an antipodally symmetric Bingham density on the unit sphere

Syntax

f = S2FunBingham([-20, -10, 0])

Concentrated about the z axis.

f = S2FunBingham(Z, a)

Concentrations along supplied principal axes.

f = S2FunBingham.fit(v)

Estimate the distribution from measured axes.

Description

For a unit direction v the density is proportional to exp(sum_i Z[i] * dot(v, a[i])**2). The normalisation gives mean one over the sphere, or integral 4*pi with respect to surface area. Directions v and -v have equal density. Equal parameters give the uniform density; one largest parameter gives an axial peak, and two equal largest parameters give a girdle. Adding a common constant to Z leaves the density unchanged.

If a frame is supplied, evaluation averages over its symmetry-equivalent axes. Parameters need not be sorted: each Z[i] belongs to a[i]. The fit method returns them in ascending order.

Input

`Z`	three concentration parameters
`a`	three orthonormal vector3d axes, or a 3-by-3 matrix with axes as columns; None selects the Cartesian axes; a referenceFrame can be supplied here instead
`frame`	referenceFrame of the directions and their symmetry, by default that of a

Output

`f`	S2FunBingham; evaluate by f(v) or f.eval(v)

Properties

`Z`	concentration along each principal axis
`a`	three principal axes as vector3d
`frame`	coordinate frame and symmetry
`antipodal`	True: the density describes unoriented axes

Members

a	the three orthonormal principal axes as vector3d, with a[i] paired with Z[i]
bandwidth	the bandwidth of the harmonic expansion, the preference maxS2Bandwidth
CS	the frame the distribution lives in, its symmetry
eval	the values of the density at directions, the mean over the symmetric copies of its axes
fit	the Bingham distribution fitted to directions or axes, with the confidence ellipse of its modal axis
mean	the mean value over the sphere, one: the distribution is normalised
rotate	the distribution turned by a rotation, its axes and its frame
S2FunBingham	an antipodally symmetric Bingham density on the unit sphere
textureIndex	the texture index of a function on SO(3) or on the sphere, the squared L2 norm of its harmonic expansion

Used on

S2Bingham

MATLAB’s S2FunBingham_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/S2FunBingham_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 S2FunBingham