27750, Advanced Characterization

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Rodrigues Vectors, Quaternions23rd January 03

• 27-750, Advanced Characterization
  Microstructural Analysis
• January 21/23, 2003
• A.D. (Tony) Rollett

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Objectives

• Introduce the Rodrigues vector as a
  representation of rotations, orientations
  (texture components) and misorientations (grain
  boundary types).
• Introduce the quaternion and its relationship to
  other representations of rotations.

French mathematician active in the early part of
the 19th C.
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References

• A. Sutton and R. Balluffi, Interfaces in
  Crystalline Materials, Oxford, 1996.
• V. Randle O. Engler (2000). Texture Analysis
  Macrotexture, Microtexture Orientation Mapping.
  Amsterdam, Holland, Gordon Breach.
• Frank, F. (1988). Orientation mapping,
  Metallurgical Transactions 19A 403-408.

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Rodrigues vectors

• Rodrigues vectors were popularized by Frank
  Frank, F. (1988). Orientation mapping.
  Metallurgical Transactions 19A 403-408., hence
  the term Rodrigues-Frank space for the set of
  vectors.
• Most useful for representation of
  misorientations, i.e. grain boundary character
  also useful for orientations (texture
  components).
• Fibers based on a fixed axis are always straight
  lines in RF space (unlike Euler space).

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Rodrigues vector, contd.

• We write the axis-angle representation as (
  ,q)
• The Rodrigues vector is defined as r
  tan(q/2)

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Orientation, Misorientation

• This lecture will discuss both orientations,
  typically denoted by g when specified by a
  matrix, and misorientations, typically denoted by
  ?g.
• Given two orientations (grains), gA and gB, the
  misorientation between them is the (matrix)
  product of the one orientation with the inverse
  of the other, ?g gBgA-1. The order in which
  the orientations are written matters (to be
  discussed!).

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Conversions matrix?RF vector

• Conversion from rotation (misorientation) matrix,
  ?ggBgA-1

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Conversion from Bunge Euler Angles

• tan(q/2) v(1/cos(F/2) cos(f1 f2)/22 1
  
• r1 tan(F/2) sin(f1 - f2)/2/cos(f1
  f2)/2
• r2 tan(F/2) cos(f1 - f2)/2/cos(f1
  f2)/2
• r3 tan(f1 f2)/2

P. Neumann (1991). Representation of
orientations of symmetrical objects by Rodrigues
vectors. Textures and Microstructures 14-18
53-58.
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Conversion from Roe Euler Angles

• tan(q/2) v(1/cosQ/2 cos(Y F)/22 1
• r1 -tanQ/2 sin(Y - F)/2/cos(Y F)/2
• r2 tanQ/2 cos(Y - F)/2/cos(Y F)/2
• r3 tan(Y F)/2

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Combining Rotations as RF vectors

• Two Rodrigues vectors combine to form a third,
  rC, as follows, where rB follows after rA. Note
  that this is not the parallelogram law for
  vectors! rC (rA, rB) rA rB - rA x
  rB/1 - rArB

vector product
scalar product
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Combining Rotations as RF vectors component form
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Quaternions

• A close cousin to the Rodrigues vector is the
  quaternion.
• It is defined as a four component vector in
  relation to the axis-angle representation as
  follows, where uvw are the components of the
  unit vector representing the rotation axis, and q
  is the rotation angle.
• As with the Rodrigues vector, trigonometric
  functions of the semi-angle are used.

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Quaternion definition

• q q(q1,q2,q3,q4) q(u sinq/2, v sinq/2, w
  sinq/2, cosq/2)
• Alternative notation puts cosine term in 1st
  positionq (cosq/2, u sinq/2, v sinq/2, w
  sinq/2).

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Historical Note

• This set of components was obtained by Rodrigues
  prior to Hamiltons invention of quaternions and
  their algebra. Some authors refer to the
  Euler-Rodrigues parameters for rotations in the
  notation (l,L) where l is equivalent to q4 and L
  is equivalent to the vector (q1,q2,q3). Yet
  another notation is (q0,q1,q2,q3), where q0 is
  equivalent to q4, i.e. cos(q/2).

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Rotations represented by Quaternions

• The particular form of the quaternion that we are
  interested in has a unit norm (vq12q22q32
  q421) but quaternions in general may have
  arbitrary length.
• Thus for representing rotations, orientations and
  misorientations, only quaternions of unit length
  are considered.

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Why Use Quaternions?

• Among many other attractive properties, they
  offer the most efficient way known for performing
  computations on combining rotations. This is
  because of the small number of floating point
  operations required to compute the product of two
  rotations.

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Conversions matrix?quaternion
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Conversions quaternion ?matrix

• The conversion of a quaternion to a rotation
  matrix is given byaij
  (q42-q12-q22-q32)dij 2qiqj
  2q4Sk1,3eijkqk
• eijk is the permutation tensor, dij the
  Kronecker delta

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Roe angles ? quaternion

• q1, q2, q3, q4 -sinQ/2 sin(Y - F)/2 ,
  sinQ/2 cos(Y - F)/2, cosQ/2 sinY F)/2,
  cosQ/2 cos(Y F)/2

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Bunge angles ? quaternion

• q1, q2, q3, q4 sinF/2 cos(f1 - f2)/2 ,
  sinF/2 sin(f1 - f2)/2, cosF/2 sinf1
  f2)/2, cosF/2 cos(f1 f2)/2

Note the occurrence of sums and differences of
the 1st and 3rd Euler angles!
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Combining quaternions

• The algebraic form for combination of quaternions
  is as follows, where qB follows qA qC qA
  qBqC1 qA1 qB4 qA4 qB1 - qA2 qB3 qA3
  qB2qC2 qA2 qB4 qA4 qB2 - qA3 qB1 qA1
  qB3qC3 qA3 qB4 qA4 qB3 - qA1 qB2 qA2
  qB1qC4 qA4 qB4 - qA1 qB1 - qA2 qB2 - qA3 qB3

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Positive vs Negative Rotations

• One curious feature of quaternions that is not
  obvious from the definition is that they allow
  positive and negative rotations to be
  distinguished. This is more commonly described
  in terms of requiring a rotation of 4p to
  retrieve the same quaternion as you started out
  with but for visualization, it is more helpful to
  think in terms of a difference in the sign of
  rotation.

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Positive vs Negative Rotations

• Lets start with considering a rotation of q
  about an arbitrary axis, r. From the point of
  view of the result one obtains the same thing if
  one rotates backwards by the complementary angle,
  q-2p (also about r). Expressed in terms of
  quaternions, however, the representation is
  different! Setting ru,v,w again,
• q(r,q) q(u sinq/2, v sinq/2, w sinq/2, cosq/2)

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Positive vs Negative Rotations

• q(r,q-2p) q(u sin(q-2p)/2, v sin(q-2p)/2, w
  sin(q-2p)/2, cos(q-2p)/2) q(-u sinq/2, -v
  sinq/2, -w sinq/2, -cosq/2) -q(r,q)

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Positive vs Negative Rotations

• The result, then is that the quaternion
  representing the negative rotation is the
  negative of the original (positive) rotation.
  This has some significance for treating dynamic
  problems and rotation angular momentum, for
  example, depends on the sense of rotation. For
  static rotations, however, the positive and
  negative quaternions are equivalent or, more to
  the point, physically indistinguishable, q ? -q.

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Quaternion acting on a vector

• The active rotation of a vector from X to x is
  given byxi (q42-q12-q22-q32)Xi
  2qiSjqjXj 2q4SjXjSkeijkqk
• eijk is the permutation tensor, dij the
  Kronecker delta

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Computation combining rotations

• The number of operations required to form the
  product of two rotations represented by
  quaternions is 16 multiplies and 12 additions,
  with no divisions or transcendental functions.
• Matrix multiplication requires 3 multiplications
  and 2 additions for each of nine components, for
  a total of 27 multiplies and 18 additions.
• Rodrigues vector, the product of two rotations
  requires 3 additions, 6 multiplies 3 additions
  (cross product), 3 multiplies 3 additions, and
  one division, for a total of 10 multiplies and 9
  additions.
• The product of two rotations (or composing two
  rotations) requires the least work with Rodrigues
  vectors.

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Negative of a Quaternion

• The negative (inverse) of a quaternion is given
  by negating the fourth component, q-1
  (q1,q2,q3,-q4) this relationship describes the
  switching symmetry at grain boundaries.

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Summary

• Rodrigues vectors allow rotations to be
  parameterized with a 3-component vector.
• Rodrigues-Frank vector space has the advantage
  that (a) a constant rotation axis is represented
  by a straight line and (b) symmetry elements
  appear as delimiting planes.
• Quaternions form a complete algebra. In the form
  of unit length quaternions, they are very useful
  for describing rotations. Calculation of
  misorientations in cubic systems is particularly
  efficient.