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27-750
Advanced Characterization and
Microstructural Analysis
Vectors, Matrices, Rotations
Spring 2007
CarnegieMellon
MRSECMost of the material in these slides
originated in lecture notes by Prof. Brent
Adams (now at BYU).
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Notation
X point
x1,x2,x3 coordinates of a point
u vector
o originbase vector (3 dirn.)
n1 coefficient of a vector
Kronecker delta
eijk permutation tensor
aij,Lij rotation matrix (passive)or, axis transformation
gij rotation matrix (active*)
u (ui) vector (row or column)
||u|| L2 norm of a vector
A (Aij) general second rank
tensor (matrix)l eigenvalue
v eigenvector
I Identity matrix
AT transpose of matrix
n, r rotation axisq rotation angle
tr trace (of a matrix)
3 3D Euclidean space
u
e3
ij
* in most texture books,gdenotes an axis transformation, or passive rotation!
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Points, vectors, tensors, dyadics
Material pointsof the crystalline sample, ofwhich xand yare examples, occupy a subsetof the three-dimensional Euclidean point
space, 3, which consists of the set of allordered triplets of real numbers, {x1,x2,x3}.The term pointis reserved for elements of 3.The numbersx1,x2,x3describe the location of
the pointx by its Cartesian coordinates.
Cartesian; from Ren Descartes, a French mathematician, 1596 to 1650.
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VECTORS
The differencebetween any two pointsdefines a vectoraccording to the
relation . As such denotes thedirected line segment with its origin at x andits terminus at y. Since it possesses both adirection and a length the vectoris an
appropriate representation for physicalquantities such as force, momentum,displacement, etc.
x,y 3v = y -x v
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Two vectors u and v compound (addition) accordingto the parallelogram law. If u and v are taken to bethe adjacent sides of a parallelogram (i.e., emanatingfrom a common origin), then a new vector, w,
is defined by the diagonal of the parallelogram whichemanates from the same origin. The usefulness ofthe parallelogram law lies in the fact that many
physical quantities compound in this way.
w u v
Parallelogram Law
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It is convenient to introduce a rectangularCartesian coordinate framefor consisting ofthe base vectors , , and and a point ocalled the origin. These base vectors haveunit length, they emanate from the commonorigin o, and they are orthogonal to each
another. By virtue of the parallelogram lawany vector can be expressed as a vectorsum of these three base vectors according tothe expressions
e1
e2
e3
v
v v1e1 v2 e2 v3e3 vieii13 vi eiCoordinate Frame
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where are real numbers calledthe componentsof in the specifiedcoordinate system. In the previous equation,the standard shorthand notation has been
introduced. This is known as the summationconvention. Repeated indices in the sameterm indicate that summation over therepeated index, from 1 to 3, is required. Thisnotation will be used throughout the textwhenever the meaning is clear.
v1, v2 andv3
Coordinate Frame, contd.
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v2 v12 v2
2 v32 vivi
The magnitude,v, of is related to its
components through the parallelogram
law:
v
Magnitude of a vector
You will also encounter this quantity as theL2 Norm in matrix-vector algebra:
v 2 v v12 v2
2 v32 vivi
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The scalar productuvof the two vectorsand whose directions are separated by theangle q is the scalar quantity
where u and v are the magnitudes of u andv respectively. Thus, uv is the product of theprojected length of one of the two vectors withthe length of the other. Evidently the scalar
product is commutative, since:
u v uvcosq uivi
u v v u
Scalar Product (Dot product)
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There are many instances where the scalarproduct has significance in physical theory.Note that if and are perpendicular then
=0, if they are parallel then =uv,
and if they are antiparallel =-uv. Also,the Cartesian coordinates of a pointx, withrespect to the chosen base vectors andcoordinate origin, are defined by the scalar
product
u vu v u v
u v
xi (x o) ei
Cartesian coordinates
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For the base vectors themselves the followingrelationships exist
The symbol is called the Kronecker delta.Notice that the components of the Kroneckerdelta can be arranged into a 3x3 matrix, I,where the first index denotes the row and the
second index denotes the column. Iis calledthe unit matrix; it has value 1 along thediagonal and zero in the off-diagonal terms.
ei ej ij 1 if i = j0 if i j
ij
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The vector product of vectors and
is the vector normal to the planecontaining and , and oriented in thesense of a right-handed screw rotating from
to . The magnitude of is givenby uvsinq, which corresponds to the area ofthe parallelogram bounded by and . Aconvenient expression for in terms of
components employs the alternating symbol,e or
u v uv u v
u v uv
u vu v
uv eijkeiujvk
Vector Product (Cross Product)
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Related to the vector and scalar products isthe triple scalar product whichexpresses the volume of the parallelipipedbounded on three sides by the vectors ,
and . In component form it is given by
eijk 1 if ijk = 123, 312 or 231 (even permutations of 12
-1 if ijk = 132, 213 or 321 (odd permutations of 12
0 if any two of ijk are equal
(u v )wu vw
(u v) w eijkuivjwk
Permutation tensor, eijk
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With regard to the set of orthonormal basevectors, these are usually selected in such amanner that .
Such a coordinate basis is termed righthanded. If on the other hand
,then the basis is left handed.
(e1 e2) e3 1
(e1 e2) e3 1
Handed-ness of Base Vectors
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CHANGES OF THE
COORDINATE SYSTEM Many different choices are possible for the
orthonormal base vectors and origin of the
Cartesian coordinate system. A vector is anexample of an entity which is independent ofthe choice of coordinate system. Its directionand magnitude must not change (and are, in
fact, invariants), although its components willchange with this choice.
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Consider a neworthonormal system consisting ofright-handed base vectors
with the same origin, o, associated with
and
The vector
is clearly expressed equally wellin either coordinate system:
Note - same vector, different values of thecomponents. We need to find a relationship betweenthe two sets of components for the vector.
v
v vi ei viei
e1, e2 and e3New Axes
e1, e2 and e3
^ e1^
e2^
e2^
e3^
e3^
e1
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The two systems are related by the ninedirection cosines, , which fix the cosine ofthe angle between the ithprimed and thejth
unprimed base vectors:
Equivalently, represent the components
of in according to the expression
aij
aij ei ejaij
ei
ejei aijej
Direction Cosines
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That the set of direction cosines are notindependent is evident from the followingconstruction:
Thus, there are sixrelationships (itakesvalues from 1 to 3, andj takes values from 1
to 3) between the ninedirection cosines, andtherefore only threeare independent.
ei
ej aikajl
ek
el aikajlkl aikajk ij
Direction Cosines, contd.
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Note that the direction cosines can be
arranged into a 3x3 matrix, L, and thereforethe relation above is equivalent to theexpression
where LT denotes the transpose of L. Thisrelationship identifies L as an orthogonalmatrix, which has the properties
LL
T I
L1 LT det L 1
Orthogonal Matrices
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When both coordinate systems are right-handed,
det(L
)=+1 andL
is a proper orthogonal matrix.
The orthogonality of L also insures that, in addition to
the relation above, the following holds:
Combining these relations leads to the following inter-
relationships between components of vectors in thetwo coordinate systems:
ej aij ei
v = LT v , vi aji vj , v = Lv , vj ajiv i
Relationships
v Lv ; ei aijej
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These relations are called the laws of
transformationfor the components of vectors.They are a consequence of, and equivalentto, the parallelogram law for addition ofvectors. That such is the case is evidentwhen one considers the scalar productexpressed in two coordinate systems:
u v uivi aji ujaki vk ajiaki uj vk jk uj vk uj vj ui vi
Transformation Law
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Thus, the transformation law as expressed preserves
the lengths and the angles between vectors. Anyfunction of the components of vectors which remainsunchanged upon changing the coordinate system iscalled an invariantof the vectors from which thecomponents are obtained. The derivations illustrate
the fact that the scalar product,
is an invariantof the vectors uand v.Other examples of invariantsinclude the vectorproduct of two vectors and the triple scalar product ofthree vectors. Note that the transformation law forvectors also applies to the components of pointswhen they are referred to a common origin.
u v
Invariants
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Rotation Matrices
Since an orthogonal matrix merely rotates a
vector but does not change its length, thedeterminant is one, det(L)=1.
L aij
a11 a12 a13
a21 a22 a23
a31 a32 a33
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A rotation matrix, L, is an orthogonal matrix,
however, because each row is mutuallyorthogonal to the other two.
Equally, each column is orthogonal to theother two, which is apparent from the fact thateach row/column contains the direction
cosines of the new/old axes in terms of theold/new axes and we are working with[mutually perpendicular] Cartesian axes.
aki
akj
ij
, aik
ajk
ij
Orthogonality
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Vector realization of rotation
The convenient way tothink about a rotationis to draw a plane that
is normal to the rotationaxis. Then project thevector to be rotated ontothis plane, and onto therotation axis itself.
Then one computes the vector product of the rotationaxis and the vector to construct a set of 3 orthogonalvectors that can be used to construct the new, rotatedvector.
vv
q
n
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Vector realization of rotation
One of the vectorsdoes not changeduring the rotation.The other two can be
used to construct thenew vector.
vv
q
n
v n v
n
n v n
n v
v gv (cosq)v (sinq)n v (1 cosq)(n v )nNote that this equation does not require any specific coordinatesystem; we will see similar equations for the action of matrices,
Rodrigues vectors and (unit) quaternions
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A rotation is commonly written as ( ,q) or
as (n,w). The figure illustrates the effectof a rotation about an arbitrary axis,
OQ (equivalent to and n) through an
anglea
(equivalent toq
andw
).
r
r
gij ij cosqeijknksinq
(1 cosq)ninj(This is an active rotation: a
passive rotation axis
transformation)
Rotations (Active): Axis- Angle Pair
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The rotation can be converted to a matrix(passive rotation) by the following expression,
where is the Kronecker delta and
is the permutation tensor; note thechange of sign on the off-diagonal terms.
aij ij cosq rirj 1 cosq
ijkrksinqk1,3
Axis Transformation from Axis-Angle Pair
Compare with
active rotation
matrix!24
A rotation is commonly written as ( ,q) oras (n,w). The figure illustrates the effect
of a rotation about an arbitrary axis,OQ (equivalent to and n) through anangle a(equivalent to q and w).
r
r
gij ij cosq eijknksinq
(1 cosq)ninj(This is anactive rotation: a
passiv e rotation axistransformation)
Rotations (Active): Axis- Angle Pai
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Rotation Matrix for Axis
Transformation from Axis-Angle Pair
gij ij cosq rirj 1 cosq
ijkrksin
q
k1,3
cosq u2 1 cosq uv 1 cosq wsinq uw 1 cosq vsinquv 1 cosq wsinq cosq v2 1 cosq vw 1 cosq usinquw 1 cosq vsinq vw 1 cosq usinq cosq w
2 1 cosq
This form of the rotation matrix is a passiverotation, appropriate to axis transformations
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Eigenvector of a Rotation
A rotation has a single (real) eigenvectorwhich is the rotation axis. Since aneigenvector must remain unchanged by the
action of the transformation, only the rotationaxis is unmoved and must therefore be theeigenvector, which we will callv. Note that
this is a different situation from other secondrank tensors which may have more than onereal eigenvector, e.g. a strain tensor.
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Characteristic Equation
An eigenvector corresponds to a solution of thecharacteristic equation of the matrix a, where l
is a scalar: av =lv
(a - lI)v
=0
det(a - lI) = 0
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Characteristic equation is a cubic and so three
eigenvaluesexist, for each of which there is acorresponding eigenvector.
Consider however, the physical meaning of arotation and its inverse. An inverse rotationcarries vectors back to where they started outand so the only feature to distinguish it from theforward rotation is the change in sign. The
inverse rotation, a-1
must therefore share thesame eigenvector since the rotation axis is thesame (but the angle is opposite).
Rotation: physical meaning
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Therefore we can write:
av = a-1v = v,
and subtract the first two quantities.
(aa-1) v = 0.
The resultant matrix, (a
a-1) clearlyhas zero determinant(required for non-trivialsolution of a set of homogeneous equations).
Forward vs. Reverse Rotation
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Eigenvalue = +1
To prove that (a - I)v = 0 (l= 1):
Multiply by aT: aT(a - I)v = 0(aTa - aT)v = 0
(I- aT
)v
=0.
Add the first and last equations:(a - I)v + (I- aT)v = 0
(a - aT)v = 0.
If a
T
aI, then the last step would not be valid. The last result was already demonstrated.
Orthogonal matrix
property
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n
(a23 a32 ),(a31 a13),(a12 a21)
(a23 a32)2 (a31 a13)
2 (a12 a21)2
One can extract the rotation axis,n,(the only real eigenvector, same asvin previous
slides, associated with the eigenvaluewhosevalue is +1) in terms of the matrix coefficients for
(a - aT
)v
=0
, with a suitable normalization to obtaina unit vector:
Rotation Axis from Matrix
Note the order (very important) of the coefficients in each subtraction;
again, if the matrix represents an active rotation, then the sign is inverted.
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R i A i f M i d
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(aa-1) =
0 a12 a21 a13 a31a21 a12 0 a23 a32a31 a13 a32 a23 0
Given this form of the difference matrix,based ona-1 = aT, the only non-zero vector that
will satisfy(aa-1) n = 0is:
n (a23 a32),(a31 a13),(a12 a21)
(a23 a32 )2 (a31 a13)
2 (a12 a21)2
Rotation Axis from Matrix, contd.
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Another useful relation gives us the magnitude of the
rotation,q, in terms of thetraceof the matrix,aii:
, therefore,
cos q= 0.5 (trace(a) 1).
aii 3cosq (1 cosq)ni2 1 2cosq
Rotation Angle from Matrix
- In numerical calculations, it can happen that tr(a)-1 is either slightly greaterthan 1 or slightly less than -1. Provided that there is no logical error, it isreasonable to truncate the value to +1 or -1 and then apply ACOS.- Note that if you try to construct a rotation of greater than 180 (which isperfectly possible using the formulas given), what will happen when you extractthe axis-angle is that the angle will still be in the range 0-180 but you willrecover the negative of the axis that you started with. This is a limitation of the
rotation matrix (which the quaternion does not share).
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(S ll) A l f
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(Small) Rotation Angle from Matrix
What this shows is that for small angles, it is safer to use a sine-based formula to extract the angle(be careful to include only a12-a21, but not a21-a12). However, this is strictly limited to angles less than90 because the range of ASIN is -/2 to +/2, in contrast to ACOS, which is 0 to , and the formulabelow uses the squares of the coefficients, which means that we lose the sign of the (sine of the)angle. Thus, if you try to use it generally, it can easily happen that the angle returned by ASIN is, infact, -q because the positive and the negative versions of the axis will return the same value.
sinqijkajk
2
2
i
gij
cosq u2 1 cosq uv 1 cosq wsinq uw 1 cosq v sinquv 1 cosq wsinq cosq v2 1 cosq vw 1 cosq usinq
uw 1 cosq v sinq vw 1 cosq usinq cosq w2 1 cosq
gI
0 uv 1 cosq wsinq uw 1 cosq v sinquv 1 cosq wsinq 0 vw 1 cosq usinquw 1 cosq v sinq vw 1 cosq usinq 0
w 2usinq 2v sinq 2wsinq
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R i A l 180
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gij
ij
cos rirj
1 cos
ijkrksin
k1,3
2u2 1 2uv 2uw
2uv 2v2 1 2vw
2uw 2vw 2w2 1
Rotation Angle = 180
A special case is when the rotation,q, is equal to 180
(=). The matrix then takes the special form:
In this special case, the axis is obtained thus:
n a11 1
2
a22 12
a33 12
However, numerically, the standard procedure is surprisingly robust
and, apparently, only fails when the angle is exactly 180.
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tr[a] cos1 cos2 sin1 sin2 cos
sin1 sin2 cos1cos2 cos cos
Trace of the (mis)orientation matri
cos (1 cos)(cos1
cos2
sin1
sin2
)
cos (1 cos) cos 1 2 Thus the cosine, v, of the rotation angle,
vcosq, expressed in terms of the Euler angles:
tr[a] 1
2
cos2 2 cos 1 2 sin2 2
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Is a Rotation a Tensor? (yes!)Recall the definition of a tensor as a quantity
that transforms according to this convention,whereLis an axis transformation,anda is a rotation:
a = LTaL
Since this is a perfectly valid method of
transforming a rotation from one set of axesto another, it follows that an active rotationcan be regarded as a tensor. (Think oftransforming the axes on which the rotation axis
is described.)
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Matrix, Miller Indices
In the following, we recapitulate some results obtained in thediscussion of texture components (where now it should be clearerwhat their mathematical basis actually is).
The general Rotation Matrix, a, can be represented as in thefollowing:
Where the Rows are the direction cosines for [100], [010], and [001]in the sample coordinate system(pole figure).
[100] direction
[010] direction
[001] direction
a11 a12 a13a21 a22 a23
a31 a32 a33
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Matrix, Miller Indices
The columnsrepresent components of three other unit vectors:
Where the Columns are the direction cosines (i.e. hklor uvw) for theRD, TD and Normal directions in the crystal coordinate system.
[uvw]RD TD ND(hkl)
a11 a12 a13
a21 a22 a23
a31 a32 a33
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Compare Matrices
aij Crystal
Sample
b1 t1 n1b2 t2 n2
b3 t3 n3
cos1 cos2
sin1sin2 cossin1 cos2
cos1sin2 cossin2 sin
cos1sin2 sin1cos2 cos
sin1sin2cos1cos2 cos
cos2 sin
sin1 sin cos1sin cos
[uvw] [uvw] (hkl)(hkl)
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Summary
The rules for working with vectors andmatrices, i.e. mathematics, especially with
respect to rotations and transformations ofaxes, has been reviewed.
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Supplemental Slides
[none]