nodes and surface-area weights for integrating functions on the unit sphere

Syntax

grid = quadratureS2Grid(16)
grid = quadratureS2Grid(16, 'GaussLegendre')
integral = np.sum(grid.weights * fun(grid.nodes()))

Description

Equally spaced azimuths are combined with a polar quadrature rule. The default is Clenshaw–Curtis; Gauss–Legendre is also supported. Weights sum to 4*pi, so divide a weighted integral by 4*pi for the spherical mean. Flattened nodes run fastest over the polar-angle axis and then over azimuth, matching grid.gridShape = (numRho, numTheta).

Input

`bandwidth`	nonnegative integer harmonic bandwidth
`args`	optional reference frame and quadrature scheme, ‘ClenshawCurtis’ or ‘GaussLegendre’
`dtype`	floating-point type for coordinates and weights; default np.float64

Output

grid quadratureS2Grid, also usable as vector3d directions

Properties

`weights`	surface quadrature weights summing to 4*pi
`thetaGrid`	polar node angles in radians
`rhoGrid`	azimuth node angles in radians
`gridShape`	azimuth and polar node counts

Members

gridShape	the shape of the nodes over the axes, rho by theta
nodes	the nodes as a plain list of directions
quadratureS2Grid	nodes and surface-area weights for integrating functions on the unit sphere
subsref
weights	the weights integrate to 4 pi, the surface measure of the sphere

Used on

S2FunQuadrature

MATLAB’s quadratureS2Grid_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/quadratureS2Grid_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 quadratureS2Grid