Vector Fields in Orientation Space

A vector field on the rotation group assigns a tangent vector to every orientation. Evaluating an SO3VectorField at an orientation \(R\) returns an SO3TangentVector attached to \(R\).

Read The Tangent Space on the Rotation Group first for the definition of tangent vectors and their left and right representations. Vector fields model orientation-dependent spin in Taylor and Sachs calculations. The gradient of an orientation distribution function (ODF) is another important example.

plottingConvention.default('y↑→x');

A first vector field: the gradient of an ODF

Consider the model ODF for a quartz specimen known as the Dubna texture. The gradient points in the direction of the fastest local increase of the ODF, and its norm gives that rate of increase.

odf = SO3Fun.dubna;
G = odf.grad

G = SO3VectorFieldHarmonic (Quartz → y↑→x)
  bandwidth: 48
  tangent space: leftVector

Evaluation at one sampled orientation returns the tangent vector attached to that orientation.

ori = odf.discreteSample(1);
G.eval(ori)

ans = SO3TangentVector (y↑→x)
 intern symmetries: Quartz → y↑→x
 tangentSpace: leftVector
     x    y    z
  60.5 -106 29.6

Plot the ODF in sigma sections and draw the gradient arrows on top.

plot(odf,'sigma')
hold on
plot(G,'linewidth',1.5,'color','black','resolution',7.5*degree)
hold off

The arrows point uphill towards the nearest local maximum. Their lengths increase where the ODF changes more steeply. This ascent direction is the basis of the steepest-descent algorithm used by max and calcComponents.

Three representations

MTEX provides three interchangeable representations. A harmonic or RBF field stores three scalar component functions using the array convention introduced in Vector-Valued Orientation Functions.

All three representations support the common SO3VectorField operations.

Left and right tangent-vector coordinates

An SO3VectorField has a requested tangent-space representation and two associated symmetries. Left coordinates are the default. The right and left methods re-express the same geometric vectors; they do not change the vectors or their base orientations.

GR = right(G)
v = GR.eval(ori);
vRight = right(G.eval(ori));
norm(v-vRight)

GR = SO3VectorFieldHarmonic (Quartz → y↑→x)
  bandwidth: 48
  tangent space: rightVector
  intern symmetries: Quartz → y↑→x
  intern tangent space: leftVector
 
ans =
     0

The zero residual shows that converting the field before evaluation and converting the evaluated tangent vector give the same result. Arithmetic also converts compatible fields to a common representation automatically.

G + GR

ans = SO3VectorFieldHarmonic (Quartz → y↑→x)
  bandwidth: 48
  tangent space: leftVector

Why the visible symmetries change

A field has an internal tangent-space representation used for storage and an external representation requested through its tangentSpace property. Evaluation first constructs the tangent vector in the internal representation and then converts it to the requested representation.

Symmetry acts differently on the two coordinate choices. For a right-represented tangent vector, evaluations at symmetry-equivalent orientations are meaningful only with respect to the left symmetry. For a left-represented tangent vector, the reverse applies.

ori = orientation.rand(G.CS,G.SS);
G.eval(ori.symmetrise)
GR.eval(ori.symmetrise)

ans = SO3TangentVector (y↑→x)
 intern symmetries: Quartz → y↑→x
 size: 6 × 1
 tangentSpace: leftVector
        x        y        z
  -0.0585   0.0209 -0.00396
  -0.0585   0.0209 -0.00396
  -0.0585   0.0209 -0.00396
  -0.0585   0.0209 -0.00396
  -0.0585   0.0209 -0.00396
  -0.0585   0.0209 -0.00396
 
ans = SO3TangentVector (Quartz)
 intern symmetries: Quartz → y↑→x
 size: 6 × 1
 tangentSpace: rightVector
         x        y        z
   0.00932 -0.00683  -0.0612
    0.0106 -0.00465   0.0612
   -0.0106 -0.00465  -0.0612
  -0.00932 -0.00683   0.0612
   0.00126   0.0115  -0.0612
  -0.00126   0.0115   0.0612

MTEX therefore keeps the original crystal and specimen symmetries in two hidden properties. The symmetry of the internal three-component SO3Fun depends on the internal tangent space. The externally visible symmetry depends on the requested tangent space. This is why G and GR have different external symmetries even though they describe the same field.

Operations on vector fields

The following operations apply to vector fields VF, VF1 and VF2:

Since a gradient is itself a vector field, MTEX can also compute its divergence, curl and scalar antiderivative.

Divergence and the Laplacian

Treating G as an orientation-space velocity field gives an intuitive reading of its divergence. Negative divergence marks a sink where nearby orientations condense. Positive divergence marks a source where they spread apart.

The divergence of a gradient equals the Laplacian of its scalar field. Plot the two calculations side by side at the same sigma section.

plot(G.div,'sigma',60*degree)
nextAxis
plot(laplace(SO3FunHarmonic(odf)),'sigma',60*degree)
mtexColorbar

The source and sink regions, contour shapes and colour scale agree in the two panels. The left panel was computed from the vector field, whereas the right panel was computed directly from the ODF.

Curl and conservative fields

Curl describes the axis of local circulation within orientation space. The curl of a gradient is zero, so the next plot should contain no nonzero arrows.

plot(G.curl,'sigma')

Vanishing curl identifies a conservative field: a field that is the gradient of a scalar potential. The antiderivative method reconstructs that potential.

odf2 = G.antiderivative

odf2 = SO3FunHarmonic (Quartz → y↑→x)
  bandwidth: 48
  weight: 0

A gradient loses the additive constant of its source function. Restoring the original mean makes the reconstructed potential coincide with odf.

odf2 = odf2 + mean(odf);
plot(odf2,'sigma')

Define a field by an evaluation formula

An anonymous function is convenient when a vector formula is known. The following rule uses the rotation axis multiplied by the rotation angle, with cubic symmetry on both sides.

cs = crystalSymmetry('432')
f = @(mori) axis(mori) .* angle(mori);
VF = SO3VectorFieldHandle(f,cs,cs)

cs = crystalSymmetry (⊙c→a)
 
  symmetry: 432    
  elements: 24     
  a, b, c : 1, 1, 1
 
 
VF = SO3VectorFieldHandle (432 → 1)
  tangent space: leftVector
  intern symmetries: 432 → 432

Evaluating a \(10^\circ\) rotation about \([1\;2\;3]\) and reducing the axis to small integers recovers the expected direction ratio \(1:2:3\).

round(VF.eval(orientation.byAxisAngle(vector3d(1,2,3),10*degree)))

ans = SO3TangentVector (crystal)
 intern symmetries: 432 → 432
 tangentSpace: leftVector
  x y z
  1 2 3

The following axis-angle plot samples the formula throughout the cubic fundamental region. Arrow count is consistent with that small region. The arrow directions have not been verified independently against eval for an SO3VectorFieldHandle, so use the plot qualitatively and treat evaluated values as authoritative.

quiver3(VF,'axisAngle','resolution',7.5*degree,'color','black',...
  'linewidth',2)

Convert a field to harmonic form

Passing any SO3VectorField to the harmonic constructor expands its three components by quadrature.

SO3VectorFieldHarmonic(VF)

ans = SO3VectorFieldHarmonic (432 → 1)
  bandwidth: 64
  tangent space: leftVector
  intern symmetries: 432 → 432

Fit harmonic components to sampled values

A second construction starts from rotations and one vector3d value at each rotation. The first array dimension again corresponds to nodes.

nodes = equispacedSO3Grid(specimenSymmetry('1'),'points',1e3);
nodes = nodes(:);
y = vector3d.byPolar(sin(3*nodes.angle),nodes.phi2+pi/2);

The approximation below produces a harmonic vector field with bandwidth 16.

SO3VF1 = SO3VectorFieldHarmonic.approximate(nodes,y,'bandwidth',16)

SO3VF1 = SO3VectorFieldHarmonic (y↑→x → y↑→x)
  bandwidth: 16
  tangent space: leftVector

Construct by quadrature of a function handle

A handle that accepts a rotation and returns a vector3d can also be passed directly to quadrature. Here the earlier cubic formula produces a harmonic vector field.

SO3VF2 = SO3VectorFieldHarmonic.quadrature(@(v) f(v))

SO3VF2 = SO3VectorFieldHarmonic (y↑→x → y↑→x)
  bandwidth: 64
  tangent space: leftVector

Construct from three scalar harmonic functions

A three-component SO3FunHarmonic can be wrapped directly. Its first, second and third entries become the \(x\), \(y\) and \(z\) components of the vector field.

SO3F = SO3FunHarmonic(rand(1e3,3))
SO3VF3 = SO3VectorFieldHarmonic(SO3F)

SO3F = SO3FunHarmonic (y↑→x → y↑→x)
  isReal: false
  size: 3 × 1
  bandwidth: 9
  weights: [0.7 0.05 0.75]
 
 
SO3VF3 = SO3VectorFieldHarmonic (y↑→x → y↑→x)
  isReal: false
  bandwidth: 9
  tangent space: leftVector

Application: orientation-dependent spin in the Taylor model

Taylor theory accommodates a prescribed strain by activating slip systems in each crystal. The antisymmetric part of the resulting deformation describes the local lattice spin, and therefore the local misorientation predicted for each orientation. Without an input orientation, calcTaylor returns this spin as an SO3VectorField.

cs = crystalSymmetry('432');
sS = slipSystem.bcc(cs)

sS = slipSystem (432)
 size: 1 × 3
 
   u    v    w  | h    k    l CRSS
   1   -1    1    0    1    1    1
  -1    1    1    2    1    1    1
  -1    1    1    3    2    1    1

Set plane strain with \(q=0\) and calculate the spin field for the symmetrised body-centred-cubic slip systems.

q = 0;
epsilon = strainTensor(diag([1 -q -(1-q)]))
[~,~,W] = calcTaylor(epsilon,sS.symmetrise)

epsilon = strainTensor (y↑→x)
  type: Lagrange 
  rank: 2 (3 × 3)
 
  1  0  0
  0  0  0
  0  0 -1
 
W = SO3VectorFieldHarmonic (1 → y↑→x)
  bandwidth: 32
  tangent space: rightSpinTensor
  intern symmetries: 432 → y↑→x
  intern tangent space: leftVector

Display the spin directions in four Euler-angle sections.

sP = phi1Sections(cs,specimenSymmetry('222'));
sP.phi1 = (10:20:70)*degree;
plot(W,sP,'resolution',7.5*degree,'layout',[2 2])

Direction and length vary with orientation, showing that the Taylor model predicts a different local misorientation across orientation space. The value at the copper orientation can be retrieved directly.

WCopper = W.eval(orientation.copper(cs))

WCopper = spinTensor (crystal)
  rank: 2 (3 × 3)
 
 *10^-2
      0      0  32.15
      0      0  32.15
 -32.15 -32.15      0

The amount of spin

The norm of the spin vector is the angle of local misorientation. Its maximum locates the orientation with the largest predicted rotation.

[~,oriMax] = max(norm(W))

oriMax = orientation (432 → y↑→x)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  268.696 45.4283 63.3496

Plot the norm at \(0.5^\circ\) resolution and overlay the more coarsely sampled vector field. The background shows magnitude, the arrows show direction, and the annotation marks oriMax.

plot(norm(W),sP,'resolution',0.5*degree,'layout',[2 2])
mtexColorMap LaboTeX
hold on
plot(W,sP,'resolution',7.5*degree,'color','black')
hold off
annotate(oriMax)

Compare spin with a crystal direction

Since W gives the rotation axis of the local misorientation, its inner product with a chosen direction measures signed alignment. Here the direction is crystal \([100]\).

plot(dot(W,Miller(1,0,0,cs)),sP,'layout',[2 2])
mtexColorMap blue2red
mtexColorbar

Positive and negative regions indicate parallel and antiparallel components along \([100]\). Values near zero indicate that the spin axis is locally perpendicular to that direction.

Sources and sinks of the Taylor spin field

Finally compute the divergence of W. As in the gradient example, negative values are sinks and positive values are sources in orientation space.

flux = W.div
plot(flux,sP,'resolution',0.5*degree,'layout',[2 2],...
  'faceAlpha',0.5)
mtexColorMap blue2red
mtexColorbar

flux = SO3FunHarmonic (432 → y↑→x)
  bandwidth: 32
  weight: 0

The alternating red and blue regions show that the Taylor spin field moves orientations towards some parts of orientation space and away from others.

close all

References

Next

Continue with Rotational Kernel Functions to understand the localized basis functions used by the RBF representation listed on this page.