- Source: 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:
= 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
Pranala luar
Hamilton, William Rowan. On quaternions, or on a new system of imaginaries in algebra. Philosophical Magazine. Vol. 25, n 3. p. 489–495. 1844.
Hamilton, William Rowan (1853), "Lectures on Quaternions". Royal Irish Academy.
Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co..
Tait, Peter Guthrie (1873), "An elementary treatise on quaternions". 2d ed., Cambridge, [Eng.]: The University Press.
Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
Maxwell, James Clerk (1873), "A Treatise on Electricity and Magnetism". Clarendon Press, Oxford.
Tait, Peter Guthrie (1886), "Quaternion di www.ugcs.caltech.edu (diarsipkan tanggal 20140808040037)". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript file)
Joly, Charles Jasper (1905), "A manual of quaternions". London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84
Macfarlane, Alexander (1906), "Vector analysis and quaternions", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
1911 encyclopedia: "Quaternions".
Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "Foundations of quaternion quantum mechanics". J. Mathematical Phys. 3, pp. 207–220, MathSciNet.
Du Val, Patrick (1964), "Homographies, quaternions, and rotations". Oxford, Clarendon Press (Oxford mathematical monographs). LCCN 64056979 //r81
Crowe, Michael J. (1967), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
Altmann, Simon L. (1986), "Rotations, quaternions, and double groups". Oxford [Oxfordshire]: Clarendon Press ; New York: Oxford University Press. LCCN 85013615 ISBN 0-19-855372-2
Altmann, Simon L. (1989), "Hamilton, Rodrigues, and the Quaternion Scandal". Mathematics Magazine. Vol. 62, No. 5. p. 291–308, Dec. 1989.
Adler, Stephen L. (1995), "Quaternionic quantum mechanics and quantum fields". New York: Oxford University Press. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 ISBN 0-19-506643-X
Trifonov, Vladimir (1995), "A Linear Solution of the Four-Dimensionality Problem", Europhysics Letters, 32 (8) 621–626, DOI: 10.1209/0295-5075/32/8/001
Ward, J. P. (1997), "Quaternions and Cayley Numbers: Algebra and Applications", Kluwer Academic Publishers. ISBN 0-7923-4513-4
Kantor, I. L. and Solodnikov, A. S. (1989), "Hypercomplex numbers, an elementary introduction to algebras", Springer-Verlag, New York, ISBN 0-387-96980-2
Gürlebeck, Klaus and Sprössig, Wolfgang (1997), "Quaternionic and Clifford calculus for physicists and engineers". Chichester ; New York: Wiley (Mathematical methods in practice; v. 1). LCCN 98169958 ISBN 0-471-96200-7
Kuipers, Jack (2002), "Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality" (reprint edition), Princeton University Press. ISBN 0-691-10298-8
Conway, John Horton, and Smith, Derek A. (2003), "On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry", A. K. Peters, Ltd. ISBN 1-56881-134-9 (review).
Kravchenko, Vladislav (2003), "Applied Quaternionic Analysis", Heldermann Verlag ISBN 3-88538-228-8.
Hanson, Andrew J. (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3
Trifonov, Vladimir (2007), "Natural Geometry of Nonzero Quaternions", International Journal of Theoretical Physics, 46 (2) 251–257, DOI: 10.1007/s10773-006-9234-9
Ernst Binz & Sonja Pods (2008) Geometry of Heisenberg Groups American Mathematical Society, Chapter 1: "The Skew Field of Quaternions" (23 pages) ISBN 978-0-8218-4495-3.
Vince, John A. (2008), Geometric Algebra for Computer Graphics, Springer, ISBN 978-1-84628-996-5.
For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by D.J. Evans, (1977), "On the Representation of Orientation Space", Mol. Phys., vol 34, p 317.
Kata Kunci Pencarian:
- Kuaternion
- Grup kuaternion
- Oktonion
- Bikuaternion
- Daftar grup simetri bola hingga
- Garis bilangan riil
- Pangkat dua
- Grup Dedekind
- Matematika
- −1 (angka)
