Added to Favorites

Related Searches

Definitions

Quaternions, in mathematics, 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. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations, such as in 3D computer graphics, although they have been superseded in many applications by vectors and matrices.

In modern language, quaternions form a 4-dimensional normed division algebra over the real numbers. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by $mathbb\{H\}$ (Unicode ℍ). It can also be given by the Clifford algebra classifications Cℓ_{0,2}(R) = Cℓ^{0}_{3,0}(R). The 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.

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton knew that the complex numbers could be viewed as points in a plane, and he was looking for a way to do the same for points in space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and multiply triples of numbers. But he had been stuck on the problem of division: He did not know how to take the quotient of two points in space.

On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. He could not divide triples, but he could divide quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:

- $i^2\; =\; j^2\; =\; k^2\; =\; ijk\; =\; -1.$

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the rest of his life to studying and teaching them. He founded a school of "quaternionists" and popularized them in several books. The last and longest, Elements of Quaternions, had 800 pages and was published shortly after his death.

After Hamilton's death, his pupil Peter Tait continued promoting quaternions. At this time, quaternions were a mandatory examination topic in Dublin, and in some American universities they were the only advanced mathematics topic taught. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described entirely in terms of quaternions. There was even a professional research association, the Quaternion Society (1899 - 1913), exclusively devoted to the study of quaternions.

From the mid 1880s, quaternions began to be displaced by vectors, which had been developed by Josiah Willard Gibbs and Oliver Heaviside. Vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics.

However, quaternions have had a revival in the late 20th century, primarily due to their utility in describing spatial rotations. Representations of rotations by quaternions are more compact and faster to compute than representations by matrices. For this reason, quaternions are used in computer graphics, control theory, signal processing, attitude control, physics, bioinformatics, and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relation to quadratic forms.

Reading works written before 1900 on the subject of classical Hamiltonian quaternions is difficult for modern readers because they frequently use Hamilton's notation and terminology, which differs from the notation and terminology used today.

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, 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.

- $i^2\; =\; j^2\; =\; k^2\; =\; i\; j\; k\; =\; -1,$

where i, j, and k are imaginary numbers, determine all the possible products of i, j, and k. For example, since

- $-1\; =\; i\; j\; k,$

- $$

k & = i j.end{align} All the other possible products can be determined by similar methods, and this gives the following table:

- $begin\{alignat\}\{2\}$

- $a\_1a\_2\; +\; a\_1b\_2i\; +\; a\_1c\_2j\; +\; a\_1d\_2k\; +\; b\_1a\_2i\; +\; b\_1b\_2i^2\; +\; b\_1c\_2ij\; +\; b\_1d\_2ik\; +\; c\_1a\_1j\; +\; c\_1b\_2ji\; +\; c\_1c\_2j^2\; +\; c\_1d\_2jk\; +\; d\_1a\_1k\; +\; d\_1b\_2ki\; +\; d\_1c\_2kj\; +\; d\_1d\_2k^2.$

- $(a\_1a\_2\; -\; b\_1b\_2\; -\; c\_1c\_2\; -\; d\_1d\_2)\; +\; (a\_1b\_2\; +\; b\_1a\_2\; +\; c\_1d\_2\; -\; d\_1c\_2)i\; +\; (a\_1c\_2\; -\; b\_1d\_2\; +\; c\_1a\_2\; +\; d\_1b\_2)j\; +\; (a\_1d\_2\; +\; b\_1c\_2\; -\; c\_1b\_2\; +\; d\_1a\_2)k.$

- $mathbf\{H\}\; =\; \{(a,\; b,\; c,\; d)\; mid\; a,\; b,\; c,\; d\; in\; mathbf\{R\}\}.$

- $1\; =\; (1,\; 0,\; 0,\; 0),$

- $i\; =\; (0,\; 1,\; 0,\; 0),$

- $j\; =\; (0,\; 0,\; 1,\; 0),$

- $k\; =\; (0,\; 0,\; 0,\; 1),$

- $(a\_1,\; b\_1,\; c\_1,\; d\_1)\; +\; (a\_2,\; b\_2,\; c\_2,\; d\_2)\; =\; (a\_1\; +\; a\_2,\; b\_1\; +\; b\_2,\; c\_1\; +\; c\_2,\; d\_1\; +\; d\_2).$

- $(a\_1,\; b\_1,\; c\_1,\; d\_1)(a\_2,\; b\_2,\; c\_2,\; d\_2)\; =\; (a\_1a\_2\; -\; b\_1b\_2\; -\; c\_1c\_2\; -\; d\_1d\_2,\; a\_1b\_2\; +\; b\_1a\_2\; +\; c\_1d\_2\; -\; d\_1c\_2,\; a\_1c\_2\; -\; b\_1d\_2\; +\; c\_1a\_2\; +\; d\_1b\_2,\; a\_1d\_2\; +\; b\_1c\_2\; -\; c\_1b\_2\; +\; d\_1a\_2).$

Hamilton called pure imaginary quaternions right quaternions and real numbers (considered as quaternions with zero vector part) scalar quaternions.

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of p is (p + p^{*})/2, and the vector part of p is (p − p^{*})/2.

The square root of the product of a quaternion with its conjugate is called its norm and is denoted ||q||. (Hamilton called this quantity the tensor of q, but this conflicts with modern usage. See tensor.) It has the formula

- $lVert\; q\; rVert\; =\; sqrt\{qq^*\}\; =\; sqrt\{a^2\; +\; b^2\; +\; c^2\; +\; d^2\}.$

- $lVertalpha\; qrVert\; =\; |alpha|lVert\; qrVert.$

- $lVert\; pq\; rVert\; =\; lVert\; p\; rVertlVert\; q\; rVert.$

- $d(p,\; q)\; =\; lVert\; p\; -\; q\; rVert.$

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion Uq called the versor of q:

- $mathbf\{U\}q\; =\; q/lVert\; qrVert.$

Using conjugation and the norm makes it possible to define the reciprocal of a quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of q and $q^*/lVert\; q\; rVert^2$ is 1. So the reciprocal of q is defined to be

- $q^\{-1\}\; =\; q^*/lVert\; q\; rVert^2.$

The norm makes the quaternions into a normed algebra, and even into a composition algebra and a unital Banach algebra. Composition algebras are very rare. Hurwitz's theorem states that there are only four composition algebras over the real numbers: R, C, H, and O. Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the commutativity of multiplication. Finite dimensional division algebras over the real numbers are also very rare. The Frobenius theorem states that there are exactly three: R, C, and H.

Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±j, ±k} forms a group under multiplication. This group is called the quaternion group and is denoted Q_{8}. The real group ring of Q_{8} is a ring RQ_{8} which is also an eight-dimensional vector space over R. It has one basis vector for each element of Q_{8}. The quaternions are the quotient ring of RQ_{8} by the ideal generated by the elements 1 − (−1), i − (−i), j − (−j), and k − (−k). Here the first term in each of the differences is one of the basis elements 1, i, j, and k, and the second term is one of basis elements −1, −i, −j, and −k, not the additive inverses of 1, i, j, and k.

For the remainder of this section, i, j, and k will denote both imaginary basis vectors of H and a basis for R^{3}. Notice that replacing i by −i, j by −j, and k by −k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.

Choose two imaginary quaternions p = b_{1}i + c_{1}j + d_{1}k and q = b_{2}i + c_{2}j + d_{2}k. Their dot product is

- $p\; cdot\; q\; =\; b\_1b\_2\; +\; c\_1c\_2\; +\; d\_1d\_2.$

- $p\; cdot\; q\; =\; textstylefrac\{1\}\{2\}(p^*q\; +\; q^*p)\; =\; textstylefrac\{1\}\{2\}(pq^*\; +\; qp^*).$

The cross product of p and q relative to the orientation determined by the ordered basis i, j, and k is

- $(c\_1d\_2\; -\; d\_1c\_2)i\; +\; (d\_1b\_2\; -\; b\_1d\_2)j\; +\; (b\_1c\_2\; -\; c\_1b\_2)k.$

- $p\; times\; q\; =\; textstylefrac\{1\}\{2\}(pq\; -\; q^*p^*).$

In general, let p and q be quaternions (possibly non-imaginary), and write

- $p\; =\; p\_s\; +\; vec\{p\}\_v,$

- $q\; =\; q\_s\; +\; vec\{q\}\_v,$

- $pq\; =\; p\_sq\_s\; -\; vec\{p\}\_vcdotvec\{q\}\_v\; +\; p\_svec\{q\}\_v\; +\; vec\{p\}\_vq\_s\; +\; vec\{p\}\_v\; times\; vec\{q\}\_v.$

Using 2×2 complex matrices, the quaternion a + bi + cj + dk can be represented as

- $begin\{pmatrix\}a+bi\; \&\; c+di\; -c+di\; \&\; a-bi\; end\{pmatrix\}$

This representation has the following properties:

- Complex numbers (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 isomorphism between S
^{3}and SU(2). The latter group is important for describing spin in quantum mechanics; see Pauli matrices.

Using 4×4 real matrices, that same quaternion can be written as

- $begin\{pmatrix\}$

a & b & c & d

-b & a & -d & c

-c & d & a & -b

-d & -c & b & aend{pmatrix}

- $=\; a$

1 & 0 & 0 & 0

0 & 1 & 0 & 0

0 & 0 & 1 & 0

0 & 0 & 0 & 1end{pmatrix} + b begin{pmatrix}

0 & 1 & 0 & 0

-1 & 0 & 0 & 0

0 & 0 & 0 & -1

0 & 0 & 1 & 0end{pmatrix} + c begin{pmatrix}

0 & 0 & 1 & 0

0 & 0 & 0 & 1

-1 & 0 & 0 & 0

0 & -1 & 0 & 0end{pmatrix} + d begin{pmatrix}

0 & 0 & 0 & 1

0 & 0 & -1 & 0

0 & 1 & 0 & 0

-1 & 0 & 0 & 0end{pmatrix}

In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. Complex numbers are block diagonal matrices with two 2×2 blocks.

Let C^{2} be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in C^{2} can be written in terms of the basis elements 1 and j as

- $(a\; +\; bi)1\; +\; (c\; +\; di)j.$

- $(a\; +\; bi,\; c\; +\; di)\; leftrightarrow\; (a,\; b,\; c,\; d).$

- $a^2\; -\; b^2\; -\; c^2\; -\; d^2\; =\; -1,$

- $2ab\; =\; 0,$

- $2ac\; =\; 0,$

- $2ad\; =\; 0.$

- $a\; +\; bsqrt\{-1\}\; mapsto\; a\; +\; bq.$

In the language of abstract algebra, this is an injective ring homomorphism from C to H.

Every non-real quaternion lies in a unique copy of C. Write q as the sum of its scalar part and its vector part:

- $q\; =\; q\_s\; +\; vec\{q\}\_v.$

- $q\; =\; q\_s\; +\; lVertvec\{q\}\_vrVertcdotmathbf\{U\}vec\{q\}\_v.$

- $a\; +\; bsqrt\{-1\}\; mapsto\; a\; +\; bmathbf\{U\}vec\{q\}\_v.$

The relationship of quaternions to each other within the complex subplanes of H can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions p and q commute (p q = q p) only if they lie in the same complex subplane of H, the profile of H as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. Ian R. Porteous's book Clifford Algebras and the Classical Groups (Cambridge, 1995) describes the role of the sphere of unit vectors in proposition 8.13 on page 60.

Let the complex function be written

- $f(x\; +\; i\; y)\; =\; u(x,y)\; +\; i\; v(x,y),!$

where u and v are real-valued functions of two real variables. According to the above profile, any quaternion can be written

- $q\; =\; a\; +\; b\; r,\; r^\{2\}\; =\; -1$.

This is called Fueter's construction.

- Non singular representation (compared with Euler angles for example)
- More compact (and faster) than matrices
- Pairs of unit quaternions represent a rotation in 4D space (see SO(4): Algebra of 4D rotations).

The set of all unit quaternions (versors) 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 image of a subgroup of S³ is a point group, and conversely, the preimage of a point group is a subgroup of S³. 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 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}.

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 with basis 1, i, j, and ij, where i^{2} = a, j^{2} = b and ij = −ji (so ij^{2} = −ab). These algebras are called quaternion algebras and are isomorphic to the algebra of 2×2 matrices over F or form division algebras over F, depending on
the choice of a and b.

The usefulness of quaternions for geometrical computations can be generalised to other dimensions, by identifying the quaternions as the even part Cℓ^{+}_{3,0}(R) of the Clifford algebra Cℓ_{3,0}(R). This is an associative multivector algebra built up from fundamental basis elements σ_{1}, σ_{2}, σ_{3} using the product rules

- $sigma\_1^2\; =\; sigma\_2^2\; =\; sigma\_3^2\; =\; 1,$

- $sigma\_i\; sigma\_j\; =\; -\; sigma\_j\; sigma\_i\; qquad\; (j\; neq\; i)$

- $r^\{prime\}\; =\; -\; w,\; r,\; w$

- $r^\{primeprime\}\; =\; sigma\_2\; sigma\_1\; ,\; r\; ,\; sigma\_1\; sigma\_2$

But this is very similar to the corresponding quaternion formula,

- $r^\{primeprime\}\; =\; -mathbf\{k\},\; r,\; mathbf\{k\}$

In fact, the two are identical, if we make the identification

- $mathbf\{k\}\; =\; sigma\_2\; sigma\_1,\; mathbf\{i\}\; =\; sigma\_3\; sigma\_2,\; mathbf\{j\}\; =\; sigma\_1\; sigma\_3$

- $mathbf\{i\}^2\; =\; mathbf\{j\}^2\; =\; mathbf\{k\}^2\; =\; mathbf\{i\}\; mathbf\{j\}\; mathbf\{k\}\; =\; -1$

In this picture, quaternions correspond not to vectors but to bivectors, quantities with magnitude and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions σ_{1} and σ_{2}, there is only one bivector basis element σ_{1}σ_{2}, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ_{1}σ_{2}, σ_{2}σ_{3}, σ_{3}σ_{1}, so three imaginaries.

With this recognition, the sequence can be continued. So in the Clifford algebra Cℓ_{4,0}(R), there are six bivector basis elements, since with four different basic vector directions, six different pairs and therefore six different linearly independent planes can be defined. Rotations in such spaces using these generalisations of quaternions, called rotors, can be very useful for applications involving homogeneous coordinates. But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as a pseudovector.

Dorst et al identify the following advantages for placing quaternions in this wider setting:

- Rotors are natural and non-mysterious in geometric algebra and easily understood as the encoding of a double reflection.
- In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods (which is required when augmenting linear algebra with quaternions).
- A rotor is universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
- In the conformal model of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitary axis, whereas quaternions are limited to an axis through the origin.
- Rotor-encoded transformations make interpolation particularly straightforward.

For further detail about the geometrical uses of Clifford algebras, see Geometric algebra.

- "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.
- "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 Theory of Relativity in 1924
- "… 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

- Rotation operator (vector space)
- 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
- Hurwitz quaternion order
- Euler Angles
- Clifford algebra
- Exterior algebra
- Slerp

- Hamilton, William Rowan (1853), " Lectures on Quaternions". Royal Irish Academy.
- 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.
- Tait, Peter Guthrie (1886), " 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
- Macfarlane, Alexander (1906), "Vector analysis and quaternions", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
- 1911 encyclopedia: " Quaternion".
- 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). 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
- 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

- Matrix and Quaternion FAQ v1.21 Frequently Asked Questions
- 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)
- 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.
- Johan E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations., arXiv General Mathematics 2005.
- Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007.
- NUI Maynooth Department of Mathematics, Hamilton Walk
- OpenGL:Tutorials:Using Quaternions to represent rotation
- D. Erickson, Derivation of rotation matrix from unitary quaternion representation in old paper:
- Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent",

- 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.
- Quaternion Calculator [Java]
- Quaternion Toolbox for Matlab
- Boost library support for Quaternions in C++
- Mathematics of flight simulation >Turbo-PASCAL software for quaternions, Euler angles and Extended Euler angles

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)

This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 10:54:02 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 10:54:02 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

Copyright © 2014 Dictionary.com, LLC. All rights reserved.