orientations and misorientations, rotations taking coordinates in the frame CS to coordinates in the frame SS; a misorientation is one between two crystal frames
Syntax
ori = orientation(rot, cs, ss)
A rotation between a crystal and a specimen frame.
ori = orientation(a, b, c, d, cs, ss)
The quaternion coefficients.
mori = orientation(rot, cs, cs, antipodal=True)
A misorientation with grain exchange.
ori = orientation(f, points=100)
Points along a fibre.
ori = orientation.byEuler(phi1, Phi, phi2, cs, ss)
Euler angles, and the other constructors.
Description
An orientation is a rotation that carries the two frames it maps between: frameA, the crystal frame CS, from which it takes the coordinates, and frameB, the specimen frame SS or the second crystal frame of a misorientation, to which it takes them. The symmetries of both frames enter every comparison, angle, mean and plot. A frame left out is the triclinic crystal frame, or the default specimen frame. Coefficients are normalised; an improper transform is refused, an orientation is proper.
An orientation given keeps its frames unless others are named; a fibre is sampled at points orientations, an embedding projected back onto the orientations.
Input
rot |
rotation, quaternion or orientation |
a, b, c, d |
the quaternion coefficients, numbers or arrays |
cs |
crystalFrame, the frame mapped from |
ss |
specimenFrame, or the second crystalFrame of a misorientation |
f |
fibre |
Keywords
| keyword | default | description |
|---|---|---|
antipodal |
a misorientation with grain exchange, between two frames of one crystal: the misorientation and its inverse are taken as one | |
dtype |
the floating point type of the coefficients | |
points |
the number of orientations sampled along a fibre |
Output
ori |
orientation |
Properties
frameA |
source frame of the mapping, usually the crystal frame |
frameB |
target frame, usually the specimen or a second crystal frame |
CS, SS |
aliases for frameA and frameB |
antipodal |
grain exchange symmetry for misorientations between the same crystal frame |
notIID |
whether the orientations come from spatially dependent map measurements |
data |
unit quaternion coefficients, with shape ori.shape + (4,) |
phi1, Phi, phi2 |
Bunge Euler angles in radians |
See also
orientation.byEuler, orientation.byAxisAngle, orientation.byMiller, orientation.byMatrix, orientation.map, rotation, crystalFrame, OrientationDefinition, MisorientationTheory
Members
| align | the orientation two frames stand in by their axes, MATLAB’s orientation.align |
| angle | the rotational angle up to symmetry, of the orientation or of the misorientation to other |
| axis | the rotational axis of a misorientation, an orientation or the misorientation between two orientations |
| Bagaryatsky | the Bagaryatsky orientation relationship between cementite and ferrite, (100) || (1-10) and [010] || [111], a misorientation from the cementite frame csT to the ferrite frame csA |
| Bain | the Bain orientation relationship from fcc to bcc, (100) || (100) and [010] || [011], a misorientation from the parent frame csG to the child frame csA |
| brass2 | the brass2 component, Bunge Euler angles (325°, 45°, 0°), in cs and the specimen frame |
| brass | the brass component, Bunge Euler angles (35°, 45°, 0°), in cs and the specimen frame |
| Burgers | the Burgers orientation relationship between bcc and hcp, (110) || (0001) and [-11-1] || [2-1-10], from csParent to csChild |
| byAxisAngle | the orientations of the rotations by omega about the axis v |
| byEuler | orientations defined by Euler angles |
| byHomochoric | the orientations of homochoric vectors |
| byMatrix | the orientations of rotation matrices |
| byMiller | the orientation that takes the crystal plane hkl to z and the crystal direction uvw to x |
| byRodrigues | the orientations of Rodrigues vectors, the axes scaled by tan(omega/2) |
| byScreenAlignment | construct the orientation that makes two frames appear aligned on screen |
| calcAngleDistribution | the distribution of the misorientation angle: of a misorientation density, of a list of misorientations, or of the uniform density of two crystal frames |
| calcBinghamODF | the Bingham distribution estimated from orientations |
| calcCluster | the clusters of directions or orientations: one cluster id per element, counted from zero, and the centre of every cluster |
| calcKernel | the kernel a density estimate of orientations or directions should use, a de la Vallee Poussin kernel of the halfwidth chosen from the sample |
| calcMIndex | the M-index of an ODF by Skemer et al., computed from the ODF instead of a discrete number of orientations |
| calcTensor | the texture average of a single crystal tensor over an ODF, a list of orientations or a map |
| cat | the list of the operands, MATLAB’s [r1, r2] |
| char | the orientation as text: the parallel planes and directions of a misorientation that low indices give exactly, else the Euler angles |
| copper2 | the second copper component, (112)[11-1], Bunge (270, 35, 45) degrees, in cs and the specimen frame |
| copper | (112)[-1-11] is Bunge (90, 35, 45), (112)[11-1] its twin (270, 35, 45) |
| CS | the frame the orientation maps from, frameA: the crystal frame |
| ctranspose | the transpose of the leading axes |
| cube | the cube component, Bunge Euler angles (0°, 0°, 0°), in cs and the specimen frame |
| cubeND22 | the cubeND22 component, Bunge Euler angles (22°, 0°, 0°), in cs and the specimen frame |
| cubeND45 | the cubeND45 component, Bunge Euler angles (45°, 0°, 0°), in cs and the specimen frame |
| cubeRD | the cubeRD component, Bunge Euler angles (0°, 22°, 0°), in cs and the specimen frame |
| dot | the dot product up to symmetry, cos(omega/2) of the misorientation angle omega reduced over the groups of both frames |
| dot_outer | the dot product up to symmetry, cos(omega/2) of the misorientation angle omega reduced over the groups of both frames |
| exp | the rotation a tangent vector at this reference points to, exp(ori, v) of MATLAB |
| export_VPSC | write a map to the file the extension names |
| fibreVolume | the volume of a fibre: the share of an orientation density within a radius of the fibre of the orientations that map the crystal direction h onto the specimen direction r |
| find | the orientations of ori nearest to w up to symmetry, MATLAB’s orientation.find |
| fit | the orientation that maps the directions u best onto the directions v in the least squares sense |
| fundamentalRegion | return the region of symmetry-distinct orientations for the input frames |
| goss | the goss component, Bunge Euler angles (0°, 45°, 0°), in cs and the specimen frame |
| GreningerTrojano | the Greninger-Trojano orientation relationship from fcc to bcc, by the Bunge Euler angles (2.7, 46.6, 7.5) degrees of the inverse, a misorientation from csG to csA |
| id | construct identity rotations between specified reference frames |
| inv | reverse the direction of an orientation mapping |
| invGoss | the invGoss component, Bunge Euler angles (90°, 45°, 0°), in cs and the specimen frame |
| isclose | equal up to the symmetry of the frames within the angular tolerance tol, between the same frames |
| isMisorientation | whether both frames are crystal frames, so the orientation is a misorientation |
| KurdjumovSachs | the Kurdjumov-Sachs orientation relationship from fcc to bcc, (111) || (011) and [-101] || [-1-11], a misorientation from the parent frame csG to the child frame csA |
| load | read orientations from a text table |
| log | the logarithm of an orientation, the tangent vector at a reference pointing to it |
| logm | logm(T) is T.logm(): the matrix logarithm |
| map | the orientation that maps one or two pairs of directions |
| mean | the mean of a list of orientations, with the principal moments and axes of inertia |
| mtimes | |
| nan | construct missing orientations with their reference frames preserved |
| NishiyamaWassermann | the Nishiyama-Wassermann orientation relationship from fcc to bcc, (111) || (011) and [11-2] || [0-11], a misorientation from the parent frame csG to the child frame csA |
| orientation | orientations and misorientations, rotations taking coordinates in the frame CS to coordinates in the frame SS |
| parents | return the distinct parent variants of a parent-to-child relationship |
| Pitsch | the Pitsch orientation relationship from fcc to bcc, (010) || (101) and [101] || [-111], a misorientation from the parent frame csG to the child frame csA |
| PitschSchrader | the Pitsch-Schrader orientation relationship between bcc and hcp, (011) || (0001) and [100] || [2-1-10], a misorientation from csParent to csChild |
| PLage2 | the PLage2 component, Bunge Euler angles (245°, 45°, 0°), in cs and the specimen frame |
| PLage | the PLage component, Bunge Euler angles (65°, 45°, 0°), in cs and the specimen frame |
| plot | plot(x, …) draws x as its kind is drawn |
| plotIPDF | inverse pole figures, one per specimen direction: of an ODF or of a list of orientations |
| plotPDF | pole figures, one per crystal direction: of an ODF, of a list of orientations or of a measured PoleFigure |
| plotSection | plotSection(odf), plotSection(odf, ‘sigma’), plotSection(ori, oS, sections=9) |
| Potter | the Potter orientation relationship between bcc and hcp, (011) || (01-11) and [11-1] || [2-1-10], a misorientation from csParent to csChild |
| project2EulerFR | return Bunge Euler angles of symmetry-equivalent representatives in the plotting box |
| project2FundamentalRegion | the symmetric equivalents in the fundamental region, or those closest to the reference ref (MATLAB’s orientation.project2FundamentalRegion, quaternion.project2FundamentalRegion, rotation.project2FundamentalRegion) |
| QLage2 | the QLage2 component, Bunge Euler angles (245°, 20°, 0°), in cs and the specimen frame |
| QLage3 | the QLage3 component, Bunge Euler angles (115°, 160°, 0°), in cs and the specimen frame |
| QLage4 | the QLage4 component, Bunge Euler angles (295°, 160°, 0°), in cs and the specimen frame |
| QLage | the QLage component, Bunge Euler angles (65°, 20°, 0°), in cs and the specimen frame |
| quiver | arrows: tangent vectors at directions, a vector or axis field on the sphere, arrows at orientations in the sections of the current figure, or directions on a map or a volume |
| quiverSection | arrows on a section plot: of a function on the sphere along one of its circles, or at orientations in the sections of orientation space |
| rand | random orientations |
| round2Miller | the lattice planes and directions of low indices an orientation or a misorientation aligns |
| scatter | orientations as points in a three dimensional space of orientations |
| ShojiNishiyama | the Shoji-Nishiyama orientation relationship from fcc to hcp, (111) || (0001) and [10-1] || [2-1-10], a misorientation from the parent frame csG to the child frame csE |
| SR2 | the SR2 component, Bunge Euler angles (233°, 35°, 63°), in cs and the specimen frame |
| SR3 | the SR3 component, Bunge Euler angles (307°, 35°, 27°), in cs and the specimen frame |
| SR4 | the SR4 component, Bunge Euler angles (127°, 35°, 27°), in cs and the specimen frame |
| SR | the SR component, Bunge Euler angles (53°, 35°, 63°), in cs and the specimen frame |
| SS | the frame the orientation maps to, frameB: the specimen frame, or the second crystal frame of a misorientation |
| symmetrise | all crystallographically equivalent orientations |
| times | |
| transformReferenceFrame | the same orientation described with another reference frame of the crystal |
| unique | the disjoint list of orientations, symmetric equivalents within a tolerance taken as one |
| variants | the variants of a parent to child orientation relationship, and the child or parent orientations they give |
| volume | the share of orientations within a radius of a centre: the integral of f over the ball, int_{angle(g, c) < r} f(g) dg, relative to the integral over all orientations |
Used on
DefinitionAsCoordinateTransform, OrientationEmbeddings, FibreODFs, PoleFigure2ODFGhostCorrection, RotationFibre, SO3FunVectorValued
MATLAB’s orientation_index
source
Citing this page.
This page is part of the documentation of
MTEX, a free and open
source MATLAB toolbox for analyzing and modeling crystallographic textures.
It was written by The MTEX Developers and is published at
https://mtex-toolbox.github.io/orientation_index_py.html.
If you use MTEX, or reuse text or figures from this page, in your research,
please cite
F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63
BibTeX
@article{bachmann2010mtex,
author = {F. Bachmann and R. Hielscher and H. Schaeben},
title = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
journal = {Solid State Phenomena},
volume = {160},
pages = {63-68},
year = {2010},
doi = {10.4028/www.scientific.net/SSP.160.63},
url = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.