a vector field on SO(3) whose three components are series of Wigner functions, in the left or the right tangent space

Syntax

SO3VF = SO3VectorFieldHarmonic(fhat, cs, ss, 'left')

From a coefficient matrix.

SO3VF = SO3VectorFieldHarmonic(F, 'right')

Another field in the right tangent space.

SO3VF = SO3VectorFieldHarmonic(F, bandwidth=8)

Another field cut to a bandwidth.

SO3VF = SO3VectorFieldHarmonic(t, bandwidth=8)

Through tangent vectors t at orientations.

SO3VF = SO3VectorFieldHarmonic([F1, F2, F3])

From its three component functions.

SO3VF = SO3VectorFieldHarmonic(lambda ori: F(ori), cs, ss, bandwidth=8)

A callable by quadrature.

Description

A field in the left tangent space returns its tangent vectors in the specimen frame, one in the right tangent space in the crystal frame; ‘left’ unless named. Another field given is converted to the side named.

The field is built from:

  • a coefficient matrix: the components as columns of MTEX’s coefficients, made equivariant under the frames unless symmetrise=False.
  • tangent vectors t at reference orientations: summed with the weights against the Wigner functions, a quadrature only for the weights of a quadrature rule.
  • a callable of orientations: sampled on a quadrature grid of the bandwidth, by default the preference defaultSO3Bandwidth.
  • three component functions: their coefficients, cut to the largest bandwidth, and their frames unless cs and ss are given.

Input

fhat (deg2dim(L + 1), 3) array, an SO3VectorField, an SO3TangentVector, three SO3FunHarmonic or a callable of orientations
cs, ss crystalFrame and specimenFrame

Keywords

keyword default description
bandwidth   the largest degree of the components
weights   the weights of the tangent vectors, equal unless given
symmetrise   make a coefficient matrix equivariant under the frames’ symmetries

Output

SO3VF SO3VectorFieldHarmonic

See also

SO3VectorFieldRBF, grad, antiderivative, mapField, SO3TangentVector, SO3FunVectorField

Members

antiderivative the function of mean zero whose gradient the field is
antiderivative the function of mean zero whose gradient the field is
approximate MATLAB’s approximate: the quadrature of a callable or of values at orientations
bandwidth the bandwidth, the largest degree of the component expansions
component one component as a function on the component frames
cross the pointwise cross product with another field, a field of the same side
cross the pointwise cross product with another field, a field of the same side
curl the curl, a field of the same side
curl the curl, a field of the same side
div the divergence, a function on SO(3)
div the divergence, a function on SO(3)
dot the inner product with a constant vector, exact, a function on the component frames
dot the inner product with a constant vector, exact, a function on the component frames
eval the tangent vectors of the field at orientations
fhat the coefficients of the components in MTEX’s basis, one column per component
frameA the frame the orientations map from, the crystal frame
frameB the frame the orientations map to, the specimen or a second crystal frame
interpolate the function in harmonic form fitted by least squares to values at nodes: on SO(3) at orientations, on the sphere at directions, a vector field for vector values or tangent vectors
inv the push-forward under R -> R^-1: a left vector v at R becomes the right vector -v at R^-1, so inv(grad f) = grad(inv f)
isclose whether two fields have the same coefficients on one side up to the relative tolerance rtol
left the field with its values in the left tangent space, as a harmonic field
left the field with its values in the left tangent space, as a harmonic field
mean the mean over SO(3), and the length of the vectors as a function, MATLAB’s norm
mean the mean over SO(3), and the length of the vectors as a function, MATLAB’s norm
minus  
mrdivide  
mtimes  
norm the length of the vectors as a function on SO(3), the same as abs
norm the length of the vectors as a function on SO(3), the same as abs
normalize the field of unit vectors in the directions of the field, by quadrature at twice the bandwidth
normalize the field of unit vectors in the directions of the field, by quadrature at twice the bandwidth
plot plot(x, …) draws x as its kind is drawn
plus  
plus  
quadrature MATLAB’s SO3VectorFieldHarmonic.quadrature(fun, cs): the field of a callable, cut to the last degree that carries more than rounding, as a polynomial field is
quiver3 arrows in space: tangent vectors in orientation space, a field on the sphere or SO(3), or arrows at positions
quiver arrows: tangent vectors at directions, a vector or axis field on the sphere, arrows at orientations in the sections of the current figure, or directions on a map or a volume
quiverSection arrows on a section plot: of a function on the sphere along one of its circles, or at orientations in the sections of orientation space
rdivide  
rdivide  
right the field with its values in the right tangent space, as a harmonic field
right the field with its values in the right tangent space, as a harmonic field
rotate F’(R) = q F(q^-1 R) for a rotation of the specimen acting on a left field, F(q^-1 R) on a right one
rotate F’(R) = q F(q^-1 R) for a rotation of the specimen acting on a left field, F(q^-1 R) on a right one
SO3VectorField a vector field on SO(3) whose three components are series of Wigner functions, in the left or the right tangent space
SO3VectorFieldHarmonic a vector field on SO(3) whose three components are series of Wigner functions, in the left or the right tangent space
symmetrise the projection onto the equivariant fields of the frames: the invariant side one component at a time, the equivariant side F -> s F(s^-1 R) on the left, F -> s F(R s) on the right, by the Kronecker products
times  
times  
truncate the field with its coefficients cut or padded to the bandwidth L
uminus  
uminus  

Used on

SO3FunVectorField

MATLAB’s SO3VectorField_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/SO3VectorField_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.