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 **R**4 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+bi+cj+dk bisa dituliskan sebagai p=a+\vec{u} di mana \vec{u} adalah vektor 3 bilangan imaginer, \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,\ 
Karena
:-1 = i j k,\ 
jika dua sisi dikalikan dengan *k*, maka
:
\begin{align}
-k & = i j k k = i j (k^2) = i j (-1), \\
 k & = i j. 
\end{align}


Persamaan-persamaan yang lainnya juga bisa didapatkan dengan tahap aljabar:
:\begin{alignat}{2}
ij & = k, & \qquad ji & = -k, \\
jk & = i, & kj & = -i, \\
ki & = j, & ik & = -j, 
\end{alignat}

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

### Pertambahan
\begin{align}
& 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{align}
### Pengurangan
\begin{align}
& 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{align}
### Perkalian
\begin{align}
& p_1 \times p_2\\
& =(a_1a_2-b_1b_2-c_1c_2-d_1d_2)+(b_1a_2+a_1b_2-d_1c_2+c_1d_2)i+(c_1a_2+d_1b_2+a_1c_2-b_1d_2)j+(d_1a_2-c_1b_2+b_1c_2+a_1d_2)k
\end{align}

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

\begin{align}
& p_1 \times p_2\\
& =(a_1+\vec{u_1}) \times (a_2+\vec{u_2})\\
& =(a_1a_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{align}
### Pembagian
\begin{align}
& p_1/p_2 \\
& =\frac{a_1a_2+b_1b_2+c_1c_2+d_1d_2}{m}+\frac{b_1a_2-a_1b_2-d_1c_2+c_1d_2}{m}i+\frac{c_1a_2+d_1b_2-a_1c_2-b_1d_2}{m}j+\frac{d_1a_2-c_1b_2+b_1c_2-a_1d_2}{m}k
\end{align}
di mana
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:

\begin{alignat}{2}
p* = a - b i - c j - d k
\end{alignat}

Persamaan-persamaan konjugasi kuaternion adalah:

\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(), bila N(p) = 1, maka:

\begin{matrix}
p &=& \cos(\theta)+\vec{u}\sin(\theta)\\
p &=& \cos(\theta)+\hat{u}\sin(\theta)
\end{matrix}\,

di mana

\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:

:\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:

:\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)


\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

Dan juga,

\begin{matrix}
N(p^*) &=& N(p)\\
N(pq) &=& N(p)N(q)
\end{matrix}
### Kebalikan
p^{-1} = \frac{p^*}{N(p)}

Dan juga,

\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

Dan juga,

\begin{matrix}
W(p) &=& (p+p^*)/2
\end{matrix}
### Skalar
Dari kuaternion p_2=\frac{p+(p^*)}{2}

Maka: Scalar(p) = a_2
### Signum
\sgn(p) = \frac{p}
### Argumen
\arg(p) = \arccos(\frac{Scalar(p)})
### Pangkat dan Logaritma
Fungsi ekponensial: \exp(p) = \exp(a) (\cos(|\vec{u}|) + \sgn(\vec{u})\sin(|\vec{u}|))

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

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

## Trigonometri
### Fungsi trigonometris
:\sin(p)=\sin(a)\cosh(|\vec{u}|)+\cos(a)\sgn(\vec{u})\sinh(|\vec{u}|)
:\cos(p)=\cos(a)\cosh(|\vec{u}|)-\sin(a)\sgn(\vec{u})\sinh(|\vec{u}|)
:\tan(p)=\frac{\sin(p)}{\cos(p)}
### Fungsi hiperbolik
:\sinh(p)=\sinh(a)\cos(|\vec{u}|)+\cosh(a)\sgn(\vec{u})\sin(|\vec{u}|)
:\cosh(p)=\cosh(a)\cos(|\vec{u}|)+\sinh(a)\sgn(\vec{u})\sin(|\vec{u}|)
:\tanh(p)=\frac{\sinh(p)}{\cosh(p)}
### Fungsi hiperbolik invers
:\operatorname{arcsinh}(p)=\ln(p+\sqrt{p^2+1})
:\operatorname{arccosh}(p)=\ln(p+\sqrt{p^2-1})
:\operatorname{arctanh}(p)=\frac{\ln(1+p)-\ln(1-p)}{2}

## Satuan
Kuaternion satuan: p = \cos(\theta)+\hat{u}\sin(\theta)
### Pangkat
\begin{align}
& p^t =(\cos(\theta)+\hat{u}\sin(\theta))^t\\
& =\exp(\hat{u}t\theta)\\
& =\cos(t\theta)+\hat{u}\sin(t\theta)\end{align}
### Logaritma
\begin{align}
&\log(p)=\log(\cos(\theta)+\hat{u}\sin(\theta))\\
& =\log(\exp(\hat{u}\theta))\\
& =\hat{u}\theta\end{align}
### Kalkulus
\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:

:
\begin{align}
& r = q v q*\\
\end{align}


di mana

:
\begin{align}
& 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{align}


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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