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 ?
The method is based on quaternion encryption in the Cipher Block Chaining (CBC) mode.
The quaternion group Q is determined by its endomorphism monoid in the class of all groups.
The rotations performed by a quaternion can be placed such as those placed by Euler angles, however, with one difference: the angle must be halved.
A quaternion moment-based scheme is suggested in [17] where Tsougenis et al.
Presentations of the unit group of an order in a non-split quaternion algebra.
T] can be estimated using orientation represented in quaternion (8)
The desired quaternion [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is then
The attitude kinematics are modeled using the quaternion differential equation
In this study, we visually aligned the sensor axis with Vicon axis and then used quaternion approach to calibrate the orientation of sensors attached to the body (Najafi et al.
n](D), where D is a quaternion division algebra, or A is trivial on k.
took] the quaternion out of my hand, in which I transcribed these things out of my table-booke" (398).
Lang, "Edge detection of colour image based on quaternion fractional differential," IET Image Processing, vol.