quaternion - определение. Что такое quaternion
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Что (кто) такое quaternion - определение

NONCOMMUTATIVE EXTENSION OF THE REAL NUMBERS
Quaternian; Quarternions; Quarternion; Quaternion physics; Quaternians; Hamiltonian numbers; Quarterion; Hamiltonian quaternions; ℍ; Quaternion conjugate; Quaternion norm; Hamilton quaternions; Quaternions; Methods of quaternions; Unit quaternions; Quaternionic; Norm of a quaternion; Hamilton product; A+ib+jc+kd; Vector quaternion; Scalar quaternion; Square roots of quaternions; Matrix representation of quaternions
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Quaternion         
·noun The number four.
II. Quaternion ·noun A word of four syllables; a quadrisyllable.
III. Quaternion ·vt To divide into quaternions, files, or companies.
IV. Quaternion ·noun A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.
V. Quaternion ·noun The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
quaternion         
[kw?'t?:n??n]
¦ noun
1. Mathematics a complex number of the form w + xi + yj + zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions.
2. rare a set of four people or things.
Origin
C19: from late L. quaternio(n-), from L. quarterni (see quaternary).
Quaternion         
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.

Википедия

Quaternion

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

Quaternions are generally represented in the form

where a, b, c, and d are real numbers; and 1, i, j, and k are the basis vectors or basis elements.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a domain. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by H . {\displaystyle \mathbb {H} .} It can also be given by the Clifford algebra classifications Cl 0 , 2 ( R ) Cl 3 , 0 + ( R ) . {\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} In fact, it was the first noncommutative division algebra to be discovered.

According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. (The sedenions, the extension of the octonions, have zero divisors and so cannot be a normed division algebra.)

The unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).