Symmetric Spherical Functions edit page

In several applications spherical functions are symmetric with respect to some symmetry group. Typically examples are inverse pole figures or properties of single crystals. In MTEX one can ensure symmetry properties of spherical functions by defining them as variables of type S2FunHarmonicSym.

Defining a S2FunHarmonic

Definition via symmetrisation

The simplest way to define a symmetric spherical function is through the symmetrisation of an ordinary S2FunHarmonic.

% the original function
sF = S2Fun.smiley;

% the symmetry
ss = specimenSymmetry('222',sF.how2plot);

% attach the symmetry to the function
sFs1 = S2FunHarmonicSym(sF, ss);

% and enforce it
sFs1 = sFs1.symmetrise

plot(sFs1)
sFs1 = S2FunHarmonicSym (y←↑x (222))
  bandwidth: 128
  antipodal: true

Note that symmetry function are by default only plotted within their representative sector, as it is typical for inverse pole figures. In order to visualize the full sphere we have to use the option 'complete'.

plot(sFs1,'complete','upper')

Definition via function handle

If you have a function handle for the function you could create a S2FunHarmonicSym via quadrature. At first lets define a symmetry and a function handle which takes vector3d as an argument and returns double:

f = @(v) 0.1*(v.theta+sin(8*v.x).*sin(8*v.y));
cs = crystalSymmetry('6/m');

Now you can call the quadrature command to get sFs2 of type S2FunHarmonicSym 

sFs2 = S2FunHarmonicSym.quadrature(f, cs)

contour(sFs2,'linewidth',2);
mtexColorMap parula
sFs2 = S2FunHarmonicSym (6/m)
  bandwidth: 128
  antipodal: true