An introduction to computations
in crystallographic textures
A.Morawiec
Polish Academy of Sciences
Institute of Metallurgy and Materials Science
Motivation
Comprehensive understanding of description of orientations
is crucial for research on polycrystalline materials.
Examples:
• Orientation relationships
• Crystal deformation mechanisms
• Some phase transformation mechanisms
http://www.museumwales.ac.uk
• Orientation mapping
Wikipedia.com
2
=20 µm; Euler3 + 3/ 10/ 20; Step=1 µm; Grid100x100
Outline
• Rotations and rotation parameterizations
• Crystal orientation and crystal symmetry
• Statistics in the orientation space
• Standard (mis)orientation distributions
• Example of texture application: effective elastic properties of polycrystals
3
Suggested reading
• H.J. Bunge,
Texture Analysis in Materials Science,
Butterworths, London, 1982.
• U. F. Kocks, C. N. Tome, H.R. Wenk,
Texture and Anisotropy,
Preferred Orientations in Polycrystals and their Effect on Materials Properties,
Cambridge University Press, Cambridge, 1998.
• A. Morawiec,
Orientations and Rotations,
Computations in Crystallographic Textures,
Springer Verlag, Berlin, 2004.
4
Rotations and rotation parameterizations
A.Morawiec
Polish Academy of Sciences
Institute of Metallurgy and Materials Science
5
Outline
• Basics: orientations vs. rotations
• Numerical representation of rotations and orientations
• Composition of rotations
• Parameterizations of rotations and orientations
Wikipedia.com
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Basics
Basics
Rotation about a point – a displacement in which the location of
the point is not changed.
Rotation about a line – a displacement in which points of a line
retain their locations.
Euler theorem:
Rotation about a point is equivalent to a rotation about a line
Wikipedia.com
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Basics
An orientation – the equivalence class
of all displacements which differ by a translation.
With a universal reference orientation,
an orientation of an object is determined by the rotation
from the reference orientation to that orientation.
object's orientations
rotations about an axis
one-to-one correspondence
Orientation – state
Rotation – process (displacement)
9
Basics
‘Mirror’ transformation with a fixed point = improper
rotation
Improper rotations change handedness
Effects of proper rotations i.c
Mirror images
Effects of proper and improper rotations i.c
10
Basics
Inversion
Inversion = half-turn about a line followed by reflection
with respect to a plane perpendicular to the line.
improper rotation = proper
rotation composed with inversion
Wikipedia.com
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Numerical representation
of rotations and orientations
To refresh memory …
Matrix – an array of m  n numbers
Aij
i  1,2,..., m
j  1,2,..., n
An m  n matrix A is: square matrix if m = n
zero if Aij = 0
transpose of B=AT if Aij = Bji
A square matrix A is:
unit if Aij = Iij = dij
anti-symmetric if Aij = -Aji
symmetric if Aij = Aji
 A11
A
A   21
 ...

 Am1
1
d ij  
0
A12
A22
...
Am 2
A1n 
A2 n 
... ... 

... Amn 
...
...
if
i j
if
i j
 1 if ijk  even permutation of (123)

e ijk   1 if ijk  odd permutation of (123)
 0 in other cases

Matrix algebra
n
Matrix product: C = AB
Cij   Aik Bkj  Ai1B1 j  Ai 2 B2 j  ...  Ain Bnj  Aik Bkj
k 1
Trace of square matrix Aij: Tr(A)= Aii = A11 + A22 + … + Ann
Det of 3  3 matrix Aij: det(A)= eijk Ai1 Aj2 Ak3
Square matrix A is invertable if det(A) is non-zero; AA-1=I
13
Matrix representation of orientations
p dim object immersed in N dim Euclidean space
Orientation is determined by p
linearly independent vectors
N=3, p=2
a  a1
a3 
a2
b  b1 b2
With orthonormal bases
b3 
a1 a2 a3 
O
 

 b1 b2 b3 
a  a a  b 1 0
OOT  
  0 1
b

a
b

b

 

OOT  I p
An orientation of a p dim object immersed in N dim Euclidean space
can be represented numerically by a p x N matrix O satisfying OOT=I.
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Matrix representation of orientations
N=p = 3
OOT  I 3
O – an orthogonal matrix
det O2  1
OT  O 1
Arbitrary rotations (including improper rotations)
N=3, p=3
OOT  I 3
det O  1
O – a special orthogonal matrix
Proper rotations
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Matrix representation of orientations
N=p = 3
Example (special) orthogonal matrix
 2 / 3 1/ 3 2 / 3 
O   2 / 3 2 / 3  1 / 3
 1 / 3 2 / 3 2 / 3 
N=3, p=3
 2 / 3  1 / 3 2 / 3   2 / 3 2 / 3  1 / 3 1 0 0
OOT   2 / 3 2 / 3  1 / 3  1 / 3 2 / 3 2 / 3   0 1 0  I 3
 1 / 3 2 / 3 2 / 3   2 / 3  1 / 3 2 / 3  0 0 1
det O  1
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Matrix representation of orientations
N=p = 3
OOT  I 3
O – an orthogonal matrix
det O2  1
OT  O 1
Arbitrary rotations (including improper rotations)
N=3, p=3
OOT  I 3
det O  1
O – a special orthogonal matrix
Proper rotations
Groups of

orthogonal matrices – O(3)
Wikipedia.com
special orthogonal matrices – SO(3)
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Composition of rotations
Groups

– all rotations
O(3)
one-to-one correspondence to
SO(3)
Wikipedia.com
– proper rotations
Composition of rotations
R'  O' OT
O
R' '  O' ' O'T
O'
O' '
R  O' ' OT
O, O’ and O’’ – orthogonal matrices
R  O' ' OT  O' ' IOT  O' ' (O'T O' )OT  (O' ' O'T )(O' OT )  R' ' R'
R  R' ' R'
CompositionWikipedia.com
of rotations corresponds to multiplication
of representing them orthogonal matrices.
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Composition of rotations
 2 / 3 1/ 3 2 / 3 
R'   2 / 3 2 / 3  1 / 3
 1 / 3 2 / 3 2 / 3 
0 1 0 
R' '  1 0 0 
0 0  1
0 1 0   2 / 3  1 / 3 2 / 3   2 / 3 2 / 3  1 / 3 
R' ' R'  1 0 0   2 / 3 2 / 3  1 / 3  2 / 3  1 / 3 2 / 3   R
0 0  1  1 / 3 2 / 3 2 / 3  1 / 3  2 / 3  2 / 3
R  R' ' R'
CompositionWikipedia.com
of rotations corresponds to multiplication
of representing them orthogonal matrices.
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Composition of rotations − algebra of quaternions
e
- basis of a 4 dimensional vector space,
  0,1,2,3
i, j, k  1,2,3
x  x e  x0e0  x1e1  x2e2  x3e3
Standard multiplication plus
the quaternion multiplication rule
{
x  x e
e e0  e0e  e
ei e j  e ijk ek  d ij e0
y  y e
xy  x0 y0  xi yi  e0  x0 yk  xk y0  e ijk xi y j  ek
xy  yx
Wikipedia.com
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Unit quaternions and rotations
q
- components of a unit quaternion
i, j, k  1,2,3
  0,1,2,3
q q  q02  q12  q22  q32  1


Aij  q0   qk qk d ij  2qi q j  2e ijk qk q0
2
A is a special orthogonal matrix
q  [ q0 , q1 , q2 , q3 ]  [ 1 / 2, 1 / 2, 1 / 2, 1 / 2 ]
0 0 1 
A  1 0 0
0 1 0
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Unit quaternions and rotations
q, q’ – unit quaternions
O, O’ – special orthogonal matrices
q O
q '  O'
qq'  OO'
There is a two-to-one corespondence between
unit quaternions
and special orthogonal matrices.
Wikipedia.com
There is a two-to-one corespondence between unit quaternions and proper rotations.
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Unit quaternions and rotations
Unit quaternion sphere in 4D
q
q02  q12  q22  q32  1
q
Quaternions q and –q represent the same orientations


Oij q   q0   qk qk d ij  2qi q j  2e ijk qk q0  Oij  q 
2
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Summary
Euler theorem.
(Proper) rotations can be represented (special) orthogonal matrices.
Composition of rotations is represented by the product of orthogonal matrices.
There is two-to-one correspondence between unit quaternions
and special orthogonal matrices.
Composition of rotations is represented by the product of unit quaternions.
Wikipedia.com
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Transformation of vector components
x
ai  a j  d ij
and
x  x j a j  x'k a 'k
a'i a' j  d ij
 a'i
x'i  x j a j  a'i  Rij x j
a'i a j  Rij
x'  Rx
Wikipedia.com
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Parameterizations
• Rodrigues parameters
• Axis and angle
• Rotation vector
• Euler angles
• Miller indices
Independent orientation parameters
Special orthogonal matrix – 9 parameters
Unit quaternion – 4 parameters
1
2 3 4
1 2 3 
4 5 6


7 8 9
Number of independent parameters - 3
Is it possible to have a „nice” global parameterization
(one-to-one, continuous, with continuous inverse),
which would map rotations into the 3 dimensional
Euclidean space?
No!
Wikipedia.com
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Cayley transformation
O - special orthogonal matrix such that
( I  O)
non-singular
R  ( I  O) 1 ( I  O)
O  ( I  R)( I  R) 1
R - an antisymmetric 3x3 matrix
3 independent parameters
( I  R)
 0
R   r3
 r2
non-singular
r3
0
 r1
 r2 
r1 
0 
29
Cayley transformation, example
 2 / 3 1/ 3 2 / 3 
O   2 / 3 2 / 3  1 / 3
 1 / 3 2 / 3 2 / 3 
1 / 3  1 / 3  0
 0
R  ( I  O) 1 ( I  O)   1 / 3
0
1 / 3    r3
 1 / 3  1 / 3
0   r2
r3
0
 r1
 r2 
r1 
0 
r  [r1 , r2 , r3 ]  [1 / 3, 1 / 3, 1 / 3]
30
Rodrigues parameters
 0
R   r3
 r2
ri  
r3
0
 r1
 r2 
r1 
0 
1
ri  e ijk R jk
2
1
e ijk O jk
1  Oll
Oij 

1
1  rk rk  d ij  2ri rj  2e ijk rk
1  rl rl
Rodrigues parameters / unit quaternion:
r O
Rij  e ijk rk
qi
ri 
q0

Rodrigues space
r '  O'
r  r '  OO'
Wikipedia.com
one-to-one corespondence
(r  r ' ) i 
ri  r 'i e ijk rj r 'k
1  rl r 'l
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Axis and angle
Euler theorem
rotation axis
magnitude of rotation – rotation angle
Or  r
- Rodrigues vector is parallel to rotation axis
The axis is represented by vector n of unit magnitude
k n  0
k k 1
k
n  r r
Ok  k  cos 

n
Ok
cos   Oii  1 2
Wikipedia.com
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Axis and angle
qi  sin( 2) ni
ri  tan( 2) ni
Oij (n, )  d ij cos   ni n j (1  cos  )  e ijk nk sin 
O(n, )  O(n,2  )
nS2
0  
n  S 2
0    2
2

0
0
33
Axis and angle, example
 2 / 3 1/ 3 2 / 3 
O   2 / 3 2 / 3  1 / 3
 1 / 3 2 / 3 2 / 3 
r  [r1 , r2 , r3 ]  [1 / 3, 1 / 3, 1 / 3]
r
n
 [1 / 3, 1 / 3, 1 / 3 ]  [1, 1, 1] / 3
r r
cos   Oii  1 2  1 / 2
   3  60
 1


(n,  )  
[1, 1, 1], 
3
 3
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Rotation vector
 1


(n,  )  
[1, 1, 1], 
3
 3
 1
 1
r  tan( / 2)  n  tan( / 6) 
[1, 1, 1]   [1, 1, 1]
 3
 3
   n 

3 3
[1, 1, 1]
  f   ni
i
Wikipedia.com
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Rotation vector
3
i  f   ni
f 0  0
f : [0,  ]  R strictly increasing,
f(w) n
0
i    i 
  :
2
1
parametric ball
3
tan( / 2)
2.5
f    
2
f(  )
1.5
f    tan / 2
sin( )
sin( / 2)
1
0.5
f    sin / 2

tan( / 4)
[(3 /( 4 2 ))(  sin( ))]1/ 3
0
0

f    3  sin   / 4
20
Wikipedia.com 2
40

1/ 3
60
80
100
120
140
160
180
 in degrees
– isochoric parameters
36
Euler angles
O  O(n,  )
x-convention
O(1 ,  , 2 )  O(ez , 1 ) O(e' x ,  ) O(e"z , 2 )
ey
e”z
ez
ez
e”y
e”x
e’x
ex
e’x
O(1 ,  , 2 )  O(ez , 2 ) O(ex ,  ) O(ez , 1 )
37
Euler angles
O(1 ,  , 2 )  O(ez , 2 ) O(ex ,  ) O(ez , 1 )
The domain of all proper rotations is covered when
0  1  2
0  
0   2  2

1
2
0  1  2
0  
0   2  2
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Euler angles, example
O  O(n,  )
O(1 ,  , 2 )  O(ez , 2 ) O(ex ,  ) O(ez , 1 )
1  90
2  90
  60
 0 1 0
O(ez ,90)   1 0 0
 0 0 1
0
1
O(ex ,60)  0
1/ 2
0  3 / 2
0
 0 1 0 1
O(90,60,90)   1 0 0  0
1/ 2
 0 0 1 0  3 / 2

(n,  )  [1, 0,
0 
3 / 2
1 / 2 
0   0 1 0   1 / 2 0

3 / 2   1 0 0   0
1
1 / 2   0 0 1  3 / 2 0
3] / 2 , 

3 / 2

0 
1 / 2 
39
Euler angles
O(1 ,  , 2 )  O(ez , 2 ) O(ex ,  ) O(ez , 1 )
Singularity:
Lattman angles:
O(1 ,0,  2 )  O(1   ,0,  2   )
O(1 ,  ,  2 )  O(1   ,  ,  2   )
   1   2  2
   1   2  2
40
Miller indices
e.g., 110001
uvw
hkl
hu  kv  lw  0
ei
aj
– i-th external basis vector
– j-th basis vector of crystal direct lattice
hkl   Ai1Oi3 , Ai 2Oi3 , Ai3Oi3 
Cubic system
Aij  ei  a j
B  A1
uvw  B1iOi1, B2iOi1, B3iOi1 
hkl  O13O23O33  uvw  O11O21O31
Wikipedia.com
41
Miller indices, example
Cubic system
hkl  O13O23O33 
 2 / 3 1/ 3 2 / 3 
O   2 / 3 2 / 3  1 / 3
 1 / 3 2 / 3 2 / 3 
uvw
hkl
uvw  O11O21O31
hkl  O13O23O33   (2 1 2)
uvw  O11O21O31  [2 2 1]
hkl uvw  (2 1 2)[2 2 1]
42
Summary
43
Summary
• Euler theorem: Rotation about a point is equivalent to a rotation about a line.
• There is one-to-one correspondence between object's orientations and
rotations about a fixed point.
• An orientation of an object can be represented an orthogonal matrix.
(Proper) rotations are represented (special) orthogonal matrices.
Composition of rotations corresponds to multiplication of representing
them orthogonal matrices.
• There is two-to-one correspondence between unit quaternions
and special orthogonal matrices.
An orientation of an object can be represented a unit quaterion.
Composition of rotations corresponds to multiplication of representing
them unit quaternions.
44
Summary
• There are a number of orientation parameterizations.
„Nice” parameterizations involve more than 3 numbers.
There is a relationship between antisymmetric matrices
and special orthogonal matrices (Cayley transformation).
Cayley transformation is a good starting point for deriving
3-number rotation parameterizations.
• Axis and angle
• Euler angles
• Rodrigues parameters
• Rotation vector
• Miller indices
45
Unit quaternions and rotations
q
- components of a unit quaternion
i, j, k  1,2,3
  0,1,2,3
q q  q02  q12  q22  q32  1


Oij  q0   qk qk d ij  2qi q j  2e ijk qk q0
2
O – special orthogonal matrix
qi   e ijk Okj 2 
qi   e ijk O jk 2 
  1  Oll 1/ 2
46

(ab)=(a = b(aa ) = be = b. −1(ab) = b = (ba)a ∈ G, (ab)c = a(bc). −1

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