In mathematics, quaternions are a non-commutative extension of complex numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, commutativity is the ability to change the order of something without changing the end result Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Ireland (pronounced /ˈaɾlənd/ Éire) is the third largest island in Europe, and the twentieth-largest island in the world A mathematician is a person whose primary area of study and research is the field of Mathematics. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Year 1843 ( MDCCCXLIII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian Calendar (or a Common Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements At first, quaternions were regarded as pathological because they disobeyed the commutative law ab = ba. In Mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive In Mathematics, commutativity is the ability to change the order of something without changing the end result Although they have been superseded in most applications by vectors and matrices, they still find uses in both theoretical and applied mathematics, in particular for calculations involving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 Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data

In modern language, quaternions form a 4-dimensional normed division algebra over the real numbers. In Mathematics, a normed division algebra A is a Division algebra over the real or complex numbers which is also a Normed vector In Mathematics, the real numbers may be described informally in several different ways The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by (Unicode ℍ). Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical It can also be given by the Clifford algebra classifications Cℓ_0,2(R) = Cℓ⁰_3,0(R). In Mathematics, Clifford algebras are a type of Associative algebra. 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. In Mathematics, more specifically in Abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877 characterizes the finite dimensional In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations

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 H as a union of complex planes 6.1 Informal Introduction 6.2 Detailed Specification 6.2.1 Isomorphisms to the imaginary unit 6.2.2 Identification of imaginary-unit isomorphisms 6.2.3 Isomorphisms to the complex plane 6.2.4 Quaternions as isomorphic complex numbers 6.2.5 Euler's Formula 6.2.6 Commutative subrings 7 Functions of a quaternion variable 8 Three-dimensional and four-dimensional rotation groups 9 Generalizations 10 History 10.1 Recent years 10.2 Quotes 11 Quaternions in fiction 12 Recent developments and research directions 12.1 Quaternions and Minkowski metric 13 See also 14 External articles and resources 14.1 Books and publications 14.2 Links and monographs 14.3 Software

Definition

The quaternions are defined as the ring:

where addition is defined by:

and multiplication is defined by expanding:

using the distributive law and then applying the defining relations:

Every quaternion is a unique and real linear combination of the basis quaternions 1, i, j, and k. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics

Properties

Basis multiplication

The set of equations

where i, j, and k are imaginary numbers, 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. In Mathematics, commutativity is the ability to change the order of something without changing the end result g. ij = k, while ji = − 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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations The equation z² + 1 = 0, for instance, has infinitely many quaternion solutions z = bi + cj + dk with b² + c² + d² = 1, 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 i and − i.

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). In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the dimension of a Vector space V is the cardinality (i 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 bi + cj + dk. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication 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, Q₈. In Group theory, the quaternion group is a non-abelian group of order 8

The quaternions are an example of a division ring, an algebraic structure similar to a field except for commutativity of multiplication. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In particular, multiplication is still associative and every non-zero element has a unique multiplicative inverse. In Mathematics, associativity is a property that a Binary operation can have

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. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible 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. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The use of quaternions can replace tensors in representation. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually 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.

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

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. In Mathematics, Clifford algebras are a type of Associative algebra. 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. Later in this article, is used to denote the conjugate.

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.

Quaternion products

Grassmann product

The most useful quaternion product is the Grassmann product, which is non-commutative. In Mathematics, commutativity is the ability to change the order of something without changing the end result There are times that the Grassmann product can be commutative and times that the Grassmann product can be anticommutative--this is because the first three operators are commutative and the cross product is anticommutative. In mathematics anticommutativity refers to the property of an operation being anticommutative, i The operation is usually denoted as the concatenation of one quaternion with another.

let

The components of Q:

It should be noted at this point that the anticommutative part of the product is the cross product of the vectors . In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which 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.

Properties:

• Non-commutative: for some and , .
• Associative:
• Left and Right Distributive:

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.

Example:

In terms of the Grassmann product:

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

Properties:

• Commutative:
• Associative:
• Distributive:

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 Grassmann product:

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

where p ^* : = a − bi − cj − dk is the conjugate of p. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Note that (q p)^* = p^* q^*, which is not in general equal to q^* p^*. The multiplicative inverse of a non-zero quaternion p can be conveniently computed as p⁻¹ = p^* / |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 R⁴) and the arithmetic operations are continuous. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined 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. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the

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 Grassmann 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 Grassmann 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 reciprocal

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

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⁻¹ q and q p⁻¹. 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:

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. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. 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.

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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix e. , quaternion-matrix homomorphisms). In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector One is to use 2×2 complex matrices, and the other is to use 4×4 real matrices. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the real numbers may be described informally in several different ways

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
• 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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n
• The conjugate of a quaternion corresponds to the conjugate transpose of the matrix. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with
• Restricted to unit quaternions, this representation provides an isomorphism between S³ and SU(2). In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n The latter group is important in quantum mechanics when dealing with spin; see also Pauli matrices. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary 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. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a The fourth power of the absolute value of a quaternion is the determinant of the corresponding matrix. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

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 + 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. A dyadic tensor in Multilinear algebra is a second rank Tensor written in a special notation formed by juxtaposing pairs of vectors i

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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

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. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis

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. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. 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). "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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_i, where the i subscript refers to the imaginary unit, or . Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

Identification of imaginary-unit isomorphisms

Membership in H_i can be specified using set notation. In Set theory and its applications to Logic, Mathematics, and Computer science, set-builder notation (sometimes simply "set notation" Two such tests are:

H_i 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 H_i 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. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis That is, for every element there exists a subset C_r of the full set of quaternions H that is isomorphic to the complex plane. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

This is the reason for using the subscript i to label the set H_i.

Quaternions as isomorphic complex numbers

The union of the complex planes generated by all elements of H_i 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 q_r 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     . This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

Commutative subrings

Finally, the relationship of quaternions to each other within i_r subplanes of H can also be identified and expressed in terms of commutative subrings. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations Specifically, since two quaternions p and q commute (p q = q p) only if they lie in the same i_r 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. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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

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 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 R³ consisting of quaternions with real part equal to zero. Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions 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 represent a rotation in 4D space (see SO(4): Algebra of 4D rotations). The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Physics and Mathematics, a sequence of n numbers can be understood as a location in an n -dimensional space In Mathematics, SO(4 is the Four-dimensional rotation group that is the group of Rotations about a fixed point in four-dimensional Euclidean

The set of all unit quaternions forms a 3-dimensional sphere S³ and a group (a Lie group) under multiplication. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group 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. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

For more details on this topic, see Point groups in three dimensions#Spin_analogs. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group

The image of a subgroup of S³ is a point group, and conversely, the preimage of a point group is a subgroup of S³. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group 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. A regular Icosahedron has 60 rotational (or orientation-preserving symmetries and a total of 120 symmetries including transformations that combine a reflection and a rotation In Mathematics, the binary icosahedral group is an extension of the Icosahedral group I of order 60 by a Cyclic group of order 2

The group S³ is isomorphic to SU(2), the group of complex unitary 2×2 matrices of determinant 1. In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n 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 integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The set A is a ring and a lattice. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of 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}. Constructions A 24-cell is given as the Convex hull of its vertices In Mathematics, the Schläfli symbol is a notation of the form {pqr

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 with basis 1, i, j, and ij, where i² = a, j² = b and ij = −ji (so ij² = −ab). In Mathematics, a quaternion algebra over a field F, is a particular kind of Central simple algebra, A, over F, namely such an algebra In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible

History

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Ireland (pronounced /ˈaɾlənd/ Éire) is the third largest island in Europe, and the twentieth-largest island in the world Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume 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). Broom Bridge, also known as Brougham Bridge is a small Bridge along Broombridge road which crosses the Royal Canal in Cabra Dublin, Ireland 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. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 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. In Mathematics, the real numbers may be described informally in several different ways Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation 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. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which 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. Classical Hamiltonian quaternions were the topic of works written before 1901 on the subject of Quaternions Quaternion Calculus was a complete system of mathematics that

The quaternions formed the theme for one of the first international mathematical associations, the Quaternion Society (1899 - 1913). A Scientific society, the Quaternion Society was an “International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics”

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. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Certain fractals can plot in quaternion coordinates. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" 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. Control theory is an interdisciplinary branch of Engineering and Mathematics, that deals with the behavior of Dynamical systems The desired output Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as The attitude of a body is its orientation as perceived in a certain Frame of reference; providing a vector along which a spacecraft is pointing is a description of its attitude Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Bioinformatics is the application of information technology to the field of molecular biology The root mean square deviation ( RMSD) is the measure of the average distance between the backbones of superimposed Proteins. Orbital mechanics or astrodynamics is the application of Celestial mechanics to the practical problems concerning the motion of Rockets and other Spacecraft 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. The National University of Ireland Maynooth ( NUIM) was founded in 1997 by the Universities Act 1997 as a constituent university of the National Murray Gell-Mann (born September 15, 1929) is an American Physicist who received the 1969 Nobel Prize in physics for his work Steven Weinberg (born May 3, 1933) is an American Physicist, and Nobel laureate in Physics for his contributions with Abdus Salam Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University The Dunsink Observatory is an astronomical Observatory established in approximately 1785 near the city of Dublin, Ireland.

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. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. " — Lord Kelvin, 1892. William Thomson 1st Baron Kelvin (or Lord Kelvin) OM, GCVO, PC, PRS, FRSE, (26 June 1824 &ndash 17 December 1907
• "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. Ludwik Silberstein (1872 – 1948 was a Polish-American physicist that helped make Special relativity and General relativity staples of university coursework Optics Book of Optics The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Decadence can refer to a personal trait or to the state of a society (or segment of it In Biology, a species is one of the basic units of Biological classification and a Taxonomic rank. "Survival of the fittest" is a Phrase which is shorthand for a concept relating to competition for survival or predominance 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. Thomas Ruggles Pynchon Jr (born May 8 1937 is an American writer based in New York City, noted for his dense and complex works of Fiction. Against the Day is a Novel by Thomas Pynchon. The Narrative takes place between the 1893 Chicago World's Fair and the time immediately 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

As 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually This once contravariant twice covariant tensor converts a one-form t and vectors a and b to a real number S(t,a,b). In Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space In Mathematics, the real numbers may be described informally in several different ways For each one-form t, S is a twice covariant tensor, which, if symmetric, is an inner product on H. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. 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 . In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity For quaternions this inner product is indefinite, its signature is independent of the one-form t, and the corresponding pseudo-Euclidean metric is Minkowski [1]. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity This metric is automatically extended over the Lie group of nonzero quaternions along its left invariant vector fields resulting in a closed FLRW metric [2] – an important solution of the Einstein equations. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the These results have some implications for the problem of compatibility between quantum mechanics and general relativity within the framework of quantum gravity [3]. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature

See also

External articles and resources

Books and publications

• Hamilton, William Rowan (1853), "Lectures on Quaternions". In Group theory, the quaternion group is a non-abelian group of order 8 In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. In Mathematics, SO(4 is the Four-dimensional rotation group that is the group of Rotations about a fixed point in four-dimensional Euclidean In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible In Ring theory, dual quaternions are a non-commutative ring constructed in the same way as the quaternions, except using Dual numbers In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic Musean hypernumbers are an algebraic concept envisioned by Charles A In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions The biquaternions are the numbers w + xi + yj + zk \ \! where w x y and z are complex numbers and the elements of {1 i j k} multiply as in the Quaternion group In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association Geometry The tesseract can be constructed in a number of different ways In Mathematics, a Hurwitz quaternion (or Hurwitz integer) is a Quaternion whose components are either all Integers or all The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant In Mathematics, Clifford algebras are a type of Associative algebra. In Computer graphics, Slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of Quaternion Interpolation Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Royal Irish Academy.
• Tait, Peter Guthrie (1873), "An elementary treatise on quaternions". Peter Guthrie Tait ( April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist, best known for the seminal energy 2d ed. , Cambridge, [Eng. ] : The University Press.
• Maxwell, James Clerk (1873), "A Treatise on Electricity and Magnetism". A Treatise on Electricity and Magnetism is an 1873 Textbook on Electromagnetism written by James Clerk Maxwell. Clarendon Press, Oxford.
• Tait, Peter Guthrie (1886), "Quaternion". Peter Guthrie Tait ( April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist, best known for the seminal energy M. A. Sec. R. S. E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. The Encyclopædia Britannica is a general English-language encyclopaedia published by Encyclopædia Britannica Inc XX, pp. 160-164. (bzipped PostScript file)
• Joly, Charles Jasper (1905), "A manual of quaternions". PostScript ( PS) is a dynamically typed concatenative Programming language created by John Warnock and Charles Geschke in 1982 London, Macmillan and co. , limited; New York, The Macmillan company. LCCN 05036137 //r84
• Macfarlane, Alexander (1906), "Vector analysis and quaternions", 4th ed. Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist 1st thousand. New York, J. Wiley & Sons; [etc. , etc. ]. LCCN es 16000048
• 1911 encyclopedia: "Quaternion". The Encyclopædia Britannica Eleventh Edition (1910–1911 is a 29-volume reference work that marked the beginning of the Encyclopædia Britannica
• 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". Patrick du Val ( March 26, 1903 &ndash January 22, 1987) was a British Mathematician, known for his work on Algebraic 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. The Princeton University Press is an independent publisher with close connections to Princeton University. ISBN 0-691-10298-8
• Conway, John Horton, and Smith, Derek A. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups (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

Links and monographs

• 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. In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under 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. The National University of Ireland Maynooth ( NUIM) was founded in 1997 by the Universities Act 1997 as a constituent university of the National
• OpenGL:Tutorials:Using Quaternions to represent rotation
• D. Erickson, Derivation of rotation matrix from unitary quaternion representation in old paper:[1]
• Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent", [2]

Software

• 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

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