Kuaternion

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Dalam matematika, Kuaternion merupakan 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 aritmatika 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[sunting | sunting sumber]

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+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[sunting | sunting sumber]

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:

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[sunting | sunting sumber]

\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[sunting | sunting sumber]

\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[sunting | sunting sumber]

\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[sunting | sunting sumber]

\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[sunting | sunting sumber]

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[sunting | sunting sumber]

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[sunting | sunting sumber]

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[sunting | sunting sumber]

Norma[sunting | sunting sumber]

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[sunting | sunting sumber]

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[sunting | sunting sumber]

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[sunting | sunting sumber]

Dari kuaternion p_2=\frac{p+(p^*)}{2}

Maka: Scalar(p) = a_2

Signum[sunting | sunting sumber]

\sgn(p) = \frac{p}{|p|}

Argumen[sunting | sunting sumber]

\arg(p) = \arccos(\frac{Scalar(p)}{|p|})

Pangkat dan Logaritma[sunting | sunting sumber]

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[sunting | sunting sumber]

Fungsi trigonometris[sunting | sunting sumber]

\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[sunting | sunting sumber]

\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[sunting | sunting sumber]

\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[sunting | sunting sumber]

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

Pangkat[sunting | sunting sumber]

\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[sunting | sunting sumber]

\begin{align}
&\log(p)=\log(\cos(\theta)+\hat{u}\sin(\theta))\\
& =\log(\exp(\hat{u}\theta))\\
& =\hat{u}\theta\end{align}

Kalkulus[sunting | sunting sumber]

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

Penerapan[sunting | sunting sumber]

Rotasi vektor grafika 3D[sunting | sunting sumber]

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.

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