertambahan, perkalian skalar dan perkalian kuaternion. Elemen-elemen kuaternion ditandakan sebagai 1, i, j dan k (i, j dan k adalah komponen imaginer), dan dapat ditulis sebagai kombinasi linear, a + bi + cj + dk (a, b, c, dan d adalah bilangan riil).

Kuaternion p = a + b i + c j + d k {\displaystyle p=a+bi+cj+dk} bisa dituliskan sebagai p = a + u → {\displaystyle p=a+{\vec {u}}} di mana u → {\displaystyle {\vec {u}}} adalah vektor 3 bilangan imaginer, u → = { b i + c j + d k } {\displaystyle {\vec {u}}=\{bi+cj+dk\}} .

Perkalian elemen dasar

Persamaan elemen kuaternion i, j, dan k adalah:

i 2 = j 2 = k 2 = i j k = − 1 ,   {\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1,\ }

Karena

− 1 = i j k ,   {\displaystyle -1=ijk,\ }

jika dua sisi dikalikan dengan k, maka

− k = i j k k = i j ( k 2 ) = i j ( − 1 ) , k = i j . {\displaystyle {\begin{aligned}-k&=ijkk=ij(k^{2})=ij(-1),\\k&=ij.\end{aligned}}}

Persamaan-persamaan yang lainnya juga bisa didapatkan dengan tahap aljabar:

i j = k , j i = − k , j k = i , k j = − i , k i = j , i k = − j , {\displaystyle {\begin{alignedat}{2}ij&=k,&\qquad ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j,\end{alignedat}}}

Persamaan-persamaan ini lalu bisa ditampilkan dengan tabel di bawah ini:

Perkalian kuaternion

×	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

Pertambahan

p 1 + p 2 = ( a 1 + b 1 i + c 1 j + d 1 k ) + ( a 2 + b 2 i + c 2 j + d 2 k ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) i + ( c 1 + c 2 ) j + ( d 1 + d 2 ) k {\displaystyle {\begin{aligned}&p_{1}+p_{2}=(a_{1}+b_{1}i+c_{1}j+d_{1}k)+(a_{2}+b_{2}i+c_{2}j+d_{2}k)\\&=(a_{1}+a_{2})+(b_{1}+b_{2})i+(c_{1}+c_{2})j+(d_{1}+d_{2})k\end{aligned}}}

Pengurangan

p 1 − p 2 = ( a 1 + b 1 i + c 1 j + d 1 k ) − ( a 2 + b 2 i + c 2 j + d 2 k ) = ( a 1 − a 2 ) + ( b 1 − b 2 ) i + ( c 1 − c 2 ) j + ( d 1 − d 2 ) k {\displaystyle {\begin{aligned}&p_{1}-p_{2}=(a_{1}+b_{1}i+c_{1}j+d_{1}k)-(a_{2}+b_{2}i+c_{2}j+d_{2}k)\\&=(a_{1}-a_{2})+(b_{1}-b_{2})i+(c_{1}-c_{2})j+(d_{1}-d_{2})k\end{aligned}}}

Perkalian

p 1 × p 2 = ( a 1 a 2 − b 1 b 2 − c 1 c 2 − d 1 d 2 ) + ( b 1 a 2 + a 1 b 2 − d 1 c 2 + c 1 d 2 ) i + ( c 1 a 2 + d 1 b 2 + a 1 c 2 − b 1 d 2 ) j + ( d 1 a 2 − c 1 b 2 + b 1 c 2 + a 1 d 2 ) k {\displaystyle {\begin{aligned}&p_{1}\times p_{2}\\&=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2})+(b_{1}a_{2}+a_{1}b_{2}-d_{1}c_{2}+c_{1}d_{2})i+(c_{1}a_{2}+d_{1}b_{2}+a_{1}c_{2}-b_{1}d_{2})j+(d_{1}a_{2}-c_{1}b_{2}+b_{1}c_{2}+a_{1}d_{2})k\end{aligned}}}

Bila kuaternion dituliskan dengan bentuk p = a + u → {\displaystyle p=a+{\vec {u}}} , maka:

p 1 × p 2 = ( a 1 + u 1 → ) × ( a 2 + u 2 → ) = ( a 1 a 2 − u 1 → ⋅ u 2 → ) + ( a 1 u 2 → + a 2 u 1 → + u 1 → × u 2 → ) {\displaystyle {\begin{aligned}&p_{1}\times p_{2}\\&=(a_{1}+{\vec {u_{1}}})\times (a_{2}+{\vec {u_{2}}})\\&=(a_{1}a_{2}-{\vec {u_{1}}}\cdot {\vec {u_{2}}})+(a_{1}{\vec {u_{2}}}+a_{2}{\vec {u_{1}}}+{\vec {u_{1}}}\times {\vec {u_{2}}})\end{aligned}}}

Pembagian

p 1 / p 2 = a 1 a 2 + b 1 b 2 + c 1 c 2 + d 1 d 2 m + b 1 a 2 − a 1 b 2 − d 1 c 2 + c 1 d 2 m i + c 1 a 2 + d 1 b 2 − a 1 c 2 − b 1 d 2 m j + d 1 a 2 − c 1 b 2 + b 1 c 2 − a 1 d 2 m k {\displaystyle {\begin{aligned}&p_{1}/p_{2}\\&={\frac {a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}}{m}}+{\frac {b_{1}a_{2}-a_{1}b_{2}-d_{1}c_{2}+c_{1}d_{2}}{m}}i+{\frac {c_{1}a_{2}+d_{1}b_{2}-a_{1}c_{2}-b_{1}d_{2}}{m}}j+{\frac {d_{1}a_{2}-c_{1}b_{2}+b_{1}c_{2}-a_{1}d_{2}}{m}}k\end{aligned}}} di mana m = a 2 2 + b 2 2 + c 2 2 + d 2 2 {\displaystyle m=a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2}}

Konjugat

Suatu kuaternion p = a + bi + cj + dk memiliki konjugat p*, dan didapatkan dengan rumus berikut:

p ∗ = a − b i − c j − d k {\displaystyle {\begin{alignedat}{2}p*=a-bi-cj-dk\end{alignedat}}}

Persamaan-persamaan konjugasi kuaternion adalah:

( p ∗ ) ∗ = p ( p q ) ∗ = q ∗ p ∗ ( p − 1 ) ∗ = p ‖ p ‖ 2 ( p ∗ ) − 1 = p ‖ p ‖ 2 ( p − 1 ) − 1 = p ( p 1 + p 2 ) ∗ = p 1 ∗ + p 2 ∗ {\displaystyle {\begin{matrix}(p^{*})^{*}&=&p\\(pq)^{*}&=&q^{*}p^{*}\\(p^{-1})^{*}&=&{\frac {p}{\|p\|^{2}}}\\(p^{*})^{-1}&=&{\frac {p}{\|p\|^{2}}}\\(p^{-1})^{-1}&=&p\\(p_{1}+p_{2})^{*}&=&p_{1}^{*}+p_{2}^{*}\\\end{matrix}}\,}

Satuan

Dengan fungsi Norma N ( ) {\displaystyle N()} , bila N ( p ) = 1 {\displaystyle N(p)=1} , maka:

p = cos ⁡ ( θ ) + u → sin ⁡ ( θ ) p = cos ⁡ ( θ ) + u ^ sin ⁡ ( θ ) {\displaystyle {\begin{matrix}p&=&\cos(\theta )+{\vec {u}}\sin(\theta )\\p&=&\cos(\theta )+{\hat {u}}\sin(\theta )\end{matrix}}\,}

di mana

‖ u → ‖ = 1 {\displaystyle \left\|{\vec {u}}\right\|=1}

Bentuk matriks

Kuaternion, seperti bilangan kompleks, bisa ditulis dalam bentuk matriks, yaitu matriks kompleks 2x2 atau matriks riil 4x4.

Bentuk matriks kompleks 2x2 untuk kuaternion a + bi + cj + dk adalah:

[ a + b i c + d i − c + d i a − b i ] = a [ 1 0 0 1 ] + b [ i 0 0 − i ] + c [ 0 1 − 1 0 ] + d [ 0 i i 0 ] {\displaystyle {\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}=a{\begin{bmatrix}\;\;1&0\\0&1\end{bmatrix}}+b{\begin{bmatrix}\;\;i&0\\0&-i\end{bmatrix}}+c{\begin{bmatrix}\;\;0&1\\-1&0\end{bmatrix}}+d{\begin{bmatrix}\;\;0&i\\i&0\end{bmatrix}}}

Bentuk matriks riil 4x4 untuk kuaternion a + bi + cj + dk adalah:

[ a b c d − b a − d c − c d a − b − d − c b a ] = a [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] + b [ 0 1 0 0 − 1 0 0 0 0 0 0 − 1 0 0 1 0 ] + c [ 0 0 1 0 0 0 0 1 − 1 0 0 0 0 − 1 0 0 ] + d [ 0 0 0 1 0 0 − 1 0 0 1 0 0 − 1 0 0 0 ] {\displaystyle {\begin{bmatrix}a&b&c&d\\-b&a&-d&c\\-c&d&a&-b\\-d&-c&b&a\end{bmatrix}}=a{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}+b{\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}+c{\begin{bmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{bmatrix}}+d{\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&1&0&0\\-1&0&0&0\end{bmatrix}}}

Selain itu juga terdapat bentuk matriks 3x3 yang digunakan dalam grafika komputer. Berikut adalah bentuk matriks kolom-utama (column-major) yang digunakan di OpenGL. (Matriks baris-utama (row-major) yang digunakan di DirectX sama dengan transposa matriks kolom-utama)

[ 1 − 2 ( c 2 + d 2 ) 2 ( b c − d a ) 2 ( b d + c a ) 2 ( b c + d a ) 1 − 2 ( b 2 + d 2 ) 2 ( c d − b a ) 2 ( b d − c a ) 2 ( c d + b a ) 1 − 2 ( b 2 + c 2 ) ] {\displaystyle {\begin{bmatrix}1-2(c^{2}+d^{2})&2(bc-da)&2(bd+ca)\\2(bc+da)&1-2(b^{2}+d^{2})&2(cd-ba)\\2(bd-ca)&2(cd+ba)&1-2(b^{2}+c^{2})\end{bmatrix}}}

Fungsi

Norma

N ( p ) = N ( a + b i + c j + d k ) = a 2 + b 2 + c 2 + d 2 {\displaystyle N(p)=N(a+bi+cj+dk)=a^{2}+b^{2}+c^{2}+d^{2}}

Dan juga,

N ( p ∗ ) = N ( p ) N ( p q ) = N ( p ) N ( q ) {\displaystyle {\begin{matrix}N(p^{*})&=&N(p)\\N(pq)&=&N(p)N(q)\end{matrix}}}

Kebalikan

p − 1 = p ∗ N ( p ) {\displaystyle p^{-1}={\frac {p^{*}}{N(p)}}}

Dan juga,

p p − 1 = p − 1 p p p − 1 = 1 ( p − 1 ) − 1 = p ( p q ) − 1 = q − 1 p − 1 {\displaystyle {\begin{matrix}pp^{-1}&=&p^{-1}p\\pp^{-1}&=&1\\(p^{-1})^{-1}&=&p\\(pq)^{-1}&=&q^{-1}p^{-1}\end{matrix}}}

Pemilihan riil

Meskipun tertetap sangat sederhana, fungsi yang hasilnya adalah bagiannya bilangan riil kuaternion ini memiliki kegunaannya tersendiri. W ( p ) = W ( a + b i + c j + d k ) = a {\displaystyle W(p)=W(a+bi+cj+dk)=a}

Dan juga,

W ( p ) = ( p + p ∗ ) / 2 {\displaystyle {\begin{matrix}W(p)&=&(p+p^{*})/2\end{matrix}}}

Skalar

Dari kuaternion p 2 = p + ( p ∗ ) 2 {\displaystyle p_{2}={\frac {p+(p^{*})}{2}}}

Maka: S c a l a r ( p ) = a 2 {\displaystyle Scalar(p)=a_{2}}

Signum

sgn ⁡ ( p ) = p | p | {\displaystyle \operatorname {sgn}(p)={\frac {p}{|p|}}}

Argumen

arg ⁡ ( p ) = arccos ⁡ ( S c a l a r ( p ) | p | ) {\displaystyle \arg(p)=\arccos({\frac {Scalar(p)}{|p|}})}

Pangkat dan Logaritma

Fungsi ekponensial: exp ⁡ ( p ) = exp ⁡ ( a ) ( cos ⁡ ( | u → | ) + sgn ⁡ ( u → ) sin ⁡ ( | u → | ) ) {\displaystyle \exp(p)=\exp(a)(\cos(|{\vec {u}}|)+\operatorname {sgn}({\vec {u}})\sin(|{\vec {u}}|))}

Logaritma natural: ln ⁡ ( | p | ) = ln ⁡ ( | p | ) + sgn ⁡ ( u → ) arg ⁡ ( p ) {\displaystyle \ln(|p|)=\ln(|p|)+\operatorname {sgn}({\vec {u}})\arg(p)}

Pangkat: p q = e q ln ⁡ ( p ) {\displaystyle p^{q}=e^{q\ln(p)}}

Trigonometri

Fungsi trigonometris

sin ⁡ ( p ) = sin ⁡ ( a ) cosh ⁡ ( | u → | ) + cos ⁡ ( a ) sgn ⁡ ( u → ) sinh ⁡ ( | u → | ) {\displaystyle \sin(p)=\sin(a)\cosh(|{\vec {u}}|)+\cos(a)\operatorname {sgn}({\vec {u}})\sinh(|{\vec {u}}|)}

cos ⁡ ( p ) = cos ⁡ ( a ) cosh ⁡ ( | u → | ) − sin ⁡ ( a ) sgn ⁡ ( u → ) sinh ⁡ ( | u → | ) {\displaystyle \cos(p)=\cos(a)\cosh(|{\vec {u}}|)-\sin(a)\operatorname {sgn}({\vec {u}})\sinh(|{\vec {u}}|)}

tan ⁡ ( p ) = sin ⁡ ( p ) cos ⁡ ( p ) {\displaystyle \tan(p)={\frac {\sin(p)}{\cos(p)}}}

Fungsi hiperbolik

sinh ⁡ ( p ) = sinh ⁡ ( a ) cos ⁡ ( | u → | ) + cosh ⁡ ( a ) sgn ⁡ ( u → ) sin ⁡ ( | u → | ) {\displaystyle \sinh(p)=\sinh(a)\cos(|{\vec {u}}|)+\cosh(a)\operatorname {sgn}({\vec {u}})\sin(|{\vec {u}}|)}

cosh ⁡ ( p ) = cosh ⁡ ( a ) cos ⁡ ( | u → | ) + sinh ⁡ ( a ) sgn ⁡ ( u → ) sin ⁡ ( | u → | ) {\displaystyle \cosh(p)=\cosh(a)\cos(|{\vec {u}}|)+\sinh(a)\operatorname {sgn}({\vec {u}})\sin(|{\vec {u}}|)}

tanh ⁡ ( p ) = sinh ⁡ ( p ) cosh ⁡ ( p ) {\displaystyle \tanh(p)={\frac {\sinh(p)}{\cosh(p)}}}

Fungsi hiperbolik invers

arcsinh ⁡ ( p ) = ln ⁡ ( p + p 2 + 1 ) {\displaystyle \operatorname {arcsinh} (p)=\ln(p+{\sqrt {p^{2}+1}})}

arccosh ⁡ ( p ) = ln ⁡ ( p + p 2 − 1 ) {\displaystyle \operatorname {arccosh} (p)=\ln(p+{\sqrt {p^{2}-1}})}

arctanh ⁡ ( p ) = ln ⁡ ( 1 + p ) − ln ⁡ ( 1 − p ) 2 {\displaystyle \operatorname {arctanh} (p)={\frac {\ln(1+p)-\ln(1-p)}{2}}}

Satuan

Kuaternion satuan: p = cos ⁡ ( θ ) + u ^ sin ⁡ ( θ ) {\displaystyle p=\cos(\theta )+{\hat {u}}\sin(\theta )}

Pangkat

p t = ( cos ⁡ ( θ ) + u ^ sin ⁡ ( θ ) ) t = exp ⁡ ( u ^ t θ ) = cos ⁡ ( t θ ) + u ^ sin ⁡ ( t θ ) {\displaystyle {\begin{aligned}&p^{t}=(\cos(\theta )+{\hat {u}}\sin(\theta ))^{t}\\&=\exp({\hat {u}}t\theta )\\&=\cos(t\theta )+{\hat {u}}\sin(t\theta )\end{aligned}}}

Logaritma

log ⁡ ( p ) = log ⁡ ( cos ⁡ ( θ ) + u ^ sin ⁡ ( θ ) ) = log ⁡ ( exp ⁡ ( u ^ θ ) ) = u ^ θ {\displaystyle {\begin{aligned}&\log(p)=\log(\cos(\theta )+{\hat {u}}\sin(\theta ))\\&=\log(\exp({\hat {u}}\theta ))\\&={\hat {u}}\theta \end{aligned}}}

Kalkulus

d d t p t = p t log ⁡ ( p ) {\displaystyle {\frac {d}{dt}}p^{t}=p^{t}\log(p)}

Penerapan

Rotasi vektor grafika 3D

Fungsi rotasi vektor dapat menggunakan operasi kuaternion daripada operasi matriks riil 4x4, dengan rumus:

r = q v q ∗ {\displaystyle {\begin{aligned}&r=qvq*\\\end{aligned}}}

di mana

v = 1 + x A i + y A j + z A k q = cos ⁡ α 2 + sin ⁡ α 2 x v i + sin ⁡ α 2 y v j + sin ⁡ α 2 z v k r = 1 + x A ′ i + y A ′ j + z A ′ k {\displaystyle {\begin{aligned}&v=1+x_{A}i+y_{A}j+z_{A}k\\&q=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}x_{v}i+\sin {\frac {\alpha }{2}}y_{v}j+\sin {\frac {\alpha }{2}}z_{v}k\\&r=1+x_{A}'i+y_{A}'j+z_{A}'k\\\end{aligned}}}

dan A adalah posisi benda yang dirotasikan, v adalah vektor poros rotasi, dan α adalah sudut rotasi berlawanan arah jarum jam.

Referensi

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