Every SO3Fun
has a left and a right symmetry. For further information on symmetries look at crystal symmetries, specimen symmetries and, Quasi symmetries.
SO3F = SO3Fun.dubna
cs = SO3F.SRight
ss = SO3F.SLeft
SO3F = SO3FunRBF (Quartz → xyz)
<strong>multimodal components</strong>
kernel: de la Vallee Poussin, halfwidth 5°
center: 19848 orientations, resolution: 5°
weight: 1
cs = crystalSymmetry
mineral : Quartz
symmetry : 321
elements : 6
a, b, c : 4.9, 4.9, 5.4
reference frame: X||a*, Y||b, Z||c*
ss = triclinic specimenSymmetry
The function values of SO3F
are equal at symmetric nodes. Since the composition of rotations is not commutative there exists a left and right symmetry.
ori = orientation.rand(cs,ss);
SO3F.eval(ori.symmetrise).'
SO3F.eval(ss*ori*cs)
ans =
0.0266 0.0266 0.0266 0.0266 0.0266 0.0266
ans =
0.0266 0.0266 0.0266 0.0266 0.0266 0.0266
The symmetries have, for example, an influence on the plot domain.
plot(SO3F,'sigma')

- Note that only the important part with respect to the symmetry is plotted
- you can plot the full rotation group using the argument
'complete'
In most subclasses of SO3Fun
the symmetries are independent from the rest of variables of the function. So one can change them very easy and only effects the function values.
SO3F.SLeft = specimenSymmetry('432')
SO3F = SO3FunRBF (Quartz → xyz (432))
<strong>multimodal components</strong>
kernel: de la Vallee Poussin, halfwidth 5°
center: 19848 orientations
The class SO3FunHarmonic
describes an rotational function by the Fourier coefficients of its harmonic series. Here it is possible to get the symmetries directly from the fourier coefficients.
Similary if we want to change the symmetry of a function it is not enough to change it. We also have to symmetries this function.
SO3F2 = SO3FunHarmonic(rand(1e3,1))
SO3F2.fhat(1:10)
SO3F2 = SO3FunHarmonic (xyz → xyz)
isReal: false
bandwidth: 9
weight: 0.66
ans =
0.6594
0.1356
0.8415
0.1153
0.9864
0.9954
0.2812
0.7690
0.2734
0.8753
plot(SO3F2)

Changing the symmetry has no effect on the Fourier coefficients. No we are only plotting the given function on some fundamental region.
SO3F2.SRight = crystalSymmetry('2')
SO3F2.fhat(1:10)
SO3F2 = SO3FunHarmonic (121 → xyz)
isReal: false
bandwidth: 9
weight: 0.66
ans =
0.6594
0.1356
0.8415
0.1153
0.9864
0.9954
0.2812
0.7690
0.2734
0.8753
plot(SO3F2)

Symmetrizing the Fourier coefficients transforms the coefficients. So we symmetrize the function and it is no longer possible to go back to the non symmetrized function from before.
SO3F2 = SO3F2.symmetrise
SO3F2.fhat(1:10)
SO3F2 = SO3FunHarmonic (121 → xyz)
isReal: false
bandwidth: 9
weight: 0.66
ans =
0.6594 - 0.0000i
0.0101 + 0.0000i
0.0000 + 0.0000i
-0.0101 - 0.0000i
0.3526 + 0.0000i
0.0000 + 0.0000i
-0.3526 - 0.0000i
-0.0531 + 0.0000i
0.0000 + 0.0000i
0.0531 - 0.0000i
plot(SO3F2)

Now the function is symmetrised on the full rotation group.
plot(SO3F2,'complete')

Note that you can expand every SO3Fun
to an SO3FunHarmonic
SO3F3 = SO3FunHarmonic(SO3F)
SO3F3 = SO3FunHarmonic (Quartz → xyz (432))
bandwidth: 48
weight: 1
and do the same as before.