quaternion


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quaternion

(kwətûr`nēən), in mathematics, a type of higher complex number first suggested by Sir William R. Hamilton in 1843. A complex number is a number of the form a+bi when a and b are real numbers and i is the so-called imaginary unit defined by the equation i2=−1. The rules for operating with complex numbers are simply those of operating with the polynomial a+bx except that i2 is replaced by −1 whenever it occurs. A quaternion, an extension of this concept, is a number of the form a+bi+cj+dk when a, b, c, and d are real numbers and i, j, and k are imaginary units defined by the equations i2=j2=k2=ijk=−1. Quaternions, as well as vectorsvector,
quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.
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 and tensorstensor,
in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates).
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 (later outgrowths of the concept of quaternions), have many important applications in mechanics.

quaternion

[kwə′ter·nē·ən]
(mathematics)
The division algebra over the real numbers generated by elements i, j, k subject to the relations i 2= j 2= k 2= -1 and ij = -ji = k, jk = -kj = i, and ki = -ik = j. Also known as hypercomplex number.
References in periodicals archive ?
In order to appreciate this geometry, quaternion algebra simplifies the game.
In this section, we will review some basic definitions of quaternion number and then discuss their implications to quaternion relativity (Q-relativity) physics [1].
The quaternion continuity equation can be written in the form
Interestingly, there is also a neat link between Maxwell equations and quaternion numbers, in particular via the Moisil-Theodoresco D operator [7, p.
It was known for quite long time that a quaternion space can be generalized to a Clifford space, and vice versa; but how to find its neat link to more convenient metric form in the General Relativity theory, has not been explored extensively [2].
The arccosines of these three normalized factors determine the quaternion angles 105.
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--the generalized quaternion group (n [greater than or equal to] 2);
They now face increasing requirements of regulators and accounting standards (such as IFRS 13) and we are excited to have extended our core risk offering to include calculation of such analytics, through this partnership with Quaternion, which has significant experience in this area.
Keywords: image processing, face recognition, fuzzy neural network classifier, quaternion correlation
Inputs for the computation were the quaternion attitude and the origin of each segment i ([q.
It was, however, not as an astronomer but as a mathematician that Hamilton made his greatest contributions to science, principally through his discovery in 1843 of quaternion operators, which enable the rotation of objects to be described mathematically.
The topics include geometric-edge random graph model for image representation, unsupervised and supervised image segmentation using graph partitioning, generative group activity analysis with quaternion descriptor, discriminating feature selection in image classification and retrieval, and region-based graph learning towards large scale image annotation.