Fibres of Orientations

A fibre is a one-dimensional curve through orientation space. Every orientation on a full fibre maps one fixed crystal direction h onto one fixed specimen direction r while leaving rotation about that direction free. MTEX stores the curve as a <fibre.fibre.html fibre> variable.

Many real textures concentrate around a line rather than one ideal orientation. Rolling textures of cubic metals are therefore commonly described by named fibres. A real texture has a finite spread around the curve; the curve itself is the ideal centreline.

This page assumes orientation construction, symmetry-equivalent orientations, and the fundamental region. The Cube and Goss components below are introduced in Standard Orientations.

A reference frame is the coordinate system in which data are expressed. The plotting convention below lays specimen Y upward and specimen X to the right. It changes only the screen layout, not the orientations.

plottingConvention.default('y↑→x');

% define crystal and specimen symmetry
cs = crystalSymmetry('432');
ss = specimenSymmetry('1');

% define two ideal orientations
ori1 = orientation.cube(cs,ss);
ori2 = orientation.goss(cs,ss);

% select the representative of Goss nearest to Cube
ori2 = ori2.project2FundamentalRegion(ori1);

A Fibre Segment Between Two Orientations

The endpoint constructor joins two orientation representatives by their shortest angular path. The result is a finite segment, not yet the full closed fibre.

segmentFibre = fibre(ori1,ori2)

segmentFibre = fibre (432 → y↑→x)
 
  h || r: (100) || (1,0,0)
 o1 → o2: (0°,0°,0°) → (0°,45°,0°)

The summary identifies the aligned directions in its h and r row. Its endpoint row runs from Cube at \((0^\circ,0^\circ,0^\circ)\) to Goss at \((0^\circ,45^\circ,0^\circ)\) in Bunge Euler angles. Both endpoints, and every orientation ori between them, satisfy

\[ \mathtt{ori} * h = r. \]

The two directions belong to different reference frames: h is in the crystal frame and r is in the specimen frame.

The Segment in Euler Space

The default three-dimensional orientation plot uses Bunge Euler coordinates.

plot(segmentFibre,'DisplayName','Fibre segment',...
  'lineWidth',4,'lineColor','green');
hold on
plot(ori1,'DisplayName','Cube','MarkerSize',12,...
  'MarkerFaceColor','darkred','MarkerEdgeColor','k');
plot(ori2,'DisplayName','Goss','MarkerSize',12,...
  'MarkerFaceColor','blue','MarkerEdgeColor','k');
hold off
legend('Location','northwest');

The green line runs from the dark-red Cube marker to the blue Goss marker. It looks straight because only \(\Phi\) changes for this pair. In general, a shortest angular path need not look straight in Euler coordinates.

The Same Segment in Axis--Angle Space

plot(segmentFibre,'lineColor','green','lineWidth',6,'axisAngle');
hold on
plot(ori1,'MarkerFaceColor','darkred','MarkerSize',15,'axisAngle');
plot(ori2,'MarkerFaceColor','blue','MarkerSize',15,'axisAngle');
hold off

This is the same orientation-space segment in different coordinates. The endpoints and angular distances have not changed. Only the coordinate map used to draw them has changed.

Extending the Segment to a Full Fibre

The option 'full' discards the finite endpoint and continues through every rotation about the aligned direction.

fullFibre = fibre(ori1,ori2,'full')

fullFibre = fibre (432 → y↑→x)
 
  h || r: (100) || (1,0,0)

The h and r row is unchanged, while the endpoint row has disappeared. Orientation space itself has no boundary, so the full fibre closes into a circle. Coordinate domains and symmetry-reduced fundamental regions do have seams and boundaries, however, and can cut that circle into arcs.

hold on
plot(fullFibre,'lineColor','gold','lineWidth',3,...
  'project2FundamentalRegion','axisAngle');
hold off

The gold curve continues the green segment to the faces of the cubic fundamental region. Its boundary points reconnect through symmetry-equivalent faces, so this boundary-to-boundary line represents one closed fibre.

Fibres in Pole Figures

Everything that can be plotted for orientations can also be plotted for fibres. plotPDF maps the fibre into pole figures, where one orientation becomes a point and a fibre becomes a curve.

h = Miller({1,1,0},{1,1,1},cs);
plotPDF(fullFibre,h,'lineWidth',3,'lineColor','orange');

Each panel contains one trace from the representative fibre. A trace may meet a projection boundary, but it still represents one continuous set of orientations.

Symmetrising a Fibre

Unlike an orientation pole-figure plot, a fibre is not automatically symmetrised. symmetrise generates the crystallographically equivalent fibres. The 'unique' option removes repeated copies.

symFibre = fullFibre.symmetrise('unique');
symmetryCopyCount = length(symFibre)

symmetryCopyCount =
     6

The six copies correspond to the six cubic equivalents of the aligned crystal direction. Plotting them adds the orange traces that were absent from the representative-only pole figures.

plotPDF(symFibre,Miller({1,1,0},{2,1,0},{1,1,1},cs),...
  'lineWidth',3,'lineColor','orange');

Fibres in Inverse Pole Figures

plotIPDF maps the fibre into inverse pole figures. It fixes specimen directions and draws the crystal directions found along the fibre. By default it restricts them to the fundamental sector.

r = [vector3d(1,1,0),vector3d(2,1,0),vector3d(1,1,1)];
plotIPDF(symFibre,r,'lineWidth',3,'lineColor','orange');

Each panel folds the symmetry copies into one fundamental sector. The orange curve is therefore the family of crystal directions that can lie along the specimen direction named above that panel.

The Complete Inverse Pole Figure

The option 'complete' removes the fundamental-sector restriction.

plotIPDF(symFibre,vector3d.Z,'complete',...
  'lineWidth',3,'lineColor','orange');

The complete plot repeats the curve across the full crystal-direction sphere. Those repeated traces are symmetry equivalents, not additional input fibres.

Defining a Fibre by Directions

A Miller direction and a vector3d specimen direction define the full fibre directly. The next fibre contains every orientation that makes the crystal c-axis \([001]\) parallel to specimen Z.

cAxisFibre = fibre(Miller(0,0,1,cs,'uvw'),vector3d.Z)

cAxisFibre = fibre (432 → y↑→x)
 
  h || r: [001] || (0,0,1)

The summary states that [001] is parallel to (0,0,1). The directions are shown in crystal and specimen coordinates, respectively.

plot(cAxisFibre,'lineColor','gold','lineWidth',4,...
  'project2FundamentalRegion','axisAngle');

The gold line is the symmetry-reduced image of every possible rotation about the aligned c-axis. Its two boundary ends continue into one another, so it represents one full fibre rather than one finite segment.

If both constructor directions are Miller variables, the fibre instead contains all misorientations that bring one crystal direction into alignment with the other.

A Fibre Through One Orientation

An initial orientation ori1 and a crystal direction h define all orientations that preserve the mapped direction of ori1:

\[ \mathtt{ori} * h = \mathtt{ori1} * h. \]

The following full fibre passes through Cube and rotates about its \([111]\) axis.

cube111Fibre = fibre(ori1,Miller(1,1,1,cs,'uvw'))

cube111Fibre = fibre (432 → y↑→x)
 
  h || r: [111] || (1,1,1)

Its summary reports that [111] is parallel to (1,1,1). Cube maps this crystal direction onto the same specimen direction.

plot(cube111Fibre,'lineColor','darkred','lineWidth',4,...
  'project2FundamentalRegion','axisAngle');

The dark-red curve passes through the Cube orientation at zero angle. Its ends meet the boundary at symmetry-equivalent continuations of the curve.

orientation samples a fibre for numerical work. angle measures the angular distance from an orientation to its nearest point on a fibre. Their geometric use without crystal symmetry is developed in Fibres in Rotation Space.

Predefined Rolling Fibres

Cubic rolling textures have named segments, as their ideal components do: alpha, beta, gamma, epsilon, eta, tau, and theta. MTEX provides each as a static fibre constructor. These names assume the conventional rolling frame and orthorhombic specimen symmetry.

ss = specimenSymmetry('orthorhombic');
beta = fibre.beta(cs,ss);

Plot the conventional endpoint segment returned by each constructor. Passing 'full' would extend that segment around its entire direction-pair circle.

plot(fibre.alpha(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(1),'DisplayName','alpha');
hold on
plot(fibre.beta(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(2),'DisplayName','beta');
plot(fibre.gamma(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(3),'DisplayName','gamma');
plot(fibre.epsilon(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(4),'DisplayName','epsilon');
plot(fibre.eta(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(5),'DisplayName','eta');
plot(fibre.tau(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(6),'DisplayName','tau');
plot(fibre.theta(cs,ss),'lineWidth',3,...
  'lineColor',ind2color(7),'DisplayName','theta');
hold off
legend('Location','best');

The coloured curves occupy different routes through Euler space and meet at some shared ideal components. Each colour marks only the conventional named segment, not every point on its full direction-pair circle.

Fibre ODFs

A fibre is a curve of zero orientation-space volume, while a real texture has a spread around one. fibreODF turns the full curve into a density with a given halfwidth. This is the model fitted against a measurement.

betaFull = fibre.beta(cs,ss,'full');
odf = fibreODF(betaFull,'halfwidth',10*degree);

The result is an SO3FunCBF with a de la Vallée Poussin kernel. Its \(10^\circ\) halfwidth is the distance where the density has fallen to half its peak, not a cutoff radius. The density is constant along the ideal fibre and decays away from it.

plot3d(odf);
hold on
plot(betaFull.symmetrise,...
  'lineColor','blue','lineWidth',4);
hold off

The blue curves are the ideal centrelines. The surrounding surface is the finite-width density, so it forms a tube rather than a line.

Evaluating an ODF Along a Fibre

plotFibre evaluates an ODF along a chosen curve. Here the beta-fibre ODF is read along the eta fibre.

plotFibre(odf,fibre.eta(cs,ss),'lineWidth',2,'figSize','small');

The vertical axis is density in multiples of a random distribution. The broad maximum marks the part of the eta segment closest to the beta-fibre ridge.

The Volume Around a Fibre

volume integrates the ODF inside a tube of a given angular radius about a fibre. This is the quantity meant when a texture is reported as "so many percent beta fibre".

volume5Percent = 100 * volume(odf,betaFull,5*degree)

volume5Percent =
    7.2570

The result is about 58 percent within \(5^\circ\) of the fibre it was built on. At the \(10^\circ\) kernel halfwidth, the numerical tube integral reaches unity.

volume10Percent = 100 * volume(odf,betaFull,10*degree)

volume10Percent =
   25.9307

The second result is 100 percent to the precision printed. It does not mean the ODF is a zero-width line or vanishes at \(10^\circ\). fibreVolume discretises the tube and clips its numerical estimate at 1, so the displayed percentage can be exactly 100.

The Maths Behind Pole Figures

Let \(f(g)\) be an ODF, \(h\) a crystal direction, and \(r\) a specimen direction. The pole density at \(r\) is

\[ P_h(r) = \int_{\{g:\,g h=r\}} f(g)\,\mathrm{d}g. \]

The integration set is exactly the full orientation fibre defined by g * h = r. This crystallographic Radon transform connects the geometry on this page to measured pole figures and ODF reconstruction; see the pole figure tutorial.

References

Next

Fibres of plain rotations, without crystal symmetry, are Fibres in Rotation Space. Density models, halfwidth, and fitting are developed in Fibre ODFs. Pole-figure integration over fibres continues in Pole Figures of an ODF.