Quaternion
51440
226161608
2008-07-17T02:40:49Z
Pmanderson
64669
/* Quaternion products */ clarify, I hope
{{expert-subject|Mathematics}}
[[Image:Quaternion2.png|thumb|400px|right|Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = -k, ij = -ji]]
{{dablink|This page describes the mathematical entity. For other senses of this word, see [[quaternion (disambiguation)]].}}
In [[mathematics]], '''quaternions''' are a [[commutative|non-commutative]] extension of [[complex number]]s. They were first described by the [[Ireland|Irish]] [[mathematician]] [[William Rowan Hamilton|Sir William Rowan Hamilton]] in [[1843]] and applied to [[mechanics]] in three-dimensional space. At first, quaternions were regarded as [[pathological (mathematics)|pathological]] because they disobeyed the [[commutativity|commutative law]] ''ab'' = ''ba''. Although they have been superseded in most applications by [[vector (spatial)|vector]]s and [[matrix (mathematics)|matrices]], they still find uses in both theoretical and applied mathematics, in particular for calculations involving [[quaternions and spatial rotation|three-dimensional rotations]], such as in 3D [[computer graphics]].
In modern language, quaternions form a 4-dimensional [[normed division algebra]] over the [[real number]]s. The algebra of quaternions is often denoted by '''H''' (for Hamilton), or in [[blackboard bold]] by '''<math>\mathbb{H}</math>''' (Unicode ℍ). It can also be given by the [[Clifford algebra]] classifications ''C''ℓ<sub>0,2</sub>('''R''') = ''C''ℓ<sup>0</sup><sub>3,0</sub>('''R'''). The algebra '''H''' holds a special place in analysis since, according to the [[Frobenius theorem (real division algebras)|Frobenius theorem]], it is one of only three finite-dimensional [[division ring]]s containing the [[real numbers]] as a [[subring]].
== Definition ==
The quaternions are defined as the [[Ring (mathematics)|ring]]:
:<math>\mathbb{H}=\{a+bi+cj+dk | a,b,c,d\in\mathbb{R}\}</math>
where addition is defined by:
:<math>(a_1+b_1i+c_1j+d_1k)+(a_2+b_2i+c_2j+d_2k)\,</math>
:::<math>=(a_1+a_2)+(b_1+b_2)i+(c_1+c_2)j+(d_1+d_2)k\,</math>
and multiplication is defined by expanding:
:<math>(a_1+b_1i+c_1j+d_1k)(a_2+b_2i+c_2j+d_2k)\,</math>
using the distributive law and then applying the defining relations:
:<math> i^2 = j^2 = k^2 = ijk = -1,\,</math>
Every quaternion is a unique and real [[linear combination]] of the basis quaternions 1, ''i'', ''j'', and ''k''.
==Properties==
===Basis multiplication===
The set of equations
:<math> i^2 = j^2 = k^2 = i j k = -1 , \,\!</math>
where ''i'', ''j'', and ''k'' are imaginary numbers, is the fundamental formula for quaternion multiplicative identities, summarized in the multiplication table of basis quaternions.
:<math>\begin{matrix}
ij & = & k, & & & & ji & = & -k, \\
jk & = & i, & & & & kj & = & -i, \\
ki & = & j, & & & & ik & = & -j.
\end{matrix}</math>
For example, since
:<math> - 1 = i j k, \,\!</math>
right-multiplying both sides by ''k'' gives
:<math>
\begin{matrix}
-k & = & i j k k, \\
-k & = & i j (-1), \\
k & = & i j.
\end{matrix}
\,\!</math>
<!--
Alternatively, left-multiplying both sides by ''i'',
:<math> i i j k = -i, </math>
:<math> (-1) j k = -i, </math>
:<math> j k = i. </math>
This last equation can consequently be left-multiplied on both sides by ''j'',
:<math> j j k = j i, </math>
:<math> -k = j i, </math>
and continuing in this fashion the rest of the multiplication table is forthwith derived.-->
The rest of the table can be verified similarly.
Unlike multiplication of real or complex numbers, multiplication of quaternions is not [[commutative]]: e.g. <math>ij = k</math>, while <math>ji = -k</math>. The non-commutativity of multiplication has some unexpected consequences, among them that [[polynomial]] equations over the quaternions can have more distinct solutions than the degree of the polynomial. The equation <math>z^2 + 1 = 0</math>, for instance, has infinitely many quaternion solutions <math>z = bi + cj + dk</math> with <math>b^2 + c^2 + d^2 = 1</math>, so that these solutions form a unitary sphere centered on zero in the three-dimensional pure imaginary subspace of quaternions, this imaginary sphere intersecting the complex plane only at the two poles <math>i</math> and <math>-i</math>.
<!--
:<math>\begin{matrix}
ij & = & k \\
ji & = & -k \\
jk & = & i \\
kj & = & -i \\
ki & = & j \\
ik & = & -j \\
\end{matrix}</math> -->
<!-- the \, forces rendering and enforces uniformity -->
===Algebras===
The set '''H''' of all quaternions is a [[vector space]] over the [[real number]]s with [[Hamel dimension|dimension]] 4 (the complex numbers have dimension 2 by comparison). While '''H''' is a four-dimensional vector space, one speaks of the [[scalar (mathematics)|scalar]] part of the quaternion as being ''a'', while the vector part is the remainder <math>bi + cj + dk</math>. Thus, in the context of quaternions, a quaternion with zero for its scalar part is a ''vector''.
Addition of quaternions is accomplished by adding corresponding coefficients, as with the complex numbers. By linearity, multiplication of quaternions is completely determined by the [[multiplication table]] above for the basis quaternions. Under this multiplication, the basis quaternions, with their negatives, form the [[quaternion group]] of order 8, <math>Q_8</math>.
The quaternions are an example of a [[division ring]], an algebraic structure similar to a [[field (mathematics)|field]] except for commutativity of multiplication. In particular, multiplication is still [[associative]] and every non-zero element has a unique multiplicative inverse.
Quaternions form a 4-dimensional [[associative algebra]] over the reals (in fact a [[division algebra]]) and contain the complex numbers, but they do not form an associative algebra over the complex numbers. The quaternions, along with the complex and real numbers, are the only finite-dimensional associative division algebras over the field of real numbers.
== Quaternion operations ==
Quaternion operations have extended applications in [[electrodynamics]], [[general relativity]], and 3D graphics programming. The use of quaternions can replace [[tensor]]s in representation. It is sometimes easier to use quaternions with complex elements, leading to a form that is not a division algebra. However, the same operations can be performed using a combination of conjugate operations. Only quaternions with real elements will be discussed here.
===Definitions used in this section===
This section, used to describe common algebraic operations on quaternions, will define three quaternions. These quaternions will be used to represent a primary operand, a secondary operand, and a resultant. These are respectively: '''A''', '''B''', and '''Q'''. Not all operations are complex enough to require their display using all three quaternions.
:<math>\begin{matrix}\mathbf A & \equiv A_t & + & A_x{\mathbf i} & + & A_y{\mathbf j} & + & A_z{\mathbf k}\end{matrix}</math>
:<math>\begin{matrix}\mathbf B & \equiv B_t & + & B_x{\mathbf i} & + & B_y{\mathbf j} & + & B_z{\mathbf k}\end{matrix}</math>
:<math>\begin{matrix}\mathbf Q & \equiv Q_t & + & Q_x{\mathbf i} & + & Q_y{\mathbf j} & + & Q_z{\mathbf k}\end{matrix}</math>
Not all representations of quaternions may define the elements in the same way. These axes are chosen to, hopefully, aid in the description. The ''t'' element represents the scalar quantity. In this situation, the number 1 can be represented by the quaternion <math>1 + 0{\mathbf i} + 0{\mathbf j} + 0{\mathbf k}</math>, such that the 1 would be in the ''t'' location.
The vector form of a quaternion may also be used. This form assumes that <math>\vec{A} \equiv A_x\mathbf i + A_y\mathbf j + A_z\mathbf k</math>.
:<math>{\mathbf A} \equiv A_t + \vec A </math>
:<math>{\mathbf B} \equiv B_t + \vec B </math>
:<math>{\mathbf Q} \equiv Q_t + \vec Q </math>
Example cases will require that the defined quaternions above have example values:
:let <math>\begin{matrix}\mathbf A & = & 3 & + & \mathbf i\end{matrix}</math>
:let <math>\begin{matrix}\mathbf B & = & 5 \mathbf i & + & \mathbf j & - & 2 \mathbf k\end{matrix}</math>
===Antiautomorphisms===
; Negation (Additive inverse)
The negation operation corresponds to the negation operation of the Clifford Algebras, in that the negation operation is mapped to all elements.
:<math>-\mathbf A \equiv -A_t - A_x \mathbf i - A_y\mathbf j - A_z\mathbf k</math>
:<math>-\mathbf A \equiv -A_t - \vec A</math>
; Conjugation (Spatial inverse)
The quaternion conjugate corresponds to the reversal operation of the [[Clifford algebras]]. The term Spatial inverse refers to the negation of each of the elements that would have a spatial representation, which are the elements in the '''i''' basis, the '''j''' basis, and the '''k''' basis.
''NOTE:'' The operator symbol for the conjugate is not standardized. This can sometimes be seen as <math>\overline{Q}\,\!</math>, <math>\tilde{Q}\,\!</math>, <math>Q^*\,\!</math>, <math>Q^t\,\!</math>, and sometimes other symbols are used. Later in this article, <math>\overline{Q}\,\!</math> is used to denote the conjugate.
:<math>\overline{\mathbf A} \equiv A_t - A_x\mathbf i - A_y\mathbf j - A_z\mathbf k</math>
:<math>\overline{\mathbf A} \equiv A_t - \vec{A}</math>
===Common binary operations===
; Addition
Addition is the simple map of the addition operator over each element in the quaternions.
:<math>\mathbf A + \mathbf B \equiv (A_t + B_t) + (A_x + B_x)\mathbf i + (A_y + B_y)\mathbf j + (A_z + B_z)\mathbf k</math>
:<math>\mathbf A + \mathbf B \equiv (A_t + B_t) + \vec A + \vec B</math>
; Subtraction
Again, subtraction is a map of the subtraction operator over each element. This is equivalent to using addition with the negation operations.
:<math>\mathbf A - \mathbf B \equiv (A_t - B_t) + (A_x - B_x)\mathbf i + (A_y - B_y)\mathbf j + (A_z - B_z)\mathbf k</math>
:<math>\mathbf A - \mathbf B \equiv (A_t - B_t) + \vec A - \vec B</math>
===Quaternion products===
{{Expert-verify|date=May 2008}}
;Hamilton product
Quaternions multiply by the Hamilton product, which is [[non-commutative]] in general. For some specific values, the Hamilton product is commutative; for others, [[anticommutative]]--this is because the first three operators are commutative and the cross product is anticommutative.
:let <math>\mathbf Q = \mathbf{AB} = A_t B_t - \vec{A}\cdot\vec{B} + A_t\vec{B} + B_t\vec{A} + \vec{A}\times\vec{B}</math>
The components of '''Q''':
:<math>\begin{matrix}Q_t & = & A_t B_t & - & A_x B_x & - & A_y B_y & - & A_z B_z\end{matrix}</math>
:<math>\begin{matrix}Q_x & = & A_t B_x & + & A_x B_t & + & A_y B_z & - & A_z B_y\end{matrix}</math>
:<math>\begin{matrix}Q_y & = & A_t B_y & - & A_x B_z & + & A_y B_t & + & A_z B_x\end{matrix}</math>
:<math>\begin{matrix}Q_z & = & A_t B_z & + & A_x B_y & - & A_y B_x & + & A_z B_t\end{matrix}</math>
It should be noted at this point that the anticommutative part of the product is the [[cross product]] of the vectors <math>\left(\vec{A}\times\vec{B}\right)</math>. The remainder of the product is the commutative portion. If there is no anticommutative part to sum, then the product is entirely commutative. Quaternion products commute when their vector parts are co-linear (for then they lie in a complex number subring of the quaternion space).
Properties:
*Non-commutative: for some <math>\mathbf A</math> and <math>\mathbf B</math>, <math>\mathbf{AB} \neq \mathbf{BA}</math>.
*Associative: <math>\mathbf {A(BC)} = \mathbf{(AB)C} = \mathbf{ABC}</math>
*Left and Right Distributive: <math>\mathbf{A(B + C)} = \mathbf{AB + AC},\quad \mathbf{(A+B)C}=\mathbf{AC+BC}</math>
; Inner product
The inner product (also called the quaternion dot-product) corresponds to the sum of the products of the individual elements. It is an entirely commutative product that returns a scalar quantity.
:<math>\mathbf A \cdot \mathbf B \equiv \mathbf B \cdot \mathbf A = A_t B_t + A_x B_x + A_y B_y + A_z B_z\,\!</math>
Example:
:<math>\mathbf A \cdot \mathbf B = (3\cdot 0) + (1\cdot 5) + (0\cdot 1) + (0\cdot -2) = 5\,\!</math>
In terms of the Hamilton product:
:<math>\mathbf A \cdot \mathbf B = \frac{\mathbf{\overline A B + \overline B A}}{2}</math>
This product is useful to isolate an element from a quaternion. For instance, the ''i'' term can be pulled out from ''p'':
:<math>\mathbf A \cdot i = A_x \,\!</math>
Properties:
*Commutative: <math>\mathbf {A\cdot B} = \mathbf{B\cdot A}\,\!</math>
*Associative: <math>\mathbf {A\cdot (B\cdot Q)} = \mathbf{(A \cdot B)\cdot Q} = \mathbf{A\cdot B\cdot Q}\,\!</math>
*Distributive: <math>\mathbf {Q\cdot(A + B)} = \mathbf{Q\cdot A} + \mathbf{Q\cdot B}</math>
; Outer-product
The outer-product is not used often; however, it is mentioned as a pair with the inner-product:
:<math>\operatorname{Outer}(\mathbf A,\mathbf B) = A_t \vec{B} - B_t\vec{A} - \vec{A}\times\vec{B}\,\!</math>
The outer-product can be rewritten using the Hamilton product:
:<math>\operatorname{Outer}(\mathbf A, \mathbf B) = \frac{\mathbf{\overline A B - \overline B A}}{2} \,\!</math>
;
and the ''[[absolute value]]'' of ''p'' is the non-negative real number defined by
:<math>|p| = \sqrt{p p^*} = \sqrt{a^2 + b^2 + c^2 + d^2}. \,\!</math>
where <math>p^* := a - bi - cj - dk</math> is the conjugate of p.
Note that (''q p'')<sup>*</sup> = ''p''<sup>*</sup> ''q''<sup>*</sup>, which is not in general equal to ''q''<sup>*</sup> ''p''<sup>*</sup>. The multiplicative inverse of a non-zero quaternion ''p'' can be conveniently computed as ''p''<sup>−1</sup> = ''p''<sup>*</sup> / |''p''|².
By using the [[distance function]] ''d''(''p'', ''q'') = |''p'' − ''q''|, the quaternions form a [[metric space]] (isometric to the usual Euclidean metric on '''R'''<sup>4</sup>) and the arithmetic operations are continuous. We also have |''p q''| = |''p''| |''q''| for all quaternions ''p'' and ''q''. Using the absolute value as norm, the quaternions form a real [[Banach algebra]].
Given quarternions
:<math>p = a+\vec{u},\quad q = t+\vec{v},</math>
with
:<math>\vec{u} = bi + cj + dk,\quad\ \vec{v} = xi + yj + zk,</math> some other products are defined as follows.
; Quaternion cross-product
The cross-product of quaternions is also known as the '''odd-product''' or the '''Grassmann outer-product'''. It is equivalent to the vector cross-product, and returns a vector quantity only:
:<math>p \times q = \vec{u}\times\vec{v} \,\!</math>
:<math>p \times q = (cz - dy)i + (dx - bz)j + (by - cx)k \,\!</math>
The cross-product can be rewritten using the Grassmann product:
:<math>p \times q = \frac{pq - qp}{2} \,\!</math>
; Quaternion even-product
The even-product of quaternions is also referred to as the '''Hamilton inner-product'''. It is also not widely used, but it mentioned due to the similarity between it and the odd-product. It is the purely symmetric product; therefore, it is completely commutative.
:<math>\operatorname{Even}(p,q) = at - \vec{u}\cdot\vec{v} + a\vec{v} + t\vec{u} \,\!</math>
:<math>\operatorname{Even}(p,q) = (at - bx - cy - dz) + (ax + bt)i + (ay + ct)j + (az + dt)k \,\!</math>
The even-product can be rewritten using the Hamilton product:
:<math>\operatorname{Even}(p,q) = \frac{pq + qp}{2} \,\!</math>
; Quaternion Euclidean product
Another multiplication between two quaternions is termed the '''Euclidean product'''. Instead of the first quaternion, its conjugate is taken:
:<math>p^*q = at + \vec{u}\cdot\vec{v} + a\vec{v} - t\vec{u} - \vec{u}\times\vec{v} \,\!</math>
Due to the non-commutative nature of the quaternion multiplication, ''p*q'' is not equivalent to ''q*p''.
:<math>q^*p = at + \vec{u}\cdot\vec{v} - a\vec{v} + t\vec{u} + \vec{u}\times\vec{v} \,\!</math>
When ''p'' = ''q'', the result is the square of the absolute value.
; Quaternion reciprocal
The inverse of a quaternion is defined in a way that ''p''<sup>−1</sup>''p'' = ''p''<span/>''p''<sup>−1</sup> = 1. It is formed the same way that the complex inverse is found:
:<math>p^{-1} = \frac{p^*}{p \cdot p^*} \,\!</math>
The inner product of a quaternion and its conjugate is a scalar. The division of a quaternion by a scalar is equivalent to multiplication by the scalar inverse, such that each element of the quaternion is divided by the divisor.
; Quaternion division
The non-commutativity of quaternions allows for two divisions of numbers ''p''<sup>−1</sup> ''q'' and ''q p''<sup>−1</sup>. This means that the notation of ''q''/''p'' is ambiguous unless ''p'' is a scalar, ''q'' is a scalar, or an explicit convention is defined, which is not normally done.
; Quaternion scalar
The scalar of a quaternion can be isolated in the same way that was described earlier with the dot-product:
:<math>1\cdot p = \frac{p + p^*}{2} = a \,\!</math>
; Quaternion vector
The vector of a quaternion can be isolated using the outer-product in the same way the inner product is used to isolate the scalar:
:<math>\operatorname{Outer}(1, p) = \frac{p - p^*}{2} = \vec{u} = bi + cj + dk \,\!</math>
; Quaternion modulus
The absolute value of a quaternion is the scalar quantity that determines the length of the quaternion from the origin.
:<math>|p| = \sqrt{p \cdot p} = \sqrt{p^*p} = \sqrt{a^2 + b^2 + c^2 + d^2} \,\!</math>
; Quaternion [[versor]]
The versor of a quaternion is what is left when the quaternion is divided by its modulus. Evidently this quaternion is on the [[unit sphere]], in this case the [[3-sphere]].
:'''U'''p = <math> \frac{p}{|p|} \,\!</math>
Identification of the versor represents a polar coordinate viewpoint. The composition of versors by concatenation of vector-arcs has been used to study quaternion multiplication since they were introduced by Hamiltion.
; Quaternion argument
The argument finds the angle of the 4-vector quaternion from the unit scalar (i.e. 1). This returns a scalar angle.
:<math>\arg(p) = \arccos\left(\frac{\operatorname{Scalar}(p)}{|p|}\right) \,\!</math>
=== Example ===
Let
:<math>\begin{matrix}
x & = & 3 + i \\
y & = & 5i + j - 2k
\end{matrix}</math>
Then
:<math>\begin{matrix}
x + y & = & 3 + 6i + j - 2k \\
\\
xy & = & (3 + i)(5i + j - 2k) \\
& = & 15i + 3j - 6k + 5i^2 + ij - 2ik \\
& = & 15i + 3j - 6k - 5 + k + 2j \\
& = & -5 + 15i + 5j - 5k \\
\\
yx & = & (5i + j - 2k)(3 + i) \\
& = & 15i + 5i^2 + 3j + ji - 6k - 2ki \\
& = & 15i - 5 + 3j - k - 6k - 2j \\
& = & -5 + 15i + j - 7k
\end{matrix}</math>
== Matrix representations ==
There are at least two ways of representing quaternions as [[matrix (mathematics)|matrices]], in such a way that quaternion addition and multiplication correspond to matrix addition and [[matrix multiplication]] (i.e., quaternion-matrix [[homomorphism]]s).
One is to use 2×2 [[complex number|complex]] matrices, and the other is to use 4×4 [[real number|real]] matrices.
Using '''2×2 complex matrices''', the quaternion ''a'' + ''b i'' + ''c j'' + ''d k'' can be represented as
: <math>\left(\begin{array}{rr} a+bi & c+di \\ -c+di & a-bi \end{array}\right)</math>
This representation has the following properties:
* [[Complex number]]s (''c'' = ''d'' = 0) correspond to diagonal matrices.
* The norm of a quaternion (the square root of a product with its conjugate, as with complex numbers) is the square root of the [[determinant]] of the corresponding matrix.
* The conjugate of a quaternion corresponds to the [[conjugate transpose]] of the matrix.
* Restricted to unit quaternions, this representation provides an [[group isomorphism|isomorphism]] between [[3-sphere|''S''<sup>3</sup>]] and [[SU(2)]]. The latter group is important in [[quantum mechanics]] when dealing with [[spin (physics)|spin]]; see also [[Pauli matrices]].
Using '''4×4 real matrices''', that same quaternion can be written as
: <math>\left(\begin{array}{rrrr}
a & b & c & d \\
-b & a & -d & c \\
-c & d & a & -b \\
-d & -c & b & a
\end{array}\right)</math>
: <math>= a
\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)
+ b
\left(\begin{array}{rrrr}
0 & \;\; 1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & \;\; 1 & 0
\end{array}\right)
+ c
\left(\begin{array}{rrrr}
0 & 0 & \;\; 1 & 0 \\
0 & 0 & 0 & \;\; 1 \\
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{array}\right)
+ d
\left(\begin{array}{rrrr}
0 & 0 & 0 & \;\; 1 \\
0 & 0 & -1 & 0 \\
0 & \;\; 1 & 0 & 0 \\
-1 & 0 & 0 & 0
\end{array}\right)
</math>
In this representation, the conjugate of a quaternion corresponds to the [[transpose]] of the matrix. The fourth power of the absolute value of a quaternion is the [[determinant]] of the corresponding matrix.
== Cayley-Dickson construction==
According to the [[Cayley-Dickson construction]], a quaternion is an ordered pair of complex numbers. Letting ''j'' be a new root of −1, different from both ''i'' and −''i'', and given ''u'' and ''v'' are a pair of complex numbers, then
:<math> q = u + j v \,</math>
is a quaternion.
If ''u'' = ''a'' + ''ib'' and ''v'' = ''c'' + ''id'', then
:<math> q = a + i b + j c + j i d \,</math>.
Moreover, let
:<math> j i = - i j \,</math>,
so that
:<math> q = a + i b + j c + i j (-d) \,</math>,
and also let the product of quaternions be associative.
With these rules, we can now derive the multiplication table for ''i'', ''j'' and ''ij'', the imaginary components of a quaternion:
:<math> i i = -1, \,</math>
:<math> i j = (i j), \,</math>
:<math> i (i j) = (i i) j = -j, \,</math>
:<math> j i = - (i j), \,</math>
:<math> j j = -1, \,</math>
:<math> j (i j) = - j (j i) = - (j j) i = i, \,</math>
:<math> (i j) i = - (j i) i = -j (i i) = j, \,</math>
:<math> (i j) j = i (j j) = -i, \,</math>
:<math> (i j) (i j) = -(i j) (j i) = -i (j j) i = i i = -1. \,</math>
Notice how the [[dyadic tensor|dyad]] ''ij'' behaves just like the ''k'' in the definition.
For any complex number ''v'' = ''c'' + ''id'', its product with ''j'' has the following property:
:<math> j v = v^* j \,</math>
since
:<math> j v = j c + j i d = j c - (i j) d = (c - i d) j = v^* j \,</math>.
Let ''p'' be the quaternion with complex components ''w'' and ''z'':
:<math> p = w + j z \,</math>.
Then the product ''qp'' is
:<math> q p = (u + j v) (w + j z) = u w + u j z + j v w + j v j z \,</math>
::<math> = u w + j u^* z + j v w + j j v^* z \,</math>
::<math> = (u w - v^* z) + j (u^* z + v w). \,</math>
Since the product of complex numbers is commutative, we have
:<math> (u + j v) (w + j z) = (u w - z v^*) + j (u^* z + w v) \,</math>
which is precisely how quaternion multiplication is defined by the Cayley-Dickson construction.
Note that if ''u = a + ib'', ''v = c + id'', and ''p = a + ib + jc + kd'' then ''p''′s construction from ''u'' and ''v'' is rather
:<math> p = u + v j = u + j v^* \,</math>.
== H as a union of complex planes ==
=== Informal Introduction ===
There exists an intriguing way of understanding '''H''' that links its structure closely to the surface of an ordinary sphere of radius 1. In mathematics such a sphere is called a unit [[2-sphere]] to emphasize that only its two-dimensional surface is being considered.
The first step is to translate the XYZ coordinates of the unit 2-sphere into the ijk coordinate system of quaternions, keeping the scalar (first) value of the quaternions set to zero. For example, the XYZ point ''<1,0,0>'' becomes the quaternion ''0 + 1i + 0j + 0k''. Since quaternion absolute lengths are calculated in the same way as XYZ radii, the resulting unit 2-sphere quaternions also all have absolute lengths (radii) of 1.
A less intuitive property of unit 2-sphere quaternions is that their squares all equal -1. This is true by definition for the three main axes of ''i'', ''j'', and ''k'', but it can also be verified easily by trial for any arbitrary unit 2-sphere quaternion.
Since a length of 1 and a square of -1 are the defining properties of ''i'', these unit 2-sphere quaternions look suspiciously like mathematical analogs to ''i''. Furthermore, since each such quaternion has an "unused" scalar value associated with it, a fascinating conjecture becomes possible:
: ''For any given ijk-only point on the quaternion unit 2-sphere, does the set of all quaternions that can be expressed as the sum of a real number and a multiple of that ijk-only point behave like a [[complex plane]]?''
Somewhat unsurprisingly, the answer is yes.
That is, '''H''' can be partitioned in such a way that it looks like an infinite set of complex planes. Each such plane has its own unique version of ''i'', although they all share the same real (scalar) axis. Furthermore, each unique ''i'' value corresponds to and is fully defined by a point on the surface of an ordinary unit-radius sphere, thus providing a strong connection between the geometry of ordinary spheres and the far less intuitive four-dimensional properties of '''H'''. Once a point on the unit 2-sphere has been selected, there is no mathematical difference in the behavior of the resulting subset of '''H''' and the more traditional concept of a single abstract complex plane.
Thus quaternions do not just extend the concept of ''i'' just to the two new axes ''j'' and ''k''. They generalize ''i'' to an infinite set of points that happen to be the same ones found on the surface of an ordinary unit-radius sphere!
A more precise mathematical profile of how '''H''' can be interpreted as a union of complex planes is provided below.
=== Detailed Specification ===
==== Isomorphisms to the imaginary unit ====
The set of quaternions of absolute length (radius) 1 has the form of a [[3-sphere]] or [[hypersphere]], which is also called '''S'''³. Within this hypersphere there exists a subset of quaternions with the additional property that their squares are equal to −1. This subset has the geometric form of an ordinary sphere, or [[2-sphere]] ('''S'''²). It can be understood as a three-dimensional "slice" of the larger hypersphere in much the same way that a circle is a two-dimensional "slice" of an ordinary sphere. For reasons explained below, this sphere-like subset of '''H''' is referred to here as '''H'''<sub>i</sub>, where the ''i'' subscript refers to the [[imaginary unit]], or <math>\sqrt{-1}</math>.
==== Identification of imaginary-unit isomorphisms ====
Membership in '''H'''<sub>i</sub> can be specified using [[set notation]]. Two such tests are:
:<math>H_i = \left\{ q : q ^2 = -1 \right\} = \left\{ q : q^* = -q\ \mbox{and}\ q q^* = 1 \right\}.</math>
'''H'''<sub>i</sub> quaternions can also be identified by looking at whether it is true both that their first (scalar) component ''a'' is zero, and that their remaining ''bi'', ''cj'', and ''dk'' components have a length of 1 if interpreted as a three-dimensional vector:
:<math>H_i = \left\{ q : a = 0 \ \mbox{and}\ \sqrt{ b^2 + c^2 + d^2 } = 1 \right\}. \,\!</math>
==== Isomorphisms to the complex plane ====
A notable feature of '''H'''<sub>i</sub> is that every element <math>i_r \in H_i</math> can be used to define a subset of '''H''' (the full set of all quaternions) that behaves identically to the [[complex plane]]. That is, for every element <math>i_r \in H_i</math> there exists a subset '''C'''<sub>r</sub> of the full set of quaternions '''H''' that is [[isomorphic]] to the complex plane.
:<math>C_r = \left\{ c_r : c_r = a_r + b_r i_r \ \mbox{and}\ a_r,b_r \in R \right\} \,\!</math>
This is the reason for using the subscript ''i'' to label the set '''H'''<sub>i</sub>.
==== Quaternions as isomorphic complex numbers ====
The union of the complex planes generated by all elements of '''H'''<sub>i</sub> is the set of all quaternions '''H'''. This means any quaternion can be expressed as an isomorphic complex number whose imaginary unit is associated with a point on the ordinary unit sphere.
That is, given a quaternion ''q = a + bi + cj + dk'', the corresponding isomorphic imaginary unit can be calculated by normalizing the ijk portion (only) of the quaternion:
:<math>b_r = \|r\| = \sqrt{b^2 + c^2 + d^2}</math>
:<math>i_r = \frac{bi + cj + dk}{b_r} = \frac{bi + cj + dk}{\|r\|}</math>
The isomorphic complex number equivalent ''q<sub>r</sub>'' of the original quaternion ''q'' then becomes:
:<math>q_r = a + b_r i_r = a + \|r\| i_r</math>
==== Euler's Formula ====
Additionally, since the general point on a circle as given by [[Euler's formula]]:
:<math>e^{\theta i}= \cos{(\theta)} + i \sin{(\theta)} \,\!</math>
The general point on the 3-sphere of all unit-length quaternions is:
:<math>e^{\theta i_r} = \cos{(\theta)} + i_r \sin{(\theta)} \,\!</math>
Where <math>i_r \in H_i\ ,</math> and <math> \sin(\theta) =\frac{ \| r \|}{\| q \|}</math> .
==== Commutative subrings ====
Finally, the relationship of quaternions to each other within ''i''<sub>r</sub> subplanes of '''H''' can also be identified and expressed in terms of [[commutative]] [[subring]]s. Specifically, since two quaternions ''p'' and ''q'' commute (''p q'' = ''q p'') only if they lie in the same ''i''<sub>r</sub> complex subplane of '''H''', the profile of '''H''' as a union of complex planes arises when one seeks to find all [[commutative]] [[subring]]s of the quaternion [[ring (mathematics)|ring]]. Ian R. Porteous's book ''Clifford Algebras and the Classical Groups'' (Cambridge, 1995) describes this derivation in proposition 8.13 on page 60.
==Functions of a quaternion variable==
Functions of a complex variable can be extended to functions of a quaternion variable as follows:
Let the complex function be written
:<math>f(z) = u(x,y) + i\ v(x,y)\,\!</math>
where ''u'' and ''v'' are real-valued functions of two real variables. According to the above profile, any quaternion can be written
:<math>q = a + b\ r,\ \ \ r^{2} = -1 \ </math>.
Then the extension is given by <math>f(q) = u(a,b) + r\ v(a,b) \,\!</math>.
This is called Fueter's method.
==Three-dimensional and four-dimensional rotation groups==
As is explained in more detail in [[quaternions and spatial rotation]], the multiplicative group of non-zero quaternions acts by conjugation on the copy of '''R'''³ consisting of quaternions with real part equal to zero. The conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(''t'') is a rotation by an angle 2''t'', the axis of the rotation being the direction of the imaginary part. The advantages of quaternions are:
# Non singular representation (compared with [[Euler angles]] for example)
# More compact (and faster) than [[matrix (mathematics)|matrices]]
# Pairs of unit quaternions represent a rotation in [[Fourth dimension|4D]] space (see [[SO(4)#Algebra of 4D rotations|''SO''(4): Algebra of 4D rotations]]).
The set of all unit quaternions forms a [[3-sphere|3-dimensional sphere]] ''S''³ and a [[group (mathematics)|group]] (a [[Lie group]]) under multiplication. ''S''³ is the [[Covering space#Elementary properties|double cover]] of the group ''SO''(3,'''R''') of real orthogonal 3×3 [[orthogonal matrix|matrices]] of [[determinant]] 1 since ''two'' unit quaternions correspond to every rotation under the above correspondence.
{{details|Point groups in three dimensions#Spin_analogs}}
The image of a subgroup of ''S''³ is a [[Point groups in three dimensions|point group]], and conversely, the preimage of a point group is a subgroup of ''S''³. The preimage of a finite point group is called by the same name, with the prefix '''binary'''. For instance, the preimage of the [[icosahedral group]] is the [[binary icosahedral group]].
The group ''S''³ is isomorphic to ''SU''(2), the group of complex [[unitary matrix|unitary]] 2×2 matrices of [[determinant]] 1. Let ''A'' be the set of quaternions of the form ''a'' + ''bi'' + ''cj'' + ''dk'' where ''a'', ''b'', ''c'', and ''d'' are either all [[integer]]s or all [[rational number]]s with odd numerator and denominator 2. The set ''A'' is a [[ring (mathematics)|ring]] and a [[Lattice (group)|lattice]]. There are 24 unit quaternions in this ring, and they are the vertices of a [[24-cell|24-cell regular polytope]] with [[Schläfli symbol]] {3,4,3}.
== Generalizations ==
{{main|quaternion algebra}}
If ''F'' is any [[field (mathematics)|field]] with characteristic different from 2, and ''a'' and ''b'' are elements of ''F'', one may define a four-dimensional unitary [[associative algebra]] over ''F'' with basis 1, ''i'', ''j'', and ''ij'', where ''i''<sup>2</sup> = ''a'', ''j''<sup>2</sup> = ''b'' and ''ij'' = −''ji'' (so ''ij''<sup>2</sup> = −''ab''). These algebras are called ''quaternion algebras'' and are isomorphic to the algebra of 2×2 [[matrix (mathematics)|matrices]] over ''F'' or form [[division algebra]]s over ''F'', depending on
the choice of ''a'' and ''b''.
== History ==
Quaternions were introduced by [[Ireland|Irish]] mathematician [[William Rowan Hamilton|Sir William Rowan Hamilton]] in 1843. Hamilton was looking for ways of extending [[complex number]]s (which can be viewed as [[Point (geometry)|point]]s on a [[plane (mathematics)|plane]]) to higher spatial dimensions. He could not do so for 3 dimensions, but 4 dimensions produce quaternions. According to the story Hamilton told, on October 16, he was out walking along the Royal Canal in [[Dublin]] with his wife when the solution in the form of the equation
:<math>i^2 = j^2 = k^2 = ijk = -1\,</math>
[[Image:Quaternion Plague on Broom Bridge.jpg|right|thumb|Quaternion plaque on Brougham (Broom) Bridge, [[Dublin]], which says: <br><small>Here as he walked by<br> on the 16th of October 1843<br> Sir William Rowan Hamilton<br> in a flash of genius discovered<br> the fundamental formula for quaternion multiplication<br> i² = j² = k² = i j k = −1<br> & cut it on a stone of this bridge.</small>]]
suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called [[Broom Bridge]]).
This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices had yet to be developed.
Not only this, but Hamilton had in a sense invented the [[cross product|cross]] and [[dot product]]s of vector algebra. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of one [[real number]] and three mutually orthogonal [[imaginary unit]]s with real coefficients, and described the first coordinate as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the [[dot product]] of the vector parts, while the vector part of the product is the [[cross product]]. But the significance of these was still to be discovered. Hamilton proceeded to popularize quaternions with several books, the last of which, ''Elements of Quaternions'', had 800 pages and was published shortly after his death.
Reading works written before 1900 on the subject of [[Classical Hamiltonian quaternions]] is difficult for modern readers because the notation used by early writers on the subject of quaternions, mostly based on the notation and vocabulary of Hamilton is different than what is used today.
The quaternions formed the theme for one of the first international mathematical associations, the [[Quaternion Society (1899 - 1913)]].
=== Recent years ===
Quaternions are often used in [[computer graphics]] (and associated geometric analysis) to represent rotations (see [[quaternions and spatial rotation]]) and orientations of objects in [[Dimension|three-dimensional]] space. Certain [[fractals]] can plot in quaternion coordinates. They are smaller than other representations such as matrices, and operations on them such as composition can be computed more efficiently. Quaternions also see use in [[control theory]], [[signal processing]], [[attitude control]], [[physics]], [[bioinformatics]] (see: [[Root mean square deviation (bioinformatics)]]), and [[orbital mechanics]], mainly for representing rotations/orientations in three dimensions. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations. There is also less overhead in using quaternions compared to using rotation matrices, because a quaternion has only four components instead of nine, so the multiplication algorithms to combine successive rotations are faster, and the result is much easier to renormalize afterwards.
Since 1989, the Department of Mathematics of the [[National University of Ireland, Maynooth]] has organized a pilgrimage, where scientists (including physicists [[Murray Gell-Mann]] in 2002 and [[Steven Weinberg]] in 2005 and mathematician [[Andrew Wiles]] in 2003) take a walk from [[Dunsink Observatory]] to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.
=== Quotes ===
* "I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc." — William Rowan Hamilton (''ed''. Quoted in a letter from Tait to Cayley.)
* "Time is said to have only one dimension, and space to have three dimensions. […] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Rowan Hamilton (Quoted in R.P. Graves, "Life of Sir William Rowan Hamilton")
* "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including [[James Clerk Maxwell|Clerk Maxwell]]." — [[Lord Kelvin]], 1892.
* "Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in every-day life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols." — [[Ludwik Silberstein]], preparing the second edition of his [[List of important publications in physics#The Theory of Relativity|Theory of Relativity]] in 1924
* "… quaternions appear to exude an air of [[nineteenth century]] [[decadence|decay]], as a rather unsuccessful [[species]] in the [[survival of the fittest|struggle-for-life]] of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist." — [[Simon L. Altmann]], 1986
== Quaternions in fiction ==
* "...the thing about a Quaternion 'is' is that we're obliged to encounter it in more than one guise. As a vector quotient. As a way of plotting complex numbers along three axes instead of two. As a list of instructions for turning one vector into another..... And considered subjectively, as an act of becoming longer or shorter, while at the same time turning, among axes whose unit vector is not the familiar and comforting 'one' but the altogether disquieting ''square root of minus one''. If ''you'' were a vector, mademoiselle, you would begin in the 'real' world, change your length, enter an 'imaginary' reference system, rotate up to three different ways, and return to 'reality' a new person. Or vector....." — [[Thomas Pynchon]], in [[Against the Day]], 2006, p. 534, a [[fictional conversation]], overheard by his [[fictional characters]] [[Kit Traverse]] and [[Umeki Tsurigane]], at a fictional gathering of "Quaternionnaires from around the globe", in Ostend, Belgium in or around the year 1905.
== Recent developments and research directions ==
===Quaternions and Minkowski metric===
{{importance-sect}}
As a linear algebra over the reals, quaternions constitute a real [[vector space]] with a [[Tensor#Tensor rank|rank]] three [[tensor]] '''S''' on it, sometimes called the structure tensor. This once [[contravariant]] twice [[covariant]] tensor converts a [[one-form]] <math>t</math> and vectors <math>a</math> and <math>b</math> to a [[real number]] '''S'''<math>(t, a, b)</math>. For each one-form <math>t</math>, '''S''' is a twice covariant tensor, which, if symmetric, is an [[inner product space| inner product]] on '''H'''. Since any real vector space can also be considered a [[affine transformation| linear manifold]], such an inner product is naturally extended to a [[tensor field]], and in case of its nondegeneracy, becomes a (pseudo- or proper-)[[Euclidean space| Euclidean]] metric . For quaternions this inner product is indefinite, its [[metric signature| signature]] is independent of the one-form <math>t</math>, and the corresponding pseudo-Euclidean metric is [[Minkowski space| Minkowski]] [http://dx.doi.org/10.1209/0295-5075/32/8/001 [1]]. This metric is automatically extended over the [[Lie group]] of nonzero quaternions along its [[left invariant vector field]]s resulting in a closed [[FLRW metric]] [http://dx.doi.org/10.1007/s10773-006-9234-9 [2]] – an important solution of the [[Einstein field equations| Einstein equations]]. These results have some implications for the problem of compatibility between [[quantum mechanics]] and [[general relativity]] within the framework of [[quantum gravity]] [http://dx.doi.org/10.1007/s10773-007-9474-3 [3]].
==See also==
<div style="-moz-column-count:3; column-count:3;">
* [[Quaternion group]]
* [[Split-quaternion]] (Coquaternion)
* [[3-sphere]]
* [[SO(4)]]
* [[Associative algebra]]
* [[Complex number]]
* [[Division algebra]]
* [[Dual quaternion]]
* [[Geometric algebra]]
* [[Hypercomplex number]]
* [[Musean hypernumber]]
* [[Octonion]]
* [[Quaternions and spatial rotation]]
* [[Biquaternion]]
* [[Hyperbolic quaternion]]
* [[Tesseract]]
* [[Hurwitz quaternion]]
* [[Euler Angles]]
* [[Clifford algebra]]
* [[Exterior algebra]]
* [[Slerp]]
</div>
==External articles and resources==
===Books and publications===
*[[William Rowan Hamilton|Hamilton, William Rowan]] (1853), "''[http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 Lectures on Quaternions]''". Royal Irish Academy.
*[[Peter Guthrie Tait|Tait, Peter Guthrie]] (1873), "''An elementary treatise on quaternions''". 2d ed., Cambridge, [Eng.] : The University Press.
*Maxwell, James Clerk (1873), "''[[A Treatise on Electricity and Magnetism]]''". Clarendon Press, Oxford.
* [[Peter Guthrie Tait|Tait, Peter Guthrie]] (1886), "''[http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 Quaternion]''". 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
*[[Alexander Macfarlane (mathematician)|Macfarlane, Alexander]] (1906), "''Vector analysis and quaternions''", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
*[[1911 encyclopedia]]: "''[http://31.1911encyclopedia.org/Q/QU/QUATERNIONS.htm Quaternion]''".
*Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "''Foundations of quaternion quantum mechanics''". J. Mathematical Phys. 3, pp207-220, MathSciNet.
*[[Patrick du Val|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). The competition between quaternions and other systems is a major theme.
*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
*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
*[http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (1995), "''A Linear Solution of the Four-Dimensionality Problem''", Europhysics Letters, '''32 (8)''' 621-626, DOI: [http://dx.doi.org/10.1209/0295-5075/32/8/001 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
*[[John Horton Conway|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 ([http://nugae.wordpress.com/2007/04/25/on-quaternions-and-octonions/ review]).
*[[Vladislav Kravchenko|Kravchenko, Vladislav]] (2003), "''Applied Quaternionic Analysis''", Heldermann Verlag ISBN 3-88538-228-8.
*[http://www.cs.indiana.edu/~hanson/quatvis/ Hanson, Andrew J.] (2006), "''Visualizing Quaternions''", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3
*[http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (2007), "''Natural Geometry of Nonzero Quaternions''", International Journal of Theoretical Physics, '''46 (2)''' 251-257, DOI: [http://dx.doi.org/10.1007/s10773-006-9234-9 10.1007/s10773-006-9234-9]
===Links and monographs===
* [http://www.j3d.org/matrix_faq/matrfaq_latest.html Matrix and Quaternion FAQ v1.21] Frequently Asked Questions
* [http://www.geometrictools.com/Documentation/Documentation.html Geometric Tools documentation] Includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.
* [http://www.chez.com/pmaillot Patrick-Gilles Maillot] Provides free Fortran and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.
* [http://www.geometrictools.com/LibFoundation/Mathematics/Mathematics.html Geometric Tools source code] Includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.
* [http://world.std.com/~sweetser/quaternions/qindex/qindex.html Doing Physics with Quaternions]
* [http://www.fho-emden.de/~hoffmann/quater12012002.pdf Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)]
* [http://arxiv.org/pdf/math-ph/0201058 The Physical Heritage of Sir W. R. Hamilton] (PDF)
* [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html Hamilton’s Research on Quaternions]
* [http://www.unpronounceable.com/julia/ Quaternion Julia Fractals] 3D Raytraced Quaternion [[Julia set|Julia Fractals]] by David J. Grossman
* [http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm Quaternion Math and Conversions] Great page explaining basic math with links to straight forward rotation conversion formulae.
* John H. Mathews, [http://math.fullerton.edu/mathews/c2003/QuaternionBib/Links/QuaternionBib_lnk_3.html Bibliography for Quaternions].
* [http://www.gamedev.net/reference/articles/article1095.asp Quaternion powers on GameDev.net]
* Andrew Hanson, [http://books.elsevier.com/companions/0120884003/vq/index.html Visualizing Quaternions home page].
* [http://ai.stanford.edu/~diebel/attitude.html Representing Attitude with Euler Angles and Quaternions: A Reference], Technical report and Matlab toolbox summarizing all common attitude representations, with detailed equations and discussion on features of various methods.
*Johan E. Mebius, [http://arxiv.org/abs/math/0501249 A matrix-based proof of the quaternion representation theorem for four-dimensional rotations.], ''arXiv General Mathematics'' 2005.
*Johan E. Mebius, [http://arxiv.org/abs/math/0701759 Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations.], ''arXiv General Mathematics'' 2007.
*[[NUI Maynooth]] [http://www.maths.nuim.ie/links/hamilton.shtml Department of Mathematics, Hamilton Walk].
* [http://gpwiki.org/index.php/OpenGL:Tutorials:Using_Quaternions_to_represent_rotation OpenGL:Tutorials:Using Quaternions to represent rotation]
* D. Erickson, Derivation of rotation matrix from unitary quaternion representation in old paper:[http://aiss.suffield.drdc-rddc.gc.ca/uploads/quaternion.pdf]
* Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent", [https://webspace.utexas.edu/aam829/1/m/NegativeMath.html]
===Software===
* [http://www.geocities.com/mak2000sw/eulerquatpro.html Euler Quaternion Pro] A free GUI based utility that converts Euler angles to Quaternions around X,Y and Z (roll, pitch and yaw) axis and performs conjugate, addition, subtraction, multiplication, great circle interpolation operations on converted Quaternions.
* [http://theworld.com/~sweetser/java/qcalc/qcalc.html Quaternion Calculator] [Java]
* [http://qtfm.sourceforge.net/ Quaternion Toolbox for Matlab]
* [http://www.boost.org/doc/libs/1_35_0/libs/math/doc/quaternion/html/index.html Boost library support for Quaternions in C++]
* [http://www.xs4all.nl/~jemebius/Eea.htm Mathematics of flight simulation >Turbo-PASCAL software for quaternions, Euler angles and Extended Euler angles]
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[[Category:Rotational symmetry]]
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