Rotation Representations

Internally MTEX stores every rotation as a quaternion, i.e., by four numbers. For many purposes it is more convenient to describe a rotation by a single three dimensional vector. All representations discussed here follow the same recipe: take the rotational axis \(\vec n\) and scale it by some function \(f(\omega)\) of the rotational angle \(\omega\),

\[ \vec v = f(\omega) \, \vec n. \]

What distinguishes them is the choice of \(f\) - and that choice decides which region of \(\mathbb R^3\) the rotations fill, and whether the mapping preserves volume. Let us start with a large set of uniformly distributed random rotations.

rot = rotation.rand(100000)

rot = rotation
  size: 100000 x 1

The Rodrigues Vector

The Rodrigues vector uses \(f(\omega) = \tan(\omega/2)\), so that

\[ \vec v = \tan\frac{\omega}{2} \, \vec n. \]

It is computed by Rodrigues and inverted by rotation.byRodrigues.

vRodrigues = rot.Rodrigues;

Its defining property is that rotations sharing a common axis lie on a straight line through the origin. The price is that the representation is unbounded: as \(\omega\) approaches \(180^\circ\) the tangent diverges, so a half turn has no finite Rodrigues vector at all.

max(norm(vRodrigues))

ans =
   4.1684e+04

This is why plots in Rodrigues space are usually restricted to a fundamental region - see Plotting Rotations.

The Homochoric Vector

The homochoric vector uses

\[ f(\omega) = \left(\frac{3}{4}\left(\omega - \sin\omega\right)\right)^{1/3} \]

and is computed by homochoric, with rotation.byHomochoric as its inverse.

vHomochoric = rot.homochoric;

This choice makes the mapping equal volume: it takes the rotation group onto a ball and does not distort densities. The ball has radius \((3\pi/4)^{1/3}\), reached exactly by the half turns.

max(norm(vHomochoric))
(0.75*pi)^(1/3)

ans =
    1.3307
ans =
    1.3307

Equal volume means that uniformly distributed rotations become uniformly distributed points in that ball. For a uniform ball the distribution of the radius \(r\) has the density \(3r^2/R^3\), which is exactly what we observe.

R = (0.75*pi)^(1/3);
histogram(norm(vHomochoric),50,'Normalization','pdf')
hold on
r = linspace(0,R,100);
plot(r,3*r.^2/R^3,'linewidth',2)
hold off
legend('homochoric radii','3r^2/R^3')
xlabel('r')

The same check applied to the Rodrigues vectors shows the contrast - the density there is strongly concentrated near the origin and has an infinite tail.

histogram(min(norm(vRodrigues),10),50,'Normalization','pdf')
xlabel('|v|, clipped at 10')

The Cubochoric Vector

The cubochoric vector, introduced by Rosca, Morawiec and De Graef (2014), composes the homochoric map with an equal volume map from the ball onto a cube. It is computed by cubochoric.

vCubochoric = rot.cubochoric;

The resulting cube has edge length \(\pi^{2/3}\).

max(abs(vCubochoric.x))
pi^(2/3)/2

ans =
    1.0725
ans =
    1.0725

Being equal volume as well, it shares the density preserving property of the homochoric vector, but on a cube rather than a ball. That makes it the natural choice for building uniform grids in orientation space, which is what homochoricSO3Grid does.

There is no rotation.byCubochoric. The inverse is obtained by first mapping the cube back onto the ball with cubo2homo.

xyz = cubo2homo([vCubochoric.x(:), vCubochoric.y(:), vCubochoric.z(:)]);
rot2 = rotation.byHomochoric(xyz);

% the reconstruction error, measured as quaternion distance
max(min(norm(rot(:)-rot2),norm(rot(:)+rot2)))

ans =
   3.2957e-15

Summary

Note that none of these is what MTEX uses to store a rotation. They are derived on demand, and the axis angle parametrisation they all build on is available directly as axis and angle.

angle(rot(1))./degree
axis(rot(1))

ans =
   77.3811
 
ans = vector3d (y↓→x)
       x      y      z
  -0.116 -0.032 -0.993