Some Information on Ouaternions Useful in Texture Calculations

Crystal orientation is described by a rotation which can be expressed by a great number of different sets of parameters. Among these, the quaternion representation is the most economic one. The basic properties of quaternions are briefly summarized and their interrelationships with other orientation coordinates such as rotation axis and angle, the orientation matrix, and the Euler angles are given.


INTRODUCTION
In texture analysis the orientation is usually described by Euler angles, sometimes by rotation axis and rotation angle or other rotation parameters.
From among the possible orientation descriptions (see e.g. Goldstein (1953) or Korn and Korn, (1961)) the most economic one from the viewpoint of calculations seems to be the description by means of quaternions. The multiplication of four-element quaternions corresponding to the composition of the rotations requires considerably smaller number of operations and the relations of quaternion elements with the rotation parameters are in general simpler than they are in the case of elements of an orthogonal matrix. We shall recall the basic relations referring to the algebra of the quaternions and write down the relations which can be applied in certain calculation operations of texture analysis.
(2) 1, wheni=j /ti O, when :/:j when (ijk) is even permutation of (123) when (ijk) is odd permutation of (123) in the other cases Hence when x x'e, and y y'e, are two arbitrary quaternions their product is xy (xy xiyi)eo + (xy k + yOxk + e,kxiyi)ek. ( The quaternion x* xeo-xie is called a quaternion conjugate to x x'e, xeo+xe. Furthermore we shall be interested in quaternions satisfying the condition xx* eo. Their set will be denoted by Q; Q {x e Q: xx* eo). It is easy to check that if x, y e Q then xy .Q, and the condition xx* =eo expressed by means of coordinates has the form x'x ' 1.
Significant is the fact of the existance of a surjective mapping h:Q---> SO(3) such that h(xy) h(y)h(x) (see Altmann (1986)). There exists also a mapping h':Q--> SO(3) defined as h'(x)= h(x*) and satisfying the condition h'(xy) h'(x)h'(y). For further consideration the first mapping, i.e. h will be selected.
the quaternions x and -x represent the same rotation. Hence the coordinates will be selected in such a way that x > 0 or when x 0 that x > 0 etc. Let us express the coordinates x of the quaternion x through the elements of the matrix g. Accepting j--i in formula (4) summing over i and utilizing x'x ' 1 we shall find gii 4(x) 2-1, and hence X 0 (gii + 1) 1/2 2 (5) From the relation (4) there follows also ekgk 4XX and assuming that x4: 0 we get X ei]kg]k 2(gu + 1) v2" (6) When x = O, then expression (4) takes the form of a system of equations gq 2xix 6i from which it is easy to find xi.

RELATION OF THE QUATERNION COORDINATES WITH THE ROTATION AXIS AND THE ROTATION ANGLE
Since the rotation angle to is connected with the trace of the rotation matrix by the relation gu--2 cos to + 1, we obtain from expression (5) x cos . (7) 214 A. MORAWIEC AND J. POSPIECH On the basis of expression (4) there is also gijx x and only when the coordinates x are not simultaneously equal to zero, i.e. when xixi4 O, then x i= ani, where n icoordinate of the rotation axis.
Since nini=l on the basis of (7) we have: cr=:l:sin(o/2). Calculating x 3 from (4) in the case when x 1= x2= n l= n 2--0 we find that the proper sign is / (the sign is opposite when accepting the mapping h'). Hence X n sin . (8) Conversely, on the basis of (7) and (8)  (10) From formulae (7) and (8) one obtains directly the quaternion forms of the rational symmetry elements, e.g. for the four-fold axis Lt4OOll we have -(eo + e3) or for the three-fold axis L111] we have 1/2(e0 + e + e2 + e3).

RELATIONS OF THE QUATERNION COORDINATES WITH EULER ANGLES
A rotation described by Euler angles tpl, t#, tp2 is composed of three successive rotations of the coordinate system about the axis z by an angle tp, about the axis x by an angle and about the axis z by an angle tp2. To these rotations, on the basis of (7) and (8) correspond the following quaternions cos -eo + sin -:-e3, cos eo + sinel, ID2 t92 coseo + sin --e3 Their product represents thus the rotation by Euler angles qg, q, Simple transformations of the above formulae allow to obtain the reverse relations cos ((x) 2 + (xa)2) ((xl) 2 / (x2)2), sin q 2X COS I01 (X0X xEx3)/X, sin I91 (X0X 2 / XlX3)/, (12) cos tp2 (xx / x2x3)/X, sin tp2 (xx 3-xx2)/X FINAL REMARKS The above considerations show that the quaternions may represent a convenient tool in computation operations of texture analysis. Use is made here of the fact that the operations which are performed on nine-element orthogonal matrices can be replaced in an equivalent manner by operations by four-element quaternions. The usage of quaternions simplifies also transformations of the rotation parameters of one kind to rotation parameters of another kind.
It should also be mentioned that the components x1, x2, x 3 of the quaternion may be used as rotation parameters. In the cartesian 216 A. MORAWIEC AND J. POSPIECH system the rotation is represented by a point of the sphere (xl) 2 + (x2) 2 + (xa) 2 -< 1. (Antipodal points of the surface represent the same rotation.) The formula for the volume element in the coordinates x 1, x2, x a 1 1 dx ax E dx 3 xO has the form dg -5-6 where (1 xkxk) 2