Thanks to crystal symmetry the orientation space can be reduced to the so called fundamental or asymmetric region. Those regions play an important role for the computation of axis and angle distributions of misorientations.

The space orientations can be imagined as a three dimensional ball with radius 180 degree. The distance of some point in the ball to the origin represent the rotational angle and the vector represents the rotational axis. In MTEX this can be represented as follows

% triclic crystal symmetry cs = crystalSymmetry('triclinic') % the corresponding orientation space oR_all = fundamentalRegion(cs); % lets plot the ball of all orientations plot(oR_all)

cs = crystalSymmetry symmetry : -1 a, b, c : 1, 1, 1 alpha, beta, gamma: 90°, 90°, 90° reference frame : X||a, Y||b*, Z||c*

Next we plot some orientations into this space

% rotation about the z-axis about 180 degree rotZ = orientation('axis',zvector,'angle',180*degree,cs); hold on plot(rotZ,'MarkerColor','b','MarkerSize',10) hold off % rotations about the x- and y-axis about 30,60,90 ... degree rotX = orientation('axis',xvector,'angle',(-180:30:180)*degree,cs); rotY = orientation('axis',yvector,'angle',(-180:30:180)*degree,cs); hold on plot(rotX,'MarkerColor','r','MarkerSize',10) plot(rotY,'MarkerColor','g','MarkerSize',10) hold off

An alternative view on the orientation space is by sections, e.g. sections along the Euler angles or along the rotational angle. The latter one should be demonstrated next:

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

In case of crystal symmetries the orientation space can divided into as many equivalent segments as the symmetry group has elements. E.g. in the case of orthorombic symmetry the orientation space is subdivided into four equal parts, the central one looking like

cs = crystalSymmetry('222') oR = fundamentalRegion(cs); close all plot(oR_all) hold on plot(oR,'color','r') hold off

cs = crystalSymmetry symmetry: 222 a, b, c : 1, 1, 1 Warning: Possible symmetry mismach!

As an example consider the following EBSD data set

`mtexdata forsterite`

we can visualize the Forsterite orientations by

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

plot 2000 random orientations out of 152345 given orientations

We see that all orientations are automatically projected inside the fundamental region. In order to compute explicitly the represent inside the fundamental region we can do

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

ori = orientation size: 152345 x 1 crystal symmetry : Forsterite (mmm) specimen symmetry: 1

There is no necessity that the fundamental region has to be centered in the origin - it can be centered at any orientation, e.g. at the mean orientation of a grain.

% segment data into grains [grains,ebsd.grainId] = calcGrains(ebsd('indexed')); % take the orientations of the largest on [~,id] = max(grains.area); largeGrain = grains(id) ori = ebsd(largeGrain).orientations % recenter the fundamental zone to the mean orientation plot(rotate(oR,largeGrain.meanOrientation)) % project the orientations into the fundamental region around the mean % orientation ori = ori.project2FundamentalRegion(largeGrain.meanOrientation) plot(ori,'axisAngle')

largeGrain = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 1 2683 Forsterite mmm boundary segments: 714 triple points: 44 Id Phase Pixels GOS phi1 Phi phi2 931 1 2683 0.0425468 171 55 262 ori = orientation size: 2683 x 1 crystal symmetry : Forsterite (mmm) specimen symmetry: 1 ori = orientation size: 2683 x 1 crystal symmetry : Forsterite (mmm) specimen symmetry: 1 plot 2000 random orientations out of 2683 given orientations

Misorientations are characterised by two crystal symmetries. A corresponding fundamental region is defined by

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

Let plot grain boundary misorientations within this fundamental region

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

plot 2000 random orientations out of 11814 given orientations

Note that for boundary misorientations between the same phase we can **not** distinguish between a misorientation and its inverse. This is not the case for misorientations between different phases or
the misorientation between the mean orientation of a grain and all other orientations. The inverse of a misorientation is
axis - angle representation is simply the one with the same angle but antipodal axis. Accordingly this additional symmetry
is handled in MTEX by the keyword **antipodal**.

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

We see that the fundamental region with antipodal symmetry has only half the size as without. In the case of misorientations between the same phase MTEX automatically sets the antipodal flag to the misorientations and plots them accordingly.

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

mori = misorientation size: 15974 x 1 crystal symmetry : Forsterite (mmm) crystal symmetry : Forsterite (mmm) antipodal: true plot 2000 random orientations out of 15974 given orientations

If you want to avoid this you can remove the anitpodal flag by

mori.antipodal = false; plot(mori)

plot 2000 random orientations out of 15974 given orientations

Again we can plot constant angle sections through the fundamental region. This is done by

`plotSection(mori,'axisAngle')`

plotting 2000 random orientations out of 15974 given orientations

Note that in the previous plot we distinguish between `mori` and `inv(mori)`. Adding antipodal symmetry those are considered as equivalent

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

plotting 2000 random orientations out of 15974 given orientations

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