Fundamental Regions

Symmetry makes one physical orientation a set of equivalent rotations, see Symmetry. A fundamental region keeps one representative of each set so orientations can be plotted and analysed without symmetry-related copies.

A fundamental region is not unique. MTEX normally chooses the compact region around the identity rotation. Points on its boundary can be tied between equivalent representatives, so "one representative" includes a consistent boundary convention.

This page assumes the crystal-to-specimen map from Theory and the axis--angle coordinates from Rotation Representations. These regions are also the domains used for ODFs, rotation-axis distributions and rotation-angle distributions.

plottingConvention.default('y↑→x');

The Space of All Rotations

Without rotational symmetry, axis--angle space is a ball of radius \(180^\circ\). The direction of a point is the rotation axis, and its distance from the centre is the rotation angle.

% triclinic crystal symmetry
cs = crystalSymmetry('triclinic');

% the corresponding orientation space
oR_all = fundamentalRegion(cs);

plot(oR_all)

Place one half turn about the z axis into the ball.

rotZ = orientation.byAxisAngle(vector3d.Z,180*degree,cs);

hold on
plot(rotZ,'MarkerColor','b','MarkerSize',10)
hold off

Add families about the x and y axes in steps of \(30^\circ\).

rotX = orientation.byAxisAngle(vector3d.X,(-180:30:180)*degree,cs);
rotY = orientation.byAxisAngle(vector3d.Y,(-180:30:180)*degree,cs);

hold on
plot(rotX,'MarkerColor','r','MarkerSize',10)
plot(rotY,'MarkerColor','g','MarkerSize',10)
hold off

Each family lies on a straight line through the centre because its rotations share an axis and differ only in angle. The \(-180^\circ\) and \(180^\circ\) ends of a line are the same rotation. Opposite points on the surface are therefore identified, which is why rotation space is not an ordinary solid ball.

Sections of constant rotation angle are a flat view of the same geometry.

plotSection(rotZ,'MarkerColor','b','axisAngle',(30:30:180)*degree)
hold on
plot(rotX,'MarkerColor','g','add2all')
plot(rotY,'MarkerColor','r','add2all')
hold off

Notice that each x- and y-axis family passes through the same point at zero angle. At \(180^\circ\), its two opposite axis directions again describe one rotation.

What Crystal Symmetry Does

Each proper symmetry operation folds the ball onto itself. A group with \(n\) proper operations therefore cuts it into \(n\) equal-volume regions. Orthorhombic 222 symmetry has four such operations.

Crystal point groups can also contain reflections or inversion. Those improper operations do not create additional rotations, so fundamentalRegion uses the proper rotation group by default.

cs = crystalSymmetry('222');

oR = fundamentalRegion(cs);

close all
plot(oR_all)
axis off
hold on
plot(oR,'color','r')
hold off

The red region is one of four equivalent parts of the complete grey ball. MTEX chooses the central representative shown here.

An orientation with specimen symmetry is reduced from the other side as well. Pass both symmetries as fundamentalRegion(ori.CS,ori.SS); the role of the second group is developed in Specimen Symmetry.

Real Data Lands Inside It

MTEX plots measured orientations in their fundamental region by default.

mtexdata forsterite silent

plot(ebsd('Fo').orientations,'axisAngle')

plot 2000 random orientations out of 152345 given orientations

No point crosses the boundary because the plotting code selects an equivalent representative inside it. This selection does not change the orientations stored in ebsd. MTEX reports that it samples 2,000 points for this dense plot. The sample changes only the figure, not the data or the fundamental region.

Apply the same selection explicitly with project2FundamentalRegion. The method returns a new orientation array.

ori = ebsd('Fo').orientations.project2FundamentalRegion;

Recentring the Region

The region need not be centred on the identity rotation. Centring it on a grain mean keeps a tight orientation cloud together when the standard region would split equivalent representatives across a boundary.

Grain reconstruction is only supporting setup here. It supplies the mean orientation of the largest grain.

[grains,ebsd] = calcGrains(ebsd);

[~,id] = max(grains.area);
largeGrain = grains(id);

% keep the indexed orientations inside the grain footprint
ori = ebsd(largeGrain).orientations;
ori = ori(~isnan(ori));

center = largeGrain.meanOrientation;

% select representatives nearest to the grain mean
ori = ori.project2FundamentalRegion(center);

% retain those explicit representatives when plotting
plot(ori,'axisAngle','ignoreFundamentalRegion','all')
hold on
plot(center,'MarkerFaceColor','r','MarkerSize',20)
hold off

% maximum angular distance from the mean, in degrees
maxSpread = max(angle(ori,center)) ./ degree

maxSpread =
    6.0553

The points remain in a compact neighbourhood of the red mean orientation. The displayed maximum is just over \(6^\circ\), rather than a spread over the whole region. The red marker need not be the Euclidean centre of the drawn coordinates; angle supplies the rotational distance.

Recentring selected equivalent representatives. It did not rotate or otherwise change the measured orientations.

Fundamental Regions of Misorientations

A misorientation carries one crystal symmetry from each crystal. Its fundamental region is reduced by both groups and is correspondingly smaller.

oR = fundamentalRegion(ebsd('Fo').CS,ebsd('En').CS);

plot(oR)

Boundary misorientations between forsterite and enstatite lie inside this two-symmetry region.

plot(grains.boundary('Fo','En').misorientation)

plot 2000 random orientations out of 11751 given orientations

The points stay inside the outline even though the region is much smaller than the full ball. The phase order remains meaningful for a two-phase boundary, so MTEX does not identify these misorientations with their inverses.

Antipodal Symmetry Between Grains of One Phase

Between two grains of the same phase there is no way to say which grain is first. A misorientation and its inverse are therefore indistinguishable. In axis--angle coordinates the inverse has the same angle and the opposite axis. MTEX records this grain-exchange symmetry with the 'antipodal' flag described for directions in Axes and Antipodal Symmetry.

oR = fundamentalRegion(ebsd('Fo').CS,ebsd('Fo').CS,'antipodal');

plot(oR)

The antipodal region has half the volume of the corresponding region without grain-exchange symmetry. MTEX sets the flag on same-phase boundary misorientations by itself, as the following object summary shows.

mori = grains.boundary('Fo','Fo').misorientation

mori = misorientation (Forsterite → Forsterite)
  size: 15872 × 1
  antipodal: true

plot(mori)

plot 2000 random orientations out of 15872 given orientations

Each boundary now contributes one representative from the pair consisting of a misorientation and its inverse. No point needs the other half of the non-antipodal region.

Removing the flag from this local variable draws the same rotations in the larger region, where the two orders count as different.

mori.antipodal = false;

plot(mori)

plot 2000 random orientations out of 15872 given orientations

The cloud now occupies the larger outline. These are not additional grain boundaries; only the choice of representative has changed.

Axis--Angle Sections

Sections of constant rotation angle flatten the larger region. The panel outline at each angle shows which rotation axes are allowed there.

plotSection(mori,'axisAngle')

plotting 2000 random orientations out of 15872 given orientations

Opposite axes remain separate in these panels because mori.antipodal is currently false.

Supplying 'antipodal' to the plot identifies every pair of opposite axes.

plotSection(mori,'axisAngle','antipodal')

plotting 2000 random orientations out of 15872 given orientations

The second gallery contains the same boundary data in smaller panel outlines. Its paired axes have collapsed onto one representative.

How MTEX Chooses a Region

MTEX chooses representatives by rotational distance. For a selected centre, it keeps the rotations that are at least as close to that centre as any of their symmetry equivalents. This nearest-representative cell is a Voronoi cell in rotation space.

Midplanes between the centre and its symmetry equivalents bound the cell. A different centre gives a different, equally valid fundamental region. A point exactly on a midplane has two equally near representatives, which is the boundary tie noted at the start of the page.

The figures on this page use axis--angle coordinates, where radial distance is the rotation angle. In Rodrigues--Frank coordinates the same bisectors are Euclidean planes, so fundamental regions appear as plane-faced polyhedra. Rotation Representations compares these coordinate choices.

References

Next

Continue with Specimen Symmetry for a second symmetry acting on orientations. The counterpart for directions is the Fundamental Sector.

The next chapter begins with Misorientations. Grain Exchange Symmetry develops the same-phase distinction used above.