a region of orientation space bounded by planes in the quaternions, mostly the fundamental region of a pair of frames

Syntax

oR = fundamentalRegion(cs)

The fundamental region of a crystal frame.

oR = fundamentalRegion(cs, cs, antipodal=True)

Of misorientations, with grain exchange.

oR = orientationRegion(N, cs, specimenFrame())

The region bounded by the quaternions N.

Description

The region holds the quaternions q with dot(q, N) <= 0 for every bounding quaternion N; q and -q are the same rotation.

The fundamental region of two frames, which fundamentalRegion builds, is the set of rotations closer to the identity than to any of their symmetric equivalents; projectIntoFundamentalRegion maps orientations into it. Where the two frames share rotations, about a common axis for instance, the planes alone do not separate the equivalents, and the region also restricts the rotation axes to a sector.

Input

`N`	quaternion, or an array (n, 4), the normals of the bounding planes
`frameA`	the first frame, the crystal frame
`frameB`	the second frame, the specimen or a second crystal frame

Keywords

keyword	default	description
`antipodal`		identify a misorientation with its inverse, the grain exchange symmetry
`common`		the rotations, as an array (n, 4), the two frames share
`sector`		sphericalRegion, the sector of rotation axes where the frames share rotations
`axisCenter`		the centre of that sector, an array of three

Output

`oR`	orientationRegion, with oR.CS and oR.SS its two frames, MATLAB’s CS1 and CS2

Members

axisSector	the sector of rotation axes the region allows at a fixed rotation angle: a bound with N = (-sin(w_i/4), cos(w_i/4) a_i) reads for q = (cos(w/2), sin(w/2) v) as dot(v, a_i) <= cot(w/2) tan(w_i/4), a small circle about the axis
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
calcAxisDistribution	the axis distribution of an SO(3) function, an S2 function of the rotation axes symmetric under the rotations the two frames share
checkInside	whether orientations lie inside the region, up to the tolerance, as a boolean array
CS	the first frame of the region, frameA: the crystal frame (MATLAB’s orientationRegion.CS1)
edges	the edges of the region, each a chain of points in axis angle coordinates in radians: every pair of planes meets in a line of Rodrigues space, kept where the other planes allow it
maxAngle	the largest rotation angle inside the region, at a vertex of it in Rodrigues space, or about a given axis
orientationRegion	a region of orientation space bounded by planes in the quaternions, mostly the fundamental region of a pair of frames
plot	a fundamental region of orientation space drawn in axis angle space
rodriguesPlanes	the region in Rodrigues space as A rho <= b: a bound dot(q, N) <= 0 reads n
rotate	rotate(x, rot) is x.rotate(rot), or rot * x for an object without that method
SS	the second frame of the region, frameB: the specimen or second crystal frame (MATLAB’s orientationRegion.CS2)
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

MATLAB’s orientationRegion_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/orientationRegion_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.

The class orientationRegion