Crystal Orientations (The Class orientation)
This sections describes the class orientation and gives an overview how to work with crystal orientation in MTEX.
Class Description
In texture analysis crystal orientations are used to describe the alignment of the crystals within the specimen. A crystal orientation is defined as the rotation that maps the specimen coordinate system onto the crystal coordinate system. Since, the crystal coordinate system and the specimen coordinate system are well defined only up to crystal symmetry and specimen symmetry, a orientation is in general represented by a class of crystallographically equivalent rotations. In MTEX the class orientation is an inheritant of the class rotation. In particular, every function that is defined for a rotation is also available for a orientation.
Defining a Crystal Orientation
In order to define a crystal orientation one has to define crystal and specimen symmetry first.
cs = crystalSymmetry('cubic'); ss = specimenSymmetry('orthorhombic');
Now a crystal orientation to a certain rotation
rot = rotation('Euler',30*degree,50*degree,10*degree);
is defined by
o = orientation(rot,cs,ss)
o = orientation size: 1 x 1 crystal symmetry : m-3m specimen symmetry: mmm Bunge Euler angles in degree phi1 Phi phi2 Inv. 30 50 10 0
In order to streamline the definition the arguments to define the rotation can be directly pass to define a orientation:
o = orientation('Euler',30*degree,50*degree,10*degree,cs,ss)
o = orientation size: 1 x 1 crystal symmetry : m-3m specimen symmetry: mmm Bunge Euler angles in degree phi1 Phi phi2 Inv. 30 50 10 0
Accordingly parameterisations of rotations are also available for orientations
- Bunge Euler Angle Convention
- Matthies Euler Angle Convention
- Axis angle parametrisation
- Fibre of orientations
- Four vectors defining a rotation
- 3 times 3 matrix
- quaternion
Have a look at rotation help page for more details. Beside these parameterisations for rotations there are also some parameterisations which are unique for orientations
Miller indice
There is also a Miller indice convention for defining crystal orientations.
o = orientation('Miller',[1 0 0],[0 1 1],cs,ss)
o = orientation size: 1 x 1 crystal symmetry : m-3m specimen symmetry: mmm Bunge Euler angles in degree phi1 Phi phi2 Inv. 135 90 90 0
Predifined Orientations
In the MTEX there is a list of predefined orientations:
o = orientation('goss',cs,ss)
o = orientation size: 1 x 1 crystal symmetry : m-3m specimen symmetry: mmm Bunge Euler angles in degree phi1 Phi phi2 Inv. 0 45 0 0
Rotating Crystal Directions onto Specimen Directions
Let
h = Miller(1,0,0,cs)
h = Miller size: 1 x 1 symmetry: m-3m h 1 k 0 l 0
be a certain crystal direction and
o = orientation('Euler',90*degree,90*degree,0*degree,cs,ss)
o = orientation size: 1 x 1 crystal symmetry : m-3m specimen symmetry: mmm Bunge Euler angles in degree phi1 Phi phi2 Inv. 90 90 0 0
a crystal orientation. Then the alignment of this crystal direction with respect to the specimen coordinate system can be computed via
r = o * h
r = vector3d size: 1 x 1 x y z 0 1 0
Conversely the crystal direction that is mapped onto a certain specimen direction can be computed via the backslash operator
o \ r
ans = Miller size: 1 x 1 symmetry: m-3m h 1 k 0 l 0
Concatenating Rotations
Let
o = orientation('Euler',90*degree,0,0,cs,ss); rot = rotation('Euler',0,60*degree,0);
be a crystal orientation and a rotation of the specimen coordinate system. Then the orientation of the crystal with respect to the rotated specimen coordinate system calculates by
o1 = rot * o
o1 = orientation size: 1 x 1 crystal symmetry : m-3m specimen symmetry: 1 Bunge Euler angles in degree phi1 Phi phi2 Inv. 0 60 90 0
Then the class of rotations crystallographically equivalent to o can be computed in two way. Either by using the command symmetrise
symmetrise(o)
ans = orientation size: 48 x 1 crystal symmetry : m-3m specimen symmetry: mmm
or by using multiplication
ss * o * cs
ans = rotation size: 8 x 48
Caclulating Missorientations
Let cs and ss be crystal and specimen symmetry and o1 and o2 two crystal orientations. Then one can ask for the missorientation between both orientations. This missorientation can be calculated by the function angle.
angle(o,o1) / degree
ans = 30.0000
This missorientation angle is in general smaller then the missorientation without crystal symmetry which can be computed via
angle(rotation(o),rotation(o1)) /degree
ans = 60.0000
Calculating with Orientations and Rotations
Beside the standard linear algebra operations there are also the following functions available in MTEX. Then rotational angle and the axis of rotation can be computed via then commands angle(o) and axis(o)
angle(o1)/degree axis(o1)
ans = 30.0000 ans = Miller size: 1 x 1 symmetry: m-3m h -1 k 0 l 0
The inverse orientation to o you get from the command inv(q)
inv(o1)
ans = inverse orientation size: 1 x 1 specimen symmetry: 1 crystal symmetry : m-3m Bunge Euler angles in degree phi1 Phi phi2 Inv. 90 60 180 0
Conversion into Euler Angles and Rodrigues Parametrisation
There are methods to transform quaternion in almost any other parameterization of rotations as they are:
- Euler(o) in Euler angle
- Rodrigues(o) % in Rodrigues parameter
[phi1,Phi,phi2] = Euler(o)
phi1 = 1.5708 Phi = 0 phi2 = 0
Plotting Orientations
The plot function allows you to visualize an quaternion by plotting how the standard basis x,y,z transforms under the rotation.
plot(o1)
Do something fancy here.

Complete Function list
BCV | biased cross validation |
KLCV | Kullback Leibler cross validation for optimal kernel estimation |
LSCV | least squares cross valiadation |
angle | calculates rotational angle between orientations |
axis | rotational axis of the orientation or misorientation |
bingham_test | bingham test for spherical/prolat/oblat case |
calcAngleDistribution | calculate angle distribution |
calcBinghamODF | calculate ODF from individuel orientations via kernel density estimation |
calcFourierODF | calculate ODF from individuel orientations via kernel density estimation |
calcKernel | compute an optimal kernel function for ODF estimation |
calcKernelODF | calculate ODF from individuel orientations via kernel density estimation |
calcODF | computes an ODF from individuel orientations |
calcTensor | compute the average tensor for a vector of orientations |
checkFundamentalRegion | checks whether a orientation sits within the fundamental region |
crossCorrelation | computes the cross correlation for the kernel density estimator |
display | standart output |
dot | compute minimum dot(o1,o2) modulo symmetry |
dot_outer | |
fibreVolume | ratio of orientations close to a certain fibre |
getFundamentalRegion | projects orientations to a fundamental region |
inv | erse of an orientation |
isMisorientation | check whether o is a misorientation |
ldivide | o .\ v |
mean | of a list of orientations, principle axes and moments of inertia |
mldivide | o \ v |
mtimes | orientation times Miller and orientation times orientation |
niceEuler | orientation to euler angle |
orientation | class representing orientations |
plot | annotate a orientation to an existing plot |
plotAngleDistribution | plot the angle distribution |
plotAxisDistribution | plot uncorrelated axis distribution |
plotIPDF | plot orientations into inverse pole figures |
plotODF | Plot EBSD data at ODF sections |
plotPDF | plot orientations into pole figures |
project2EulerFR | projects orientation to a fundamental region |
project2FundamentalRegion | projects orientation to a fundamental region |
project2ODFsection | project orientation to ODF sections used by plotODF |
qqplot | quantilequantile of misorientation angle against random angular |
scatter | plots ebsd data as scatter plot |
sphereVolume | ratio of orientations with a certain orientation |
symmetrise | all crystallographically equivalent orientations |
times | vec = ori .* Miller |
transformReferenceFrame | only applicable for crystal symmetry |
unique | disjoint list of quaternions |
volume | ratio of orientations with a certain orientation |
MTEX 4.0.10 |