the quadrature grid on SO(3) of a bandwidth: the nodes and weights that integrate every harmonic function up to that degree exactly

Syntax

G = quadratureSO3Grid(L)

Clenshaw-Curtis in beta, no symmetry.

G = quadratureSO3Grid(bandwidthFor(L, cs), cs, ss, 'GaussLegendre')

Gauss-Legendre, reduced by the frames.

G = quadratureSO3Grid(15, rA=4, rB=2)

Reduced by fourfold and twofold folds about z.

ori, w = G.nodes(), G.weights

The nodes and their weights.

Description

The grid integrates exactly every function of harmonic degree up to the bandwidth, which is what SO3FunHarmonic.quadrature needs to compute the coefficients of a function from its values. The weights sum to one, the Haar measure.

The Euler angles alpha and gamma are equispaced, 2L + 2 of each; the scheme chooses the rule for beta:

‘ClenshawCurtis’, the default: 2L + 1 nodes.
‘GaussLegendre’: L + 1 nodes, about half as many for the same degree.

The rotations about z of the frames shorten the grid: the crystal fold rA the gamma range and the specimen fold rB the alpha range. Both must divide 2L + 2; bandwidthFor(L, cs, ss) gives the next bandwidth that they divide, and any other is refused.

Unlike MATLAB’s, the grid is not itself an orientation: G.nodes() gives the orientations, beta fastest, then alpha, then gamma, and G.weights their weights.

Input

`bandwidth`	int, the degree up to which the rule is exact
`CS, SS`	crystalFrame and specimenFrame (positional, optional)
`scheme`	‘ClenshawCurtis’ (default) or ‘GaussLegendre’ (positional, optional)

Keywords

keyword	default	description
`rA`	`None`	the fold about z on the crystal side, without frames
`rB`	`None`	the fold about z on the specimen side, without frames
`dtype`	`np.float64`	the floating point type of the angles and weights

Output

`G`	quadratureSO3Grid

Used on

SO3FunQuadrature

MATLAB’s quadratureSO3Grid.quadratureSO3Grid

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/quadratureSO3Grid.quadratureSO3Grid_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.

quadratureSO3Grid.quadratureSO3Grid