By Adam Morawiec
The ebook is set mathematical and computational foundations of texture research. Numerical recommendations are imperative in texture research, so the ebook is essentially addressed to researchers and scholars utilizing those innovations in practice.
Orientations and Rotations is particularly various from different books on textures in its content material and point of interest. significant a part of the booklet is dedicated to orientations and rotations normally. establishing chapters comprise an intensive and thorough advent to rotations in 3 dimensions (including parameterizations and geometry of the rotation space). extra chapters also are basic yet they are going to be of curiosity for readers facing orientations of symmetric gadgets. This topic is vital for crystallographic textures because so much crystal constructions are symmetric. the ultimate chapters situation simpler facets of textures (such because the choice of orientations from diffraction styles and the calculation of potent elastic homes of polycrystals).
The e-book can be of curiosity for scientists engaged on plasticity, grain limitations, recrystallisation, grain development, and various different concerns within which textures has to be taken under consideration. as a result wide components on rotations ordinarily, the booklet will be helpful for everybody facing rotations
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Extra resources for Orientations and Rotations: Computations in Crystallographic Textures
30), and t' ----+ t" = (a' t' + b') / (c' t' + d') [~~] is assigned (a'd' - b'c' = 1) corresponds to then the appropriate matrix for t ----+ t" is [~: ~] [~~]. t:::::J f(Jl1 InAntihomographies analogy, the so-called antihomographies can be considered. An antihomography is a transformation of the complex plane defined by t ~ t' = a! + b ct + d ' ad-bci-O. The composition of two antihomographies is a homography. The composition of homography and antihomography (in arbitrary order) is an antihomography.
Thus, the above formulas are consistent, and correctly determine the angles 'PI, c/J and 'P2· If 0 33 = COS c/J = 1, then sin('P1 + 'P2) = 0 12 , COS('P1 + 'P2) = 0 11 . Similarly, if 0 33 = COS c/J = -1, then This shows that the correspondence between the matrices and Euler angles is unambiguous only if c/J i= 0 and c/J i= 7r. If c/J = 0, only the sum of 'PI and 'P2 is uniquely related to the rotation. One can write O( 'PI, 0, 'P2) = O( 'PI +0:,0, 'P2 -0:), where 0: is arbitrary, and addition and subtraction of 0: are modulo 27r.
Let us consider a composition of two rotations with axes and angles (n, w) and (n', w'). The next step is to decompose each of the rotations into two reflections O( n, w) = (I - 2L) (I - 2K) and O( n', w') = (I - 2L') (I - 2K'). ) Now, the arbitrariness in the choice of the reflecting planes allows the second reflection plane in decomposition of (n, w) to be the same as the first mirror of (n', w'). Geometrically, this means that the plane contains both n and n'. The choice can be expressed as K' = L.
Orientations and Rotations: Computations in Crystallographic Textures by Adam Morawiec