Rotation Representations

The same rotation can be described by different coordinates. The best choice depends on whether the coordinates will be plotted, sampled, or used to build a grid.

This page assumes the axis--angle and quaternion descriptions introduced in Defining Rotations. MTEX stores the proper part of every rotation as a unit quaternion. The coordinates below are computed from that quaternion on demand.

Rodrigues and homochoric coordinates are scaled-axis representations. Their direction is the rotation axis \(\vec n\). Their length is a function \(f(\omega)\) of the principal rotation angle \(0 \leq \omega \leq \pi\):

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

Cubochoric coordinates take one further step and map the homochoric ball onto a cube. They therefore do not, in general, point along the rotation axis.

rot = rotation.rand(100000);

Rodrigues--Frank Coordinates

The Rodrigues--Frank vector scales the axis by \(f(\omega)=\tan(\omega/2)\). MTEX computes it with Rodrigues.

The following output puts the angle in degrees in the first column and the length of its Rodrigues vector in the second.

sampleAngle = [0 60 120 170 179] * degree;
sampleRot = rotation.byAxisAngle(vector3d.Z,sampleAngle);
vRodrigues = sampleRot.Rodrigues;
rodriguesLength = norm(vRodrigues);

[sampleAngle(:)./degree,rodriguesLength(:)]

ans =
         0         0
   60.0000    0.5774
  120.0000    1.7321
  170.0000   11.4301
  179.0000  114.5887

The length grows rapidly near a half turn. At exactly \(180^\circ\) it is infinite, so the full rotation space is unbounded in Rodrigues coordinates. Rotations about one fixed axis nevertheless form a straight line, and symmetry boundaries become planes. These properties make Rodrigues coordinates useful for visualizing fundamental regions.

rotation.byRodrigues performs the inverse conversion. The displayed angular error confirms the round trip.

rotFromRodrigues = rotation.byRodrigues(vRodrigues);
max(angle(sampleRot,rotFromRodrigues))./degree

ans =
     0

Homochoric Coordinates

Rodrigues coordinates simplify geometry but distort volume. Homochoric coordinates instead scale the rotation axis by

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

homochoric maps all rotations into a ball. Its radius is \(R=(3\pi/4)^{1/3}\), reached by the half turns.

vHomochoric = rot.homochoric;
R = (0.75*pi)^(1/3);

halfTurn = rotation.byAxisAngle(vector3d.Z,pi);
[norm(halfTurn.homochoric),R]

ans =
    1.3307    1.3307

A volume check

rotation.rand samples the uniform, or Haar, distribution on the rotation group. Equal-volume homochoric coordinates turn that sample into a uniform distribution in the ball.

A uniform ball does not have a uniform distribution of radii. A shell at radius \(r\) has more volume than a shell of the same thickness near the centre, so the radial density is \(3r^2/R^3\).

figure;
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('sampled radii','uniform-ball density','Location','northwest')
xlabel('homochoric radius')
ylabel('probability density')

Reading the volume check

The sampled bars follow the increasing theoretical curve. This agreement is the visible consequence of preserving Haar volume. It does not mean that homochoric coordinates preserve angles, shapes, or distances between arbitrary rotations.

Cubochoric Coordinates

Cubochoric coordinates compose the homochoric map with an equal-volume map from the ball to a cube. The cube has edge length \(\pi^{2/3}\) and the same volume, \(\pi^2\), as the homochoric ball.

MTEX computes these coordinates with cubochoric. The next figure maps the same half turns first to the homochoric boundary and then to the cubochoric boundary.

halfTurnAxis = equispacedS2Grid('points',2000);
boundaryRot = rotation.byAxisAngle(halfTurnAxis,pi);
homochoricBoundary = boundaryRot.homochoric;
cubochoricBoundary = boundaryRot.cubochoric;

figure;
tiledlayout(1,2)

nexttile
scatter3(homochoricBoundary.x,homochoricBoundary.y,...
  homochoricBoundary.z,4,'filled')
axis equal
xlabel('h_1'); ylabel('h_2'); zlabel('h_3')
title('homochoric boundary')

nexttile
scatter3(cubochoricBoundary.x,cubochoricBoundary.y,...
  cubochoricBoundary.z,4,'filled')
axis equal
xlabel('c_1'); ylabel('c_2'); zlabel('c_3')
title('cubochoric boundary')

Reading the coordinate domains

The spherical boundary on the left becomes the six faces of the cube on the right. Opposite axes describe the same \(180^\circ\) rotation, so opposite boundary locations are identified. Neither domain is an ordinary solid with independent points everywhere on its boundary.

The cube is convenient for Cartesian grids. This is why homochoricSO3Grid constructs its internal grid in cubochoric coordinates despite the class name.

Inverting Cubochoric Coordinates

There is no rotation.byCubochoric constructor. First map the cube back to the homochoric ball with cubo2homo, then use rotation.byHomochoric.

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

max(angle(rot,rotFromCubochoric))./degree

ans =
   3.4151e-06

The displayed maximum is the angular round-trip error in degrees. Its small nonzero value comes from floating-point evaluation of the two nonlinear maps.

Choosing a Representation

Equal volume refers to the invariant volume measure on the rotation group. It is not a claim about Euclidean distance. Use angle, rather than coordinate-vector distance, when the physical angular separation between two rotations is required.

Do not average any of these vectors to obtain a mean rotation. The mean must respect rotation geometry; use mean on the rotations themselves.

The Maths Behind Equal Volume

For Haar-uniform rotations, the radial part of the volume element is proportional to \(\sin^2(\omega/2)\,\mathrm d\omega\). The homochoric definition gives

\[ r^3=\frac34(\omega-\sin\omega), \]

and differentiation gives

\[ 3r^2\,\mathrm dr=\frac32\sin^2(\omega/2)\,\mathrm d\omega. \]

Thus equal intervals of Euclidean volume \(r^2\,\mathrm dr\) correspond to equal intervals of rotation-group volume, up to one constant factor. At \(\omega=\pi\) the ball volume is \(4\pi R^3/3=\pi^2\), which also equals the volume of the cubochoric cube.

References

Next

Improper Rotations explains why reflections and inversions need an additional handedness flag. Then Operations covers composition, inversion, and rotation distance. Plotting Rotations uses the coordinate domains introduced here, while Orientation Fundamental Regions adds crystal and specimen symmetry.