Embeddings of Orientations

An orientation embedding represents each physical orientation by a unique tensor in a Euclidean space. Symmetrically equivalent rotations therefore have the same embedding. This makes ordinary linear operations available without first choosing one rotation from each equivalence class.

This page assumes the symmetry equivalence developed in Orientation Symmetry and the representative selection explained in Fundamental Regions. The misorientation angle supplies the intrinsic distance used below.

Why Fundamental-Region Coordinates Jump

A rotation matrix is a tensorial representation of a rotation, but it is not a unique representation of an orientation with crystal symmetry. Restricting rotations to a fundamental region selects one representative from each equivalence class. Near the boundary, however, two nearby orientations can be assigned representatives on opposite sides of the region. Their coordinate vectors are then far apart even though their misorientation angle is small.

The following experiment compares three distances for random pairs of cubic orientations.

% cubic proper rotation symmetry
cs = crystalSymmetry('432');

% select representatives in the fundamental region
ori1 = project2FundamentalRegion(orientation.rand(20000,cs));
ori2 = project2FundamentalRegion(orientation.rand(20000,cs));

% intrinsic misorientation angle in degrees
omega = angle(ori1,ori2) ./ degree;

% Euclidean distances between matrix and Rodrigues representatives
distMat = norm(tensor(ori1) - tensor(ori2));
distRV = norm(Rodrigues(ori1) - Rodrigues(ori2));

figure('position',[200 200 1200 400])
subplot(1,3,1)
scatter(omega,distMat,3,'.')
xlabel('\(\omega(\mathtt{ori}_1,\mathtt{ori}_2)\) in degrees',...
  'Interpreter','latex')
ylabel('\(\Vert\mathtt{tensor(ori_1)}-\mathtt{tensor(ori_2)}\Vert_2\)',...
  'Interpreter','latex')
title('rotation matrices')

subplot(1,3,2)
scatter(omega,distRV,3,'.')
xlabel('\(\omega(\mathtt{ori}_1,\mathtt{ori}_2)\) in degrees',...
  'Interpreter','latex')
ylabel('\(\Vert\mathtt{R(ori_1)}-\mathtt{R(ori_2)}\Vert_2\)',...
  'Interpreter','latex')
title('Rodrigues vectors')

subplot(1,3,3)
scatter(distMat,distRV,3,'.')
xlabel('\(\Vert\mathtt{tensor(ori_1)}-\mathtt{tensor(ori_2)}\Vert_2\)',...
  'Interpreter','latex')
ylabel('\(\Vert\mathtt{R(ori_1)}-\mathtt{R(ori_2)}\Vert_2\)',...
  'Interpreter','latex')
title('two coordinate distances')

In the first two panels, points near the left edge can still have a large coordinate distance. The third panel shows that matrix and Rodrigues Frank coordinates jump differently. Neither Euclidean coordinate distance is the geometry of orientation space.

This is why averaging fundamental-region coordinates is unsafe. Consider the orientations with Bunge Euler angles \((44^{\circ},0^{\circ},0^{\circ})\) and \((46^{\circ},0^{\circ},0^{\circ})\).

ori = project2FundamentalRegion(...
  orientation.byEuler([44 46]*degree,0,0,cs));

% average the selected Rodrigues representatives
naiveMean = orientation.byRodrigues(mean(ori.Rodrigues),cs)

naiveMean = orientation (432 → y↓→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
     0    0    0

The displayed result is the identity orientation. It is \(45^{\circ}\) from the expected mean because the two selected Rodrigues vectors lie on opposite sides of a fundamental-region boundary. In contrast, mean handles crystal symmetry.

intrinsicMean = mean(ori)

intrinsicMean = orientation (432 → y↓→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
    45    0    0

The symmetry-aware mean is the physical orientation represented by \((45^{\circ},0^{\circ},0^{\circ})\). More generally, this coordinate jump affects any vector-space method applied directly to fundamental-region matrices, Rodrigues vectors, or Euler angles.

Constructing an Embedding

The embedding class replaces a chosen coordinate representative by a higher-dimensional tensor that is invariant under the crystal symmetry. Its Euclidean metric is locally isometric to the misorientation metric. It therefore agrees with the misorientation angle for small separations, but its chord distance is not the geodesic distance for arbitrary pairs.

e1 = embedding(ori1);
e2 = embedding(ori2)

e2 = embedding
 symmetry: 432
 ranks: 4
 dim: 9
 size: 20000 × 1

The summary identifies the cubic symmetry, tensor rank, packed dimension, and number of embedded orientations. This output is useful because the required tensor rank and dimension depend on the crystal symmetry.

Compare the Euclidean embedding distance with the misorientation angle. Division by degree expresses the locally isometric distance in degrees.

distE = norm(e1-e2) ./ degree;

close all
scatter(omega,distE,3,'.')
xlabel('\(\omega(\mathtt{ori}_1,\mathtt{ori}_2)\) in degrees',...
  'Interpreter','latex')
ylabel('\(\Vert\mathcal E(\mathtt{ori}_1)-\mathcal E(\mathtt{ori}_2)\Vert_2\)',...
  'Interpreter','latex')

Near the origin the point cloud follows the diagonal: embedding distance closely approximates the misorientation angle. The increasing curvature at larger angles is the difference between an ambient chord and a geodesic on orientation space.

Averaging in Embedding Space

An arithmetic mean of embeddings lies in the ambient Euclidean space and generally is not itself the embedding of an orientation. Project it back with orientation.

e = embedding(ori);
meanEmbedding = mean(e);
embeddingMean = orientation(meanEmbedding)

embeddingMean = orientation (432 → y↓→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
   315    0    0

MTEX may print this cubic orientation with Euler angle \(315^{\circ}\) instead of \(45^{\circ}\). Those are symmetry-equivalent representatives, not different physical means. The intrinsic angular error confirms the result.

targetMean = orientation.byEuler(45*degree,0,0,cs);
embeddingMeanError = angle(embeddingMean,targetMean) ./ degree

embeddingMeanError =
     0

Constant Norm and Dispersion

All orientations of one symmetry have the same embedding norm. The raw radius depends on the symmetry; the 'normalized' option divides by that radius. The following output is therefore a row of ones.

normalizedNorms = norm(embedding(orientation.rand(5,cs)),...
  'normalized').'

normalizedNorms =
    1.0000    1.0000    1.0000    1.0000    1.0000

Normalized embeddings lie on the unit sphere. Their arithmetic mean lies inside the unit ball, so its norm can summarize concentration. A value close to one indicates a tight cluster. A value near zero indicates strong cancellation; it can accompany orientations far apart in the orientation space, but does not by itself prove that a maximally separated pair is present.

Write the normalized mean-embedding norm as

\[ n=\frac{1}{\rho}\left\Vert\frac{1}{N}\sum_{i=1}^N \mathcal E(\mathtt{ori}_i)\right\Vert, \qquad \rho=\Vert\mathcal E(\mathtt{ori})\Vert.\]

Compare it with the angular standard deviation

\[ \sigma=\left(\frac{1}{N}\sum_{i=1}^N \omega(\mathtt{ori}_i,\mathtt{mori})^2\right)^{1/2},\]

where \(\mathtt{mori}\) is the mean orientation. The first samples come from one family of unimodal de la Vallee Poussin distributions with varying halfwidth.

n = []; sigma = [];
for hw = logspace(-1,1.75,40)*degree

  psi = SO3DeLaValleePoussinKernel('halfwidth',hw);
  odf = unimodalODF(orientation.rand(cs),psi);
  ori = discreteSample(odf,round(1000*(hw*6)^3));

  n(end+1) = norm(mean(embedding(ori)),'normalized');
  sigma(end+1) = std(ori);

end

plot(sigma,sqrt(max(0,1-n)),'linewidth',2)
xlabel('standard deviation \(\sigma\)','Interpreter','latex')
ylabel('\(\sqrt{1-n}\)','Interpreter','latex')

The smooth curve can suggest that the mean-embedding norm determines the standard deviation. That apparent relationship comes from varying only one distribution family. Bingham distributions form a broader family and expose the ambiguity.

n = []; sigma = [];
for k = 1:2:600

  kappa = rand(4,1);
  kappa = k * kappa ./ sum(kappa);
  odf = BinghamODF(kappa,cs);
  ori = discreteSample(odf,1000);

  n(end+1) = norm(mean(embedding(ori)),'normalized');
  sigma(end+1) = std(ori);

end

hold on
scatter(sigma,sqrt(max(0,1-n)),12,'filled')
legend('de la Vallee Poussin','Bingham','Location','best')
hold off

The Bingham points do not collapse onto the first curve. Thus the norm of the mean embedding is a concentration summary, not a one-to-one proxy for angular standard deviation. The figure is also a warning against calibrating one dispersion measure from a single distribution family.

Operations

Embeddings support the following linear-space operations:

Packed Numeric Coordinates

The tensor representation stores repeated components. The double method packs the independent components into a numeric matrix while preserving Euclidean distances exactly. For cubic symmetry the following output compares the full tensor component count with the packed dimension.

fullDimension = size(double(e1,'full'),2)
packedDimension = size(double(e1),2)

fullDimension =
    81
packedDimension =
     9

Each row of the packed matrix now represents one orientation and can be passed to numerical or machine-learning code. This packing is not a learned dimensionality reduction, and an arbitrary row in the ambient space need not correspond to a valid orientation.

distD = vecnorm(double(e1) - double(e2),2,2) ./ degree;
packingError = max(abs(distE-distD))

close all
scatter(omega,distD,3,'.')
xlabel('\(\omega(\mathtt{ori}_1,\mathtt{ori}_2)\) in degrees',...
  'Interpreter','latex')
ylabel('\(\Vert\mathtt{double}(\mathcal E_1)-\mathtt{double}(\mathcal E_2)\Vert_2\)',...
  'Interpreter','latex')

packingError =
   4.2633e-14

This plot reproduces the earlier embedding-distance plot, and the printed packing error is at floating-point roundoff. Packing removes redundant tensor entries without changing the metric.

References

Next

Continue with Misorientations to study relative orientations and their symmetry. For distributions of whole orientation populations, continue with Orientation Density Functions.