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 ?
Liu, "Pitt's inequality and the uncertainty principle associated with the quaternion fourier transform," Journal of Mathematical Analysis and Applications, vol.
A quaternion is expressed by q = [q.sub.0] + [q.sub.1]i + [q.sub.2]j + [q.sub.3]k using real numbers [q.sub.0], [q.sub.1], [q.sub.2], and [q.sub.3].
Throughout the paper, we denote the real number field by R, the set of all m x n matrices over the quaternion algebra
Quaternion analysis offers another possibility of generalizing complex function theory to higher dimension, and it has wide applications in mathematical physics, Dirichlet BVPs of higher-order partial differential equations, and other fields.
[delta][[theta].sub.1] is the attitude error computed from the error quaternion [delta]Q which is defined as
Consider the quaternion group [Q.sub.16] = (x, y|x = [y.sup.2], xyx = y) of order 16.
Simplifying, we can notice that the rotation's definition of vector v post by quaternion is calculated as qvq-1 where q-1 is the conjugate of the quaternion q.
A quaternion moment-based scheme is suggested in [17] where Tsougenis et al.
where [cross product] denotes the quaternion product which can be determined using the Hamilton rule [25]:
Shoemake, "Animating rotation with quaternion curves," in Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '85), pp.
Unit quaternion is a 4D parameterization, which allows such singularities to be avoided [22].
An alternative way is using quaternion interpolation.