Quaternions


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Quaternions

 

a number system proposed in 1843 by the British scientist W. Hamilton.

Quaternions were the result of attempts to generalize the complex numbers x+iy, where x and y are real numbers and i is a basis element satisfying the condition i2 = — 1. As is well know, complex numbers are depicted geometrically as points in a plane, and operations on them correspond to simple geometrical transformations of the plane (translation, rotation, dilation, or contraction, and their combinations). The search for a number system that would be geometrically realizable with the aid of points in three-dimensional space led to the discovery that it is impossible to “construct” from the points of a space of three or more dimensions a number system in which algebraic operations retain all the properties of addition and multiplication of real or complex numbers. However, if one of the properties— commutativity of multiplication—is dispensed with while the remaining properties of addition and multiplication are retained, then it is possible to construct a number system out of points of a space of four dimensions. (It is impossible to construct even such a number system out of points of a space of three, five, or more dimensions.) These numbers, realizable in four-dimensional space, are called quaternions.

Quaternions are linear combinations of the four basis elements 1, i,j, and k:

Xx0 · l + x1i + x2j + x3k

where x0, x1, x2, and x3 are real numbers. One operates on quaternions according to the rules of operation on polynomials in 1,i, j, and k except that one must not use the commutative law of multiplication and one multiplies the basis element as in Table 1. From Table 1 it is evident that 1 plays the role of ordinary unity and may therefore by omitted when writing down the quaternion:

(1) X = x0 + x1i + x2j + x3k

The quaternion (1) may be separated into a scalar part x0 and a vector part

V - x1i + x2j + x3k

so that

X = x0 + V

If x0 = 0, then the quaternion V is called a vector; it can be identified with ordinary three-dimensional vectors.

Table 1
 1ijk
11ijk
ii-1k-i
jj-k-1i
kkj-i-1

In the middle of the 19th century, quaternions were perceived as a generalization of the concept of number that was destined to play as important a role in science as the complex numbers. This viewpoint was reinforced by the fact that quaternions were applied in electrodynamics and mechanics. However, vector calculus in its modern form has displaced quaternions from these areas. It is clear that the role of quaternions can in no way be compared to the role of complex numbers, which have numerous and varied applications in different branches of science and technology.

References in periodicals archive ?
The arccosines of these three normalized factors determine the quaternion angles 105.
Inputs for the computation were the quaternion attitude and the origin of each segment i ([q.
Daniilidis, Hand-Eye Calibration Using Dual Quaternions, International Journal of Robotics Research 18, 286-298 (1999).
The dual complex numbers defined as the dual quaternions were considered as a generalization of complex numbers by Ata and Yayli [3].
This is called The Fundamental Theorem of Algebra for Quaternions by the two mentioned authors.
Generalizing from rotation around one axis to rotations around three orthogonal axes (as in 3-dimensional space) leads Penrose from complex numbers to hyper-complex numbers (such as quaternions and octonions), and to the hypercomplex algebras discovered by Clifford, Cayley, and Grassman when spaces of more dimensions (and arbitrary signatures) are considered.
It also shows that the addition of quaternions is commutative while the multiplication is not.
The clear, structured, and rational users interface of Calc 3D Pro makes the handling with vectors, complex numbers, quaternions and coordinates easy and intuitive.
Therefore, in accordance with point 2) the real numbers field (R) is extended to the complex numbers field (R), and in accordance with point 3) the complex numbers field is expanded to the quaternions field (K), and point 4) expands the quaternions fields to the octavians field (O).
This is the quaternion algebra H, first considered by Hamilton in 1843 (it may be remarked, however, that the formulae for calculating with quaternions appeared earlier in works by Euler in 1748 and Gauss in 1819).
Unit dual quaternions are also used both as rotation and screw operators.