Symmetry of Orientation-Dependent Functions

A rotational function can repeat under symmetry operations on either side of its argument. Every SO3Fun therefore stores a right symmetry SRight and a left symmetry SLeft. For an orientation distribution function (ODF), the right side is the crystal symmetry and the left side is the specimen symmetry.

Symmetry is the point group under which the data is invariant. It is not a reference frame. Each stored symmetry is attached to the reference frame of its side, so replacing a symmetry may also change which frame the function reports.

plottingConvention.default('y↑→x');
SO3F = SO3Fun.dubna

cs = SO3F.SRight
ss = SO3F.SLeft

SO3F = SO3FunRBF (Quartz → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 1
 
 
cs = crystalSymmetry (⊙c→a)
 
  mineral        : Quartz           
  color          : lightblue        
  symmetry       : 321              
  elements       : 6                
  a, b, c        : 4.9, 4.9, 5.4    
  reference frame: X||a*, Y||b, Z||c
 
 
ss = triclinic specimenSymmetry (y↑→x)

CS and SS are convenient aliases for SRight and SLeft. The Dubna ODF has quartz crystal symmetry 321 on the right and no specimen symmetry on the left.

Left and right actions

If \(s_L\) belongs to SLeft and \(s_R\) belongs to SRight, a symmetric function satisfies

\[ f(s_L R s_R)=f®. \]

Rotation composition is not commutative, so the two point groups cannot be exchanged. symmetrise constructs the complete orbit \(s_L R s_R\) of an orientation.

ori = orientation.rand(cs,ss);
equivalentOrientations = ori.symmetrise;
equivalentValues = SO3F.eval(equivalentOrientations);
maximumOrbitDifference = ...
  max(abs(equivalentValues - SO3F.eval(ori)))

manualOrbit = ss * ori * cs;
manualOrbitValues = SO3F.eval(manualOrbit);
manualConstructionDifference = ...
  max(abs(sort(equivalentValues(:)) - sort(manualOrbitValues(:))))

maximumOrbitDifference =
   1.2768e-15
manualConstructionDifference =
     0

Both printed differences are at numerical precision. The first verifies that all symmetry-equivalent orientations have the same density. The second verifies that left multiplication by ss and right multiplication by cs construct the same orbit as ori.symmetrise.

Symmetry reduces the plot domain

By default, MTEX plots only one fundamental region. A fundamental region contains one representative from every symmetry-equivalent orbit.

plot(SO3F,'sigma')

The displayed sigma sections stop at the boundary of the quartz fundamental region. This smaller domain avoids showing repeated copies; it does not discard part of the ODF. Every orientation left outside the picture has an equivalent inside it, carrying the same density.

A symmetry label is not a projection

In most SO3Fun representations, the symmetry properties are stored separately from coefficients, centres, or other model parameters. This makes reassignment easy, but it does not mean that arbitrary stored data automatically has the newly claimed invariance.

The effect of reassignment depends on the representation. For an RBF model it changes which symmetry-related kernel copies contribute to an evaluation. For a harmonic model it leaves the Fourier coefficients untouched until the function is explicitly symmetrised.

On an ODF, assign a specimenSymmetry to the left side. A general rotational function may instead describe a relation between two crystal sides, for which an assignment such as the following is meaningful:

SO3F.SLeft = crystalSymmetry('432');

Applying that line to the Dubna ODF would change its physical meaning. It would no longer describe quartz orientations relative to an unsymmetric specimen frame.

Relabel harmonic coefficients

Construct a reproducible real-valued harmonic function without a nontrivial point-group symmetry. Its random coefficients make violations of a proposed twofold symmetry easy to detect.

rng(1)
SO3F2 = SO3FunHarmonic(randn(1000,1));
SO3F2.isReal = true;
coefficientsBefore = SO3F2.fhat;

twoFold = crystalSymmetry('2');
SO3F2.SRight = twoFold;
coefficientChangeAfterRelabelling = ...
  norm(SO3F2.fhat - coefficientsBefore)

coefficientChangeAfterRelabelling =
     0

The zero coefficient change confirms that assignment only relabelled the existing series. Test the claimed invariance at a random orientation and its twofold orbit.

probe = orientation.rand(twoFold,SO3F2.SLeft);
probeOrbit = probe.symmetrise;
valuesBeforeProjection = SO3F2.eval(probeOrbit);
orbitSpreadBeforeProjection = ...
  max(abs(valuesBeforeProjection - valuesBeforeProjection(1)))

plot(SO3F2)

orbitSpreadBeforeProjection =
    4.7063

The nonzero orbit spread shows that the relabelled coefficients do not yet define a twofold-symmetric function. The plot uses the newly labelled fundamental region, so a smaller plot domain is not evidence of actual invariance.

Project onto symmetric functions

SO3FunHarmonic.symmetrise averages the function over its left and right point groups. In coefficient space this projects the Fourier coefficients onto the subspace with the requested invariance.

SO3F2Sym = SO3F2.symmetrise;
relativeCoefficientChange = ...
  norm(SO3F2Sym.fhat - SO3F2.fhat) / norm(SO3F2.fhat)

valuesAfterProjection = SO3F2Sym.eval(probeOrbit);
orbitSpreadAfterProjection = ...
  max(abs(valuesAfterProjection - valuesAfterProjection(1)))

plot(SO3F2Sym)

relativeCoefficientChange =
    0.7446
orbitSpreadAfterProjection =
   6.0396e-14

The coefficients now change, while the orbit spread falls to numerical precision. Features that disagreed between symmetry-related orientations have been averaged. This projection loses their differences, so the original nonsymmetric function cannot be recovered from SO3F2Sym.

plot(SO3F2Sym,'complete')

In the complete plot, every twofold-related position now carries the same value. This visual repetition and the small printed orbit spread test the same property in complementary ways.

Harmonic invariance is encoded directly in the Fourier coefficients. Changing only SRight or SLeft does not encode it; applying symmetrise does.

Convert another representation before projection

Every SO3Fun can be expanded as an SO3FunHarmonic. The constructor uses the quadrature procedure from Quadrature of Orientation-Dependent Functions when the representation offers nothing quicker. The Dubna ODF is an SO3FunRBF and takes its own route, from its centres and kernel. Either way the same explicit projection can then be applied.

SO3F3 = SO3FunHarmonic(SO3F,'bandwidth',14)

SO3F3 = SO3FunHarmonic (Quartz → y↑→x)
  bandwidth: 14
  weight: 1

SO3F3 inherits the left and right symmetries of the Dubna ODF. Because the source is already invariant, its computed coefficients are symmetrised during construction. Calling SO3F3.symmetrise again would therefore leave it unchanged apart from numerical accuracy.

The maths behind symmetrisation

Let \(G_L\) and \(G_R\) be the proper rotations in the left and right point groups. Symmetrisation replaces a function \(f\) by the group average

\[ f_{\mathrm{sym}}(R)=\frac{1}{\lvert G_L\rvert\lvert G_R\rvert} \sum_{s_L\in G_L}\sum_{s_R\in G_R}f(s_L R s_R). \]

Applying any member of either group merely permutes the terms in the sum. The averaged function therefore has the required left and right invariance. Components that are incompatible with the point groups cancel, which explains both the coefficient change and the loss of information.

References

Next

Continue with Convolution to combine rotational functions. The left and right sides introduced here determine whether two functions can be convolved and which symmetries the result inherits.