Orientation-Dependent Functions

An orientation-dependent function assigns a numerical value to every rotation or crystal orientation. The set of rotations is the rotation group \(SO(3)\), which gives the MTEX class SO3Fun its name.

An important example is the orientation density function (ODF). It assigns a density to each crystal orientation in a specimen. Other examples include the Schmid factor and the Taylor factor as functions of crystal orientation.

A First Orientation-Dependent Function

MTEX represents a scalar function on \(SO(3)\) by an object of type SO3Fun. Its symmetries determine which rotations represent the same argument.

Consider the smallest rotational angle between an orientation and the identity, including its cubic symmetry equivalents. The angle command returns this angle in radians. Dividing by degree makes the function value a number in degrees.

% define cubic crystal symmetry
cs = crystalSymmetry('432');

% wrap the angle formula in an SO3Fun
SO3F = SO3FunHandle(@(ori) angle(ori) ./ degree,cs)

SO3F = SO3FunHandle (432 → y↓→x)
  eval: @(ori)angle(ori)./degree

Evaluate the Function

SO3FunHandle turns an anonymous function into an SO3Fun. The variable SO3F now stores the formula together with its symmetry.

Use eval to evaluate it at one or many orientations. Here the input is fixed so that the result is reproducible.

ori = orientation.byEuler(20*degree,30*degree,10*degree,cs)
angleInDegrees = SO3F.eval(ori)

ori = orientation (432 → y↓→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
    20   30   10
 
angleInDegrees =
   42.1812

The returned scalar is the smallest angle, in degrees, among the symmetrically equivalent representatives of ori. The next page develops this construction and the other ways to define an SO3Fun.

Plot Euler-Angle Sections

A scalar function on \(SO(3)\) depends on three coordinates. MTEX can show it as a stack of sections at fixed third Euler angle \(\varphi_2\).

plotSection(SO3F,'sections',4)
mtexColorbar

Each panel covers the first two Euler angles at one value of \(\varphi_2\). The colour changes within and between panels because the smallest symmetry-reduced angle depends on all three Euler angles.

Plot Axis--Angle Sections

The same function is especially simple in axis--angle coordinates. Each section fixes the rotational angle and varies the rotational axis.

constantContourWarning = warning('off','MATLAB:contour:ConstantData');
plotSection(SO3F,'axisAngle',(15:15:60)*degree,'upper')
warning(constantContourWarning)
mtexColorbar
mtexColorMap parula

Every panel has one colour because SO3F returns the angle that labels that panel. The two section plots show the same function in different coordinates; neither changes the underlying data.

Analyse an Orientation-Dependent Function

The common SO3Fun interface provides arithmetic, integration, differentiation and searches for extrema. For this angle function, a local maximum is an orientation farthest from a cubic symmetry equivalent of the identity.

The max command below requests up to ten distinct local maxima. The accuracy option controls the final angular search tolerance.

[value,oriMax] = max(SO3F,'numLocal',10,'accuracy',0.001*degree)

value =
   62.7994
   62.7993
   62.7993
   62.7993
   62.7993
   62.7993
 
oriMax = orientation (432 → y↓→x)
  size: 6 × 1
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  90.0002 45.0001     225
      180      45     225
  270.001 44.9999 44.9994
  225.001 44.9998 89.9993
  125.264 59.9998 215.265
  134.999 44.9999     180

The calculation finds exactly six symmetrically inequivalent maxima. Their function values are about 62.799 degrees, and their positions are the vertices of the fundamental region in orientation space.

close all
color = ind2color(repmat(1:length(oriMax),numSym(cs),1));
plot(oriMax.symmetrise,color,'axisAngle','filled','markerSize',20,...
  'restrict2FundamentalRegion')

The six colours distinguish the six maxima. The markers lie on the outer vertices because those points have the greatest possible distance from the identity after cubic symmetry has been taken into account.

Representations of Orientation-Dependent Functions

MTEX can store a function in several ways. The representation controls how the function is constructed and how expensive an operation is, but all representations share the SO3Fun interface.

Thus functions with different internal representations can be added, multiplied, averaged, integrated or differentiated through the same API.

Related Function Types

SO3VectorField assigns a vector instead of a scalar to each rotation. SO3Kernel represents a radial function whose value depends only on rotational angle.

References

Next

Continue with Defining Orientation-Dependent Functions to construct functions from formulas, harmonic coefficients and sampled values.