Quaternion

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This page describes the mathematical entity. For other senses of this word, see quaternion (disambiguation).

In mathematics, quaternions are a non-commutative extension of complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. At first, quaternions were regarded as pathological, because they disobeyed the commutative law ab = ba. Although they have been superseded in most applications by vectors, they still find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.

In modern language, quaternions form a 4-dimensional normed division algebra over real numbers. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by . This algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only three finite-dimensional division rings containing the real numbers as a subring.

Contents 1 Definition 2 Properties 2.1 Basis multiplication 2.2 Algebras 3 Quaternion operations 3.1 Definitions used in this section 3.2 Antiautomorphisms 3.3 Common binary operations 3.4 Quaternion products 3.5 Example 4 Matrix representations 5 Cayley-Dickson construction 6 Profile 7 Functions of a quaternion variable 8 Rotation group 9 Generalizations 10 History 10.1 Use controversy 10.2 Recent years 10.3 Quotes 11 See also 12 External articles and resources

[edit] Definition

Quaternions are a generalization of complex numbers, obtained by adding the elements i, j, and k to the real numbers, where i, j, and k satisfy

and where multiplication is assumed to be associative. Every quaternion is a real linear combination of the basis quaternions 1, i, j, and k. Thus every quaternion is uniquely expressible in the form a + b i + c j + d k where a, b, c, and d are real numbers.

[edit] Properties

[edit] Basis multiplication

The set of equations

is the fundamental formula for quaternion multiplicative identities, summarized in the multiplication table of basis quaternions.

For example, since

right-multiplying both sides by k gives

The rest of the table can be verified similarly.

Unlike multiplication of real or complex numbers, multiplication of quaternions is not commutative: e.g. i j = k, while j i = –k. 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 z² + 1 = 0, for instance, has infinitely many quaternion solutions z = b i + c j + d k with b² + c² + d² = 1.

[edit] Algebras

The set H of all quaternions is a vector space over the real numbers with dimension 4 (the complex numbers have dimension 2 by comparison). While H is a four-dimensional vector space, one speaks of the scalar part of the quaternion as being a, while the vector part is the remainder b i + c j + d k. Thus in the context of quaternions a vector is a quaternion with zero for its scalar part.

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, Q₈.

The quaternions are an example of a division ring, an algebraic structure similar to a 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.

[edit] Quaternion operations

Quaternion operations have extended applications in electrodynamics, general relativity, and 3D video game programming. The use of quaternions can replace tensors 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.

[edit] 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.

All representations of quaternions may not 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 , such that the 1 would be in the t location.

The vector form of a quaternion may also be used. This form assumes that .

Example cases will require that the defined quaternions above have example values:

let

let

[edit] 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.

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 , , , , and sometimes other symbols are used.

[edit] Common binary operations

Addition

Addition is the simple map of the addition operator over each element in the quaternions.

Subtraction

Again, subtraction is a map of the subtraction operator over each element. This is equivalent to using addition with the negation operations.

[edit] Quaternion products

Grassman product

The most useful quaternion product is the Grassman product, which is a non-commutative product of two quaternions. There are times that the Grassman product can be commutative and times that the Grassman product can be anticommutative

let

The elements of Q:

It should be noted at this point that the anticommutative part of the product is the cross product of the vectors . The remainder of the product is the commutative portion. If there is no anticommutative part to sum, then the product is entirely commutative. An example of a commutative product with a quaternion is any scalar value multiplied by a quaternion.

Examples:

Quaternion dot-product

The dot-product is also referred to as the Euclidean inner-product, and is equivalent to a 4-vector dot product. The dot product is the sum of the quantity of each element of p multiplied by each element of q. It is a commutative product between quaternions, and returns a scalar quantity.

The dot-product can be rewritten using the Euclidean product:

This product is useful to isolate an element from a quaternion. For instance, the i term can be pulled out from p:

and the absolute value of z is the non-negative real number defined by

Note that (w z)^* = z^* w^*, which is not in general equal to w^* z^*. The multiplicative inverse of a non-zero quaternion z can be conveniently computed as z⁻¹ = z^* / |z|².

By using the distance function d(z, w) = |z − w|, the quaternions form a metric space (isometric to the usual Euclidean metric on R⁴) and the arithmetic operations are continuous. We also have |z w| = |z| |w| for all quaternions z and w. Using the absolute value as norm, the quaternions form a real Banach algebra.

Quaternion Euclidean product

Another multiplication between two quaternions is termed the Euclidean product. Instead of the first quaternion, its conjugate is taken:

Due to the non-commutative nature of the quaternion multiplication, p*q is not equivalent to q*p.

When p = q, the result is the conjugate.

Quaternion dot-product

The dot-product is also referred to as the Euclidean inner-product, and is equivalent to a 4-vector dot product. The dot product is the sum of the quantity of each element of p multiplied by each element of q. It is a commutative product between quaternions, and returns a scalar quantity.

The dot-product can be rewritten using the Euclidean product:

This product is useful to isolate an element from a quaternion. For instance, the i term can be pulled out from p:

Quaternion outer-product

The Euclidean outer-product is not used often; however, it is mentioned as a pair with the inner-product:

The outer-product can be rewritten using the Euclidean product:

Quaternion cross-product

The cross-product of quaternions is also known as the odd-product or the Grassman outer-product. It is equivalent to the vector cross-product, and returns a vector quantity only:

The cross-product can be rewritten using the Grassman product:

Quaternion even-product

The even-product of quaternions is also referred to as the Grassman 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.

The even-product can be rewritten using the Grassman product:

Quaternion reciprocal

The inverse of a quaternion is defined in a way that p⁻¹p = 1. It is formed the same way that the complex inverse is found:

The 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⁻¹ q and q p⁻¹. This means that the notation of q/p cannot be used unless p is a scalar or q is a scalar.

Quaternion scalar

The scalar of a quaternion can be isolated in the same way that was described earlier with the dot-product:

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:

Quaternion modulus

The absolute value of a quaternion is the scalar quantity that determines the length of the quaternion from the origin.

Quaternion sign

The sign of a complex number finds the complex number of the same direction found on the unit circle. The unit quaternion is defined similarly as the quaternion in the same direction on the unit 4-dimensional hypersphere. The quaternion sign function produces the unit quaternion:

Quaternion argument

The argument finds the angle of the 4-vector quaternion from the unit scalar (i.e. 1). This returns a scalar angle.

[edit] Example

Let

Then

[edit] Matrix representations

There are at least two ways of representing quaternions as matrices, in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication (i.e., quaternion-matrix homomorphisms). One is to use 2×2 complex matrices, and the other is to use 4×4 real matrices.

In the first way, the quaternion a + b i + c j + d k is represented as

This representation has several nice properties.

• Complex numbers (c = d = 0) correspond to diagonal matrices.
• The square of the absolute value of a quaternion is 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 the isomorphism between S³ and SU(2). The latter group is important in quantum mechanics when dealing with spin; see also Pauli matrices.

In the second way, the quaternion a + b i + c j + d k is represented as

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.

[edit] 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

is a quaternion.

If u = a + i b and v = c + i d, then

.

Moreover, let

ji = − ij,

so that

q = a + ib + jc + ij( − d),

and also let the product of quaternions be associative.

With these rules, we can now derive the multiplication table for i, j and i j, the imaginary components of a quaternion:

ii = − 1,

ij = (ij),

i(ij) = (ii)j = − j,

ji = − (ij),

jj = − 1,

j(ij) = − j(ji) = − (jj)i = i,

(ij)i = − (ji)i = − j(ii) = j,

(ij)j = i(jj) = − i,

(ij)(ij) = − (ij)(ji) = − i(jj)i = ii = − 1.

Notice how the dyad i j behaves just like the k in the definition.

For any complex number v = c + i d, its product with j has the following property:

jv = v ^* j

since

jv = jc + jid = jc − (ij)d = (c − id)j = v ^* j.

Let p be the quaternion with complex components w and z:

p = w + jz.

Then the product q p is

qp = (u + jv)(w + jz) = uw + ujz + jvw + jvjz

= uw + ju ^* z + jvw + jjv ^* z

= (uw − v ^* z) + j(u ^* z + vw).

Since the product of complex numbers is commutative, we have

(u + jv)(w + jz) = (uw − zv ^* ) + j(u ^* z + wv)

which is precisely how quaternion multiplication is defined by the Cayley-Dickson construction.

Note that if u = a + i b, v = c + i d, and p = a + i b + j c + k d then p′s construction from u and v is rather

p = u + vj = u + jv ^* .

[edit] Profile

The set of quaternions that square to −1 is the set of vectors of absolute value 1, that is

From this set equality one can view H as the union of complex planes sharing the same real line and taking an imaginary unit from the set. Furthermore, the unit sphere in H, the 3-sphere, is formed by the collection of unit circles in these complex planes. As the general point on a circle is

(known as Euler's formula), the general point on the 3-sphere is e^ar where r is a unit vector of H, . Since two quaternions p and q do not commute ( p q ≠ q p) unless they lie in the same complex subplane extending the real line (there is an r ∈ S² and there are a, b, c, d ∈ R such that p = a + b r and q = c + d r), this profile arises when we seek commutative subrings of the quaternion ring. This profile is given a proposition 8.13 on page 60 of Ian R. Porteous's book Clifford Algebras and the Classical Groups, Cambridge, 1995.

[edit] 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

where u and v are real-valued functions of two real variables. According to the above profile, any quaternion can be written q = a + b r, where r ² = −1. Then the extension is given by

.

This is called Fueter's method.

[edit] Rotation group

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 R³ 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 2t, the axis of the rotation being the direction of the imaginary part. The advantages of quaternions are:

1. Non singular representation (compared with Euler angles for example)
2. More compact (and faster) than matrices
3. Pairs of unit quaternions can represent a rotation in 4d space.

The set of all unit quaternions forms a 3-dimensional sphere S³ and a group (a Lie group) under multiplication. S³ is the double cover of the group SO(3,R) of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. The group S³ is isomorphic to SU(2), the group of complex 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 integers or all rational numbers with odd numerator and denominator 2. The set A is a ring and a lattice. There are 24 unit quaternions in this ring, and they are the vertices of a 24-cell regular polytope with Schläfli symbol {3,4,3}.

[edit] Generalizations

Main article: quaternion algebra

If F is any 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 by using two generators i and j and the relations i² = a, j² = b and ij = −ji. These algebras are called quaternion algebras either isomorphic to the algebra of 2×2 matrices over F, or they are division algebras over F.

[edit] History

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 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

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 and dot products of vector algebra. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of real numbers, 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.

[edit] Use controversy

Even by this time there was controversy about the use of quaternions. Some of Hamilton's supporters, like Cargill Gilston Knott, vociferously opposed the growing fields of vector algebra and vector calculus (developed by Oliver Heaviside and Willard Gibbs among others), maintaining that quaternions provided a superior notation. While this is debatable in three dimensions, quaternions cannot be directly applied in higher dimensions (though extensions like octonions and Clifford algebras may be more applicable). Vector notation had nearly universally replaced quaternions in science and engineering by the mid-20th century.

Some early formulations of Maxwell's equations used a quaternion-based notation (although Maxwell's original formulation simply used 20 equations in 20 variables), but it proved unpopular compared to the vector-based notation of Heaviside. (All of these formulations were mathematically equivalent.)

[edit] 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 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, 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, avoiding such phenomena as gimbal lock, which can occur when Euler angles are used. Using quaternions also reduces overhead from that when rotation matrices are used, because one carries only four components, not nine, the multiplication algorithms to combine successive rotations are faster, and it is far easier to renormalize the result afterwards.

Since 1989, 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.

[edit] 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 Clerk Maxwell. -- Lord Kelvin, 1892.
• . . .quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the 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

[edit] See also

• Quaternion group
• Coquaternion (split-quaternion)
• 3-sphere
• Associative algebra
• Complex number
• Division algebra
• Geometric algebra
• Hypercomplex number
• Hypernumber
• Octonion
• Quaternions and spatial rotation
• Biquaternion
• Hyperbolic quaternion
• Tesseract
• Hurwitz quaternion
• Euler Angles

[edit] External articles and resources

Links

• Geometric Tools documentation Includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.
• Patrick-Gilles Maillot Provides free Fortran and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.
• Geometric Tools source code Includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.
• Doing Physics with Quaternions
• Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)
• Quaternion Calculator [Java]
• The Physical Heritage of Sir W. R. Hamilton (PDF)
• Hamilton’s Research on Quaternions
• Quaternion Julia Fractals 3D Raytraced Quaternion Julia Fractals by David J. Grossman
• Quaternion Math and Conversions Great page explaining basic math with links to straight forward rotation conversion formulae.
• John H. Mathews, Bibliography for Quaternions.
• Quaternion powers on GameDev.net
• Andrew Hanson, Visualizing Quaternions home page.
• 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.

Encyclopedias

• "Quaternion". 1911 encyclopedia.
• Tait, Peter Guthrie, "Quaternion". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160-164. (bzipped PostScript file)

Books and publications

• Tait, Peter Guthrie, "An elementary treatise on quaternions". 2d ed., Cambridge, [Eng.] : The University Press, c. 1873.
• Macfarlane, Alexander, "Vector analysis and quaternions", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.] 1906. LCCN es 16000048
• Joly, Charles Jasper, "A manual of quaternions". London, Macmillan and co., limited; New York, The Macmillan company, 1905. LCCN 05036137 //r84
• Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser, "Foundations of quaternion quantum mechanics". J. Mathematical Phys. 3 1962 207--220, MathSciNet.
• Du Val, Patrick, "Homographies, quaternions, and rotations". Oxford, Clarendon Press, 1964 (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.
• Adler, Stephen L., "Quaternionic quantum mechanics and quantum fields". New York : Oxford University Press, 1995. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 ISBN 0-19-506643-X (alk. paper)
• Altmann, Simon L., "Rotations, quaternions, and double groups". Oxford [Oxfordshire] : Clarendon Press ; New York : Oxford University Press, c1986. LCCN 85013615 ISBN 0-19-855372-2
• Ward, J. P. (1997). Quaternions and Cayley Numbers: Algebra and Applications, Kluwer Academic Publishers. ISBN 0-7923-4513-4.
• Gürlebeck, Klaus and Wolfgang Sprössig, "Quaternionic and Clifford calculus for physicists and engineers". Chichester ; New York : Wiley, c1997 (Mathematical methods in practice; v. 1) LCCN 98169958 ISBN 0-471-96200-7 (acid-free paper)
• 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.
• Hanson, Andrew J., "Visualizing Quaternions". Elsevier : Morgan Kaufmann ; San Fransisco : (2006). ISBN 0-12-088400-3

Software 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://www.geocities.com/mak2000sw/eulerquatpro.html