This sections describes the class **orientation** and gives an overview how to work with crystal orientation in MTEX.

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, an orientation is in general represented by a class of crystallographically equivalent rotations. In MTEX the class
**orientation** is an inheritance of the class rotation. In particular, every function that is defined for a rotation is also available for an 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. Besides these parameterisations for rotations there are also some parameterisations which are unique for orientations

**Miller indices**

There is also a Miller indices 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

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

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 ways. Either by using the command symmetrise

symmetrise(o)

ans = orientation size: 384 x 1 crystal symmetry : m-3m specimen symmetry: mmm

or by using multiplication

ss * o * cs

ans = orientation size: 8 x 48 crystal symmetry : m-3m specimen symmetry: mmm

Let cs and ss be crystal and specimen symmetry and o1 and o2 two crystal orientations. Then one can ask for the misorientation between both orientations. This misorientation can be calculated by the function angle.

angle(rot * o1,o1) / degree

ans = 30.0000

This misorientation angle is, in general, smaller than the misorientation without crystal symmetry which can be computed via

angle(rotation(o),rotation(o1)) /degree

ans = 60.0000

Besides 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: 1, X||a, Y||b, Z||c 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

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

The plot function allows you to visualize an orientation in axis angle space in relation to its fundamental region.

oR = fundamentalRegion(o1.CS,o1.SS) plot(oR) hold on % plot the orientation as it is plot(o1,'markercolor','b','markerSize',10) % plot the orientation within the fundamental zone plot(o1.project2FundamentalRegion,'markercolor','r','markerSize',10) hold off

oR = orientationRegion crystal symmetry: 432 face normales: 1 x 14 vertices: 1 x 24

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 an misorientation or two orientations |

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 |

calcInvTaylor | Taylor factor from orientation gradient |

calcKernel | compute an optimal kernel function for ODF estimation |

calcKernelODF | calculate ODF from individuel orientations via kernel density estimation |

calcMDF | computes an MDF from individuel orientations or misorientations |

calcMIndex | TODO!!! |

calcODF | computes an ODF from individuel orientations |

calcTensor | compute the average tensor for a vector of orientations |

cluster | sort orientations into clusters |

crossCorrelation | computes the cross correlation for the kernel density estimator |

ctranspose | inverse orientation |

display | standart output |

dot | compute minimum dot(o1,o2) modulo symmetry |

dot_outer | |

export_VPSC | export individual orientations to the VPSC format |

fibreVolume | ratio of orientations close to a certain fibre |

getFundamentalRegion | projects orientations to a fundamental region |

isMisorientation | check whether o is a misorientation |

ldivide | o .\ v |

log | the inverse of the exponential map |

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 |

plotIPDF | plot orientations into inverse pole figures |

plotPDF | plot orientations into pole figures |

plotSection | plot orientations to ODF sections |

power | ori.^n |

project2EulerFR | projects orientation to a fundamental region |

project2FundamentalRegion | projects orientation to a fundamental region |

qqplot | quantilequantile of misorientation angle against random angular |

rVariant | calculates rotational angle between orientations |

round2Miller | find h1,h2,d1,d2 such that ori * h1 = h2 and ori*d1=d2 |

scatter | plots orientations in 3d |

sphereVolume | ratio of orientations with a certain orientation |

symmetrise | all crystallographically equivalent orientations |

times | vec = ori .* Miller |

transformReferenceFrame | change reference frame of an orientation |

unique | disjoint list of quaternions |

volume | ratio of orientations with a certain orientation |

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