A method of vector3d.

weights that integrate a function from samples at the directions, summing to one; MATLAB’s calcQuadratureWeights

Syntax

w = calcQuadratureWeights(r)

A regular grid exactly, scattered points by Voronoi areas.

w = calcQuadratureWeights(r, method='harmonic')

The least norm correction that integrates every harmonic the count allows.

Description

On a regular grid of rings, a polar ladder from the pole in equal steps, the weights are the Clenshaw-Curtis weights of the rings, shared by their points. They are exact on the sphere and, with the equator ring halved, on a hemisphere.

For scattered directions the method chooses:

  • ‘voronoi’, the default and MATLAB’s only method: the areas of the spherical Voronoi cells, an axis list counted on both hemispheres, duplicates sharing a cell. No cell counts more than twice the 80 % quantile of the areas, so isolated points do not dominate.
  • ‘harmonic’: equal weights corrected by the least change that integrates every spherical harmonic the number of points allows.

Where neither works, as for fewer than four distinct directions, the weights are equal.

Input

r vector3d

Keywords

keyword default description
method 'voronoi' ‘voronoi’ or ‘harmonic’, for scattered directions

Output

w ndarray of the shape of r, summing to one

See also

PoleFigure.mean, S2FunHarmonic.quadrature

MATLAB’s vector3d.calcQuadratureWeights

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/vector3d.calcQuadratureWeights_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.