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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 Although they have been superseded in most applications by vectors and matrices, they still find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations, such as in 3D computer graphics.
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 ' (Unicode H). It can also be given by the Clifford algebra classifications Cl0,2(R) = Cl03,0(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.
Definition As a set, the quaternions H are equal to R4, a four-dimensional vector space over the real numbers. The sum of two elements of H is defined to be their sum as elements of R4, and similarly the product of an element of H by a real number is defined to be the same as the product in R4. To define multiplication in H, choose a basis for R4. The elements of this basis are customarily denoted as 1, i, j, and k. Every element of H can be uniquely written as a linear combination of these basis elements, that is, as a1 + bi + cj + dk, where a, b, c, and d are real numbers. The basis element 1 will be the identity element of H, meaning that multiplication by 1 does nothing, and for this reason, elements of H are usually written a + bi + cj + dk, suppressing the basis element 1. Given two elements a1 + b1i + c1j + d1k and a2 + b2i + c2j + d2k, their Hamilton product (a1 + b1i + c1j + d1k)(a2 + b2i + c2j + d2k) is defined as follows:
- Apply the distributive law to write the product as a sum of products of basis elements. This gives the following expression:
- Apply the following rule for the product of basis elements:
The other products of basis elements can be determined from this rule (see below). The result is:
Using the basis 1, i, j, k of H makes it possible to write H as a set of quadruples:
Then the basis elements are:
and the formulas for addition and multiplication are:
A number of the form a + 0i + 0j + 0k, where a is a real number, is called real, and a number of the form 0 + bi + cj + dk, where b, c, and d are real numbers, is called imaginary. If a + bi + cj + dk is any quaternion, then a is called its scalar part and bi + cj + dk is called its vector part. The scalar part of a quaternion is always real, and the vector part is always imaginary. Even though every quaternion is a vector in a four-dimensional vector space, it is common to define a vector to mean an imaginary quaternion. With this convention, a vector is the same as an element of the vector space R3. This identification leads to many applications.
Multiplication of basis elements The equations
where i, j, and k are imaginary numbers, determine all the possible products of i, j, and k. For example, since
right-multiplying both sides by k gives
All the other possible products can be determined by similar methods, and this gives the following table:
Unlike multiplication of real or complex numbers, multiplication of quaternions is not commutative: For example, , while . 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 , for instance, has infinitely many quaternion solutions with , so that these solutions form a two-dimensional sphere centered on zero in the three-dimensional pure imaginary subspace of quaternions. This sphere intersects the complex plane at the two poles and .
Conjugation, the norm, and division Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let q = a +bi +cj + dk be a quaternion. The conjugate of q is the quaternion a − bi − cj − dk. It is denoted by q*, , qt, or . Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq)* = q*p*, not p*q*.
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 product of a quaternion with its conjugate is called its norm and is denoted ||q||. It has the formula
This is always a non-negative real number, and it is same as the Euclidean norm on H considered as the vector space R4. The norm is multiplicative, meaning that
This is a consequence of the formula for the conjugate of a product. This norm makes it possible to define the distance d(p, q) between p and q as the norm of their difference:
This makes H into a metric space. Addition and multiplication are continuous in the metric topology.
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 is 1. So the reciprocal of q is defined to be
This makes it possible to divide two quaternions p and q in two different ways. That is, their quotient can be either pq−1 or q−1p. The notation p/q is ambiguous because it does not specify whether q divides on the left or the right.
Algebraic properties The set H of all quaternions is a vector space over the real numbers with dimension 4. (In comparison, the real numbers have dimension 1, the complex numbers have dimension 2, and the octonions have dimension 8.) They are an associative algebra over the real numbers. Even though they contain copies of the complex numbers, they are not an associative algebra over the complex numbers because multiplying a quaternion by a complex number is not always commutative.
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 forms a group under multiplication. This group is called the quaternion group and is denoted Q8. The quaternions are the real group ring of Q8.
Quaternions and the geometry of R3 Because the vector part of a quaternion is a vector in R3, the geometry of R3 is reflected in the algebraic structure of the quaternions. Many operations on vectors can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. For instance, this is true in electrodynamics, general relativity, and 3D computer graphics.
For the remainder of this section, i, j, and k will denote both imaginary basis vectors of H and a basis for R3. 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 = b1i + c1j + d1k and q = b2i + c2j + d2k. Their dot product is
This is equal to the scalar parts of p*q, qp*, pq*, and q*p. (Note that the vector parts of these four products are different.) It also has the formulas
The cross product of p and q relative to the orientation determined by the ordered basis i, j, and k is
(Recall that the orientation is necessary to determine the sign.) This is equal to the product of p and q as quaternions.
In general, let p and q be quaternions (possibly non-imaginary), and write
where ps and qs are the scalar parts of p and q and and are the vector parts of q. Then we have the formula
This shows that the non-commutativity of quaternion multiplication comes from the multiplication of imaginary quaternions. It also shows that the product of two quaternions is commutative if and only if their vector parts are collinear.
Quaternion operations 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.
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 , 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
Quaternion products
Outer-product
The 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 Hamilton product:
Given quarternions
with
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:
The cross-product can be rewritten using the Grassmann product:
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.
The even-product can be rewritten using the Hamilton product:
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 square of the absolute value.
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.
- Up =
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.
Example
Let
Then
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.
Using 2×2 complex matrices, the quaternion a + b i + c j + d k can be represented as
-
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 S3 and SU(2). The latter group is important in quantum mechanics when dealing with spin; see also Pauli matrices.
Using 4×4 real matrices, that same quaternion can be written 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.
Cayley-Dickson constructionAccording 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 + ib and v = c + id, then
.
Moreover, let
,
so that
,
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:
Notice how the 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:
since
.
Let p be the quaternion with complex components w and z:
.
Then the product qp is
Since the product of complex numbers is commutative, we have
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
.
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 S³. 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 (S²). 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 Hi, where the i subscript refers to the imaginary unit, or .
Identification of imaginary-unit isomorphisms Membership in Hi can be specified using set notation. Two such tests are:
Hi 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:
Isomorphisms to the complex plane
A notable feature of Hi is that every element 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 there exists a subset Cr of the full set of quaternions H that is isomorphic to the complex plane.
This is the reason for using the subscript i to label the set Hi.
Quaternions as isomorphic complex numbers
The union of the complex planes generated by all elements of Hi 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:
The isomorphic complex number equivalent qr of the original quaternion q then becomes:
Euler's formula
Additionally, since the general point on a circle as given by Euler's formula:
The general point on the 3-sphere of all unit-length quaternions is:
where and
Commutative subrings
Finally, the relationship of quaternions to each other within ir 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 ir 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 this derivation in proposition 8.13 on page 60.
Functions of a quaternion variableFunctions 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
.
Then the extension is given by .
This is called Fueter's method.
Three-dimensional and four-dimensional rotation groupsAs 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:
- 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 ).
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 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 .
Generalizations
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 i2 = a, j2 = b and ij = -ji (so ij2 = -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.
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 one real number and three mutually orthogonal imaginary units 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 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 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
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 metricAs a linear algebra over the reals, quaternions constitute a real vector space with a rank three tensor S on it, sometimes called the structure tensor. This once contravariant twice covariant tensor converts a one-form and vectors and to a real number S. For each one-form , S is a twice covariant tensor, which, if symmetric, is an inner product on H. Since any real vector space can also be considered a 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 metric . For quaternions this inner product is indefinite, its signature is independent of the one-form , and the corresponding pseudo-Euclidean metric is Minkowski ]. This metric is automatically extended over the Lie group of nonzero quaternions along its left invariant vector fields resulting in a closed FLRW metric ] – an important solution of the Einstein equations. These results have some implications for the problem of compatibility between quantum mechanics and general relativity within the framework of quantum gravity ].
See also
External articles and resources
Books and publications- Hamilton, William Rowan (1853), "". 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), "". 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: "".
- Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "Foundations of quaternion quantum mechanics". J. Mathematical Phys. 3, pp207-220, 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
- (1995), "A Linear Solution of the Four-Dimensionality Problem", Europhysics Letters, 32 (8) 621-626, DOI:
- 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.
- Kravchenko, Vladislav (2003), "Applied Quaternionic Analysis", Heldermann Verlag ISBN 3-88538-228-8.
- (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3
- (2007), "Natural Geometry of Nonzero Quaternions", International Journal of Theoretical Physics, 46 (2) 251-257, DOI:
Links and monographs- Frequently Asked Questions
- Includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.
- Provides free Fortran and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.
- Includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.
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- (PDF)
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- 3D Raytraced Quaternion Julia Fractals by David J. Grossman
- Great page explaining basic math with links to straight forward rotation conversion formulae.
- John H. Mathews, .
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- Andrew Hanson, .
- , Technical report and Matlab toolbox summarizing all common attitude representations, with detailed equations and discussion on features of various methods.
- Johan E. Mebius, , arXiv General Mathematics 2005.
- Johan E. Mebius, , arXiv General Mathematics 2007.
- NUI Maynooth .
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- 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", [https://webspace.utexas.edu/aam829/1/m/NegativeMath.html]
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.
- [Java]
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