Axes and Antipodal Symmetry

A direction distinguishes its two ends: north is not south. An axis does not. A plane normal is an axis when the two sides of the plane are physically equivalent. Reversing the axis of a twofold rotation likewise describes the same \(180^\circ\) rotation.

MTEX represents directions and axes with vector3d. The logical property antipodal records the difference. When it is true, v and -v represent the same axis.

This page assumes the vector construction and angle conventions from Defining Three-Dimensional Vectors and Vector Operations. See Spherical Projections if upper- and lower-hemisphere plots are new to you.

plottingConvention.default('y↑→x');

Plotting Directions and Axes

Take two directions that differ only in the sign of their z coordinate.

v1 = vector3d(1,1,2);
v2 = vector3d(1,1,-2);

plot([v1,v2],'label',{'v_1','v_2'},'grid','on')

The labels appear on different hemisphere plots. Selecting only the upper hemisphere would hide v2. It would not turn either direction into an axis.

plot([v1,v2],'label',{'v_1','v_2'},'antipodal','grid','on')

The plotting option represents both inputs on the upper hemisphere, using -v2 for the second one. The two points remain separated because v1 and v2 are different axes. This option affects this plot only; it does not change either input variable.

Angles and Axial Means

The angle between directed vectors ranges from \(0^\circ\) to \(180^\circ\). The angle between axes is measured to whichever end is closer and is therefore never greater than \(90^\circ\).

directedAngle = angle(v1,v2) ./ degree

directedAngle =
  109.4712

axisAngle = angle(v1,v2,'antipodal') ./ degree

axisAngle =
   70.5288

The directed angle is \(109.4712^\circ\), whereas the axial angle is \(70.5288^\circ\). The two add to \(180^\circ\). Forgetting 'antipodal' produces a plausible but obtuse answer instead of an error.

Signs matter for an ordinary mean as well. Two opposite unit directions cancel, while the same observations interpreted as axes have a mean axis.

directedMeanLength = norm(mean([vector3d.X,-vector3d.X]))

directedMeanLength =
     0

axisMeanMisfit = angle(mean([vector3d.X,-vector3d.X],'antipodal'),...
  vector3d.X,'antipodal') ./ degree

axisMeanMisfit =
     0

The directed mean has length 0. The axial mean is aligned with the X axis, so its axial misfit is \(0^\circ\). The mean of axes uses their unoriented lines rather than averaging signed components.

Attaching the Flag to the Data

Repeating an option at every call is unnecessary. Attach the flag by assigning the property. The vector3d constructor also accepts 'antipodal'.

v2.antipodal = true;

storedAxisAngle = angle(v1,v2) ./ degree

storedAxisAngle =
   70.5288

sameAxis = (v2 == -v2)

sameAxis =
  logical
   1

The stored flag gives the same \(70.5288^\circ\) angle without an option and makes v2 == -v2 true. For a binary operation, an antipodal flag on either operand requests the axial interpretation.

One vector3d variable carries one antipodal value for its entire array. Do not mix directed observations and axes in the same variable. Separate them before calculating angles, means, or densities.

Densities of Axes

Density Estimation turns a list of directions into a continuous function on the sphere. These 100 deterministic directions form a short band in the upper hemisphere.

rho = linspace(20,70,100) * degree;
theta = linspace(10,30,100) * degree;
v = vector3d.byPolar(theta,rho);

directionDensity = v.calcDensity;
plot(directionDensity,'complete')

The upper hemisphere contains the concentration band. The lower one has no point-reflected copy because these observations are still directions.

axisDensity = v.calcDensity('antipodal');
plot(axisDensity,'complete')

The lower hemisphere now repeats the upper pattern through the centre of the sphere. An axial density must satisfy \(f(v)=f(-v)\).

Measured Pole Figures

Under Friedel's law, opposite reflections have equal intensities when the crystal is centrosymmetric or resonant scattering is absent. Conventional pole-figure measurements normally use this axial model. Resonant scattering can distinguish the signs. Antipodal symmetry is therefore an experimental assumption rather than a property of every ODF.

The Pole Figure Tutorial introduces measured pole figures and the experiment behind them.

mtexdata dubna silent

CS = pf.CS;

plot(pf({1}))

MTEX draws this measured pole figure on the upper hemisphere because its specimen directions already represent axes. The lower hemisphere would repeat the same measurements.

annotate(vector3d(1,0,-1),'labeled','backgroundColor','w')

Although the annotated direction points downwards, its equivalent upper endpoint is labelled in the plot.

Pole Figures Computed from an ODF

A pole figure computed from an ODF need not be antipodal. The quartz point group 321 does not contain inversion, and the model below does not impose Friedel's law. The Pole Figures of an ODF page develops this calculation.

center = orientation.byEuler(20*degree,30*degree,0,'ZYZ',CS);
odf = unimodalODF(center);
h = Miller(1,2,2,CS);

plotPDF(odf,[h,-h])

The two pole figures have different intensity patterns. The ODF distinguishes the \((122)\) plane normal from its opposite for this non-Laue point group.

plotPDF(odf,h,'antipodal')

With antipodal symmetry imposed, MTEX draws only the upper hemisphere. The omitted lower hemisphere is now a point-reflected copy.

Inverse Pole Figures

An inverse pole figure fixes a specimen direction and displays crystal directions. The Inverse Pole Figures of an ODF page explains this complementary view.

plotIPDF(odf,[vector3d.Y,-vector3d.Y],'complete','noLabel')

The complete inverse pole figures for Y and -Y differ. Without an antipodal assumption, reversing the specimen direction changes the question.

plotIPDF(odf,vector3d.Y,'antipodal','complete','noLabel')

The complete antipodal plot repeats the same crystal-direction pattern on opposite sides of the sphere.

Fundamental Sectors

Inverse pole figures are usually reduced to the fundamental sector. This is the patch of the sphere that crystal symmetry leaves inequivalent.

plotIPDF(odf,vector3d.Y)

Without antipodal symmetry, MTEX must retain the larger fundamental sector of point group 321.

plotIPDF(odf,vector3d.Y,'antipodal')

Identifying opposite directions reduces the region further. Every omitted crystal direction is equivalent to one inside the smaller plotted sector.

Further Reading

Next

Continue with Density Estimation to work with c-axes from an EBSD map. Crystal axes written as Miller indices are treated in Miller Indices. The same flag records grain-exchange symmetry in Grain Exchange Symmetry.