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 ?
Through the noncommutativity of the quaternion algebra when difficulties arise already in determining the quaternion determinant, the problem of the determinantal representation of generalized inverses only now can be solved due to the theory of column-row determinants introduced in [28, 29].
To study the continuity of the estimated quaternions, our recently developed fast complementary filter (FCF, [33]) is adopted to fuse the gyrostabilized gravity and magnetic field together via various algorithms.
A space portion of the w component of the quaternions related to each frame can be dedicated for keeping some basic important information such as number of audio samples, text string length, start location of audio samples and start location of text characters as well.
Let [M.sub.2](Z) be the ring of 2 x 2 matrices with entries in Z and H(Q[[square root of (2)]]) the quaternion field.
Thus, this sphere becomes an example of how to calculate the square root of -1 in the set H made of quaternions.
Two quaternions are acquired from sensors adjacent to the joint J.
The algebra of quaternions H is not included in this investigation since there are already publications with algorithms for finding all zeros of unilateral polynomials with quaternionic coefficients; see [3, 14, 24].
Assume the reference quaternion is [sup.A.sub.B][??]; the current quaternion is [sup.A.sub.C][??]; and the orientation between [sup.A.sub.B][??] and [sup.A.sub.C][??] is [sup.B.sub.C][??].
Kremer, Quaternions and SLERP, University of Saarbrucken, Department for Computer Science Seminar Character Animation, Saarbrucken, German, 2008.
In practice, the most common orientation representations are the rotation matrix, Euler angles, and unit quaternion. These constitute the basis of most attitude and orientation estimation techniques, and their properties and group structures are of great importance [22].
Given two quaternions p and q, we define the right [C.sub.r] and left [C.sub.l] carrier operators as
Due to the noncommutativity of quaternions, there are two types of eigenvalues and linear maps: left and right eigenvalues of quaternion matrices and left and right quaternion linear maps.