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Rodrigues Space

On the previous page we studied quaternions as a representation of rotation and derived the misorientation distribution for objects with no symmetry beyond the identity. Here we introduce another important representation of rotations — Rodrigues space — and the concept of “folding” by rotational symmetry.

What is Rodrigues Space?

Three-dimensional Rodrigues space is a space that encodes both the rotation axis direction and the rotation amount. A point in this space is written as $\mathbf{r}=(x,y,z)$ with norm $\rho=\|\mathbf{r}\|=\sqrt{x^2+y^2+z^2}$. The coordinate $(x,y,z)$ encodes:

rotation axis direction: $(x,y,z)$
rotation angle $\theta$: $2\arctan(\rho)$, i.e. $\tan\frac{\theta}{2}=\rho$

For example, $(1,0,0)$ means “rotation by 90° about the X-axis”, and $(1,1,1)$ means “rotation by 120° ($=2\arctan\sqrt{3}$) about the [111] axis”.

The relationship between a Rodrigues coordinate $\mathbf{r}$ and the corresponding unit quaternion $q$ is elegantly expressed as:
$$\mathbf{r}=(x,y,z) \quad\leftrightarrow\quad q=\frac{1}{\sqrt{1+\rho^2}}(1,x,y,z)=\frac{1}{\sqrt{1+\rho^2}}(1,\mathbf{r}),$$
or equivalently,
$$\mathbf{r}=\tan\frac{\theta}{2}(x,y,z) \quad\leftrightarrow\quad q=\left(\cos\frac{\theta}{2},\,x\sin\frac{\theta}{2},\,y\sin\frac{\theta}{2},\,z\sin\frac{\theta}{2}\right)=\cos\frac{\theta}{2}(1,\mathbf{r}), \quad x^2+y^2+z^2=1,$$
or in scalar-vector notation,
$$\mathbf{r}=\frac{\mathbf{v}}{s} \quad\leftrightarrow\quad q=(s,\mathbf{v}), \quad s^2+\|\mathbf{v}\|^2=1.$$

Arithmetic in Rodrigues Space

Multiplication of two Rodrigues coordinates corresponds to composition of rotations. With $\mathbf{r}_1=(x_1,y_1,z_1)$, $\mathbf{r}_2=(x_2,y_2,z_2)$ and $\rho_i=\|\mathbf{r}_i\|$, the corresponding unit quaternions are
$$q_i=\frac{1}{\sqrt{1+\rho_i^2}}(1,\mathbf{r}_i).$$
Using quaternion multiplication:
$$q=q_1q_2=\frac{1}{\sqrt{(1+\rho_1^2)(1+\rho_2^2)}}\left(1-\mathbf{r}_1\cdot\mathbf{r}_2,\;\mathbf{r}_1+\mathbf{r}_2+\mathbf{r}_1\times\mathbf{r}_2\right),$$
so the composed Rodrigues coordinate is
$$\mathbf{r}=\mathbf{r}_1\otimes\mathbf{r}_2=\frac{\mathbf{r}_1+\mathbf{r}_2+\mathbf{r}_1\times\mathbf{r}_2}{1-\mathbf{r}_1\cdot\mathbf{r}_2}.$$
When $1-\mathbf{r}_1\cdot\mathbf{r}_2=0$, the scalar part of the composed quaternion vanishes (rotation angle = 180°) and $\rho\to\infty$.

The inverse corresponds to quaternion conjugation $q^{-1}=q^*$, giving simply $\mathbf{r}^{-1}=-\mathbf{r}$: same rotation amount, opposite axis. The misorientation (angular difference) between $\mathbf{r}_1$ and $\mathbf{r}_2$ is therefore
$$\Delta\mathbf{r}=\mathbf{r}_1^{-1}\otimes\mathbf{r}_2=(-\mathbf{r}_1)\otimes\mathbf{r}_2=\frac{-\mathbf{r}_1+\mathbf{r}_2-\mathbf{r}_2\times\mathbf{r}_1}{1+\mathbf{r}_1\cdot\mathbf{r}_2}.$$

Distribution of $SO(3)$ in Rodrigues Space

Rodrigues space, like unit quaternion space, represents $SO(3)$, but with an important caveat: $SO(3)$ is not uniformly distributed in Rodrigues space¹ — the 180° rotation is mapped to the point at infinity, which makes this intuitively clear. The density is obtained by computing the Jacobian of the mapping from the uniform distribution on $S^3$ (unit quaternions) to Rodrigues space (see next page²). The result is
$$P(\rho)=\frac{1}{\pi^2(1+\rho^2)^2}.$$

$P(\rho)$ in the $z=0$ plane³

The surface area of the sphere of radius $\rho$ in Rodrigues space is $4\pi\rho^2$, so the misorientation distribution function is
$$F(\rho)=\frac{4\rho^2}{\pi(1+\rho^2)^2}.$$
Substituting $\rho=\tan(\theta/2)$ and using $d\rho/d\theta=\frac{1}{2}(1+\rho^2)$:
$$F(\theta)=F(\rho)\frac{d\rho}{d\theta}=\frac{2\rho^2}{\pi(1+\rho^2)}=\frac{2}{\pi}\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{\pi},$$
in agreement with the result from the previous page.

Folding of Rodrigues Space by Symmetry Elements

We now examine how rotational symmetry affects the density function in Rodrigues space.

2-fold Rotation

The 2-fold rotation (180° rotation) about the X-axis in Rodrigues space, $\mathbf{C}_2=\lim_{A\to\infty}(A,0,0)$, maps a point $\mathbf{r}=(x,y,z)$ to
$$\mathbf{r}’=(u,v,w)=\mathbf{C}_2\otimes\mathbf{r}=\left(-\frac{1}{x},\frac{z}{x},-\frac{y}{x}\right).$$
When $|x|\ge 1$, we have $|u|=1/|x|\le 1$⁴. Thus the region $|x|\ge 1$ (red in the figure⁵) is folded into the region $|x|\le 1$ (green) by the symmetry operation, and need not be considered separately.

Does folding disturb the density $P(\rho)$ inside $|x|\le 1$? The Jacobian of $\mathbf{r}\to\mathbf{r}’$ is⁶
$$\det\left[\frac{\partial(u,v,w)}{\partial(x,y,z)}\right]=\frac{1}{x^4}.$$
The change in density function between $P(\rho)$ and $P(\rho’)$ is
$$\frac{(1+\rho^2)^2}{(1+\rho’^2)^2}=\left(\frac{1+x^2+y^2+z^2}{1+\frac{1+y^2+z^2}{x^2}}\right)^2=x^4.$$
The volume element scales by $x^{-4}$, so the density scales by $x^4$, which exactly matches the change in $P(\rho)$. Since the folding is injective (no overlap), the density $P(\rho)$ inside $|x|\le 1$ is multiplied by 2 uniformly, with its shape unchanged.

$n$-fold Rotation

Generalising to $n$-fold rotation about the X-axis: the operation rotating by $2k\pi/n$ maps $(x,y,z)\to(u,v,w)$ with $\theta_k=k\pi/n$ as
$$(u,v,w)=\left(\frac{x+\tan\theta_k}{1-x\tan\theta_k},\;\frac{y-z\tan\theta_k}{1-x\tan\theta_k},\;\frac{z+y\tan\theta_k}{1-x\tan\theta_k}\right).$$
The Jacobian⁷ is $\frac{(1+\tan^2\theta_k)^2}{(1-x\tan\theta_k)^4}$, and the density ratio is $P(\rho)\frac{(1-x\tan\theta_k)^4}{(1+\tan^2\theta_k)^2}$. As in the 2-fold case, the density changes by a uniform constant factor.

Substituting $x=\tan\alpha$ and using the tangent addition formula:
$$u=\frac{\tan\alpha+\tan\theta_k}{1-\tan\alpha\tan\theta_k}=\tan\!\left(\alpha+\frac{k\pi}{n}\right).$$
The $n$ equivalent X-coordinates are $\tan(\alpha),\tan(\alpha+\pi/n),\ldots,\tan(\alpha+(n-1)\pi/n)$, a sequence of tangents at equal spacing $\pi/n$. Among them, exactly one falls in $[-\tan(\pi/2n),\,\tan(\pi/2n)]$. Taking that as the representative, all points with $|x|\ge\tan(\pi/2n)$ can be folded in, and only the region $|x|\le\tan(\pi/2n)$ need be retained.

Summary: an $n$-fold rotation about some axis folds Rodrigues space into the slab between two planes perpendicular to that axis, centred symmetrically about the origin at distance $\tan(\pi/2n)$. The orientation density function within the slab is multiplied by $n$ uniformly.

On the next page, we apply this to specific point groups and derive misorientation distribution functions.

More precisely, the Haar measure is not uniform in Rodrigues space. As explained on the previous page, $SO(3)$ is uniformly distributed as unit quaternions on $S^3$. Four dimensions are inherently required for a uniform parameterisation. ↩︎
Intuitively: via $\mathbf{r}\leftrightarrow q=(1+\|\mathbf{r}\|^2)^{-1/2}(1,\mathbf{r})$, a volume element near $\mathbf{r}$ maps to a volume element on $S^3$ (uniform Haar measure) scaled by the 4th power (4 dimensions) of $(1+\|\mathbf{r}\|^2)^{-1/2}$. ↩︎
Mathematica: Plot3D[Pxy[x,y],{x,-3,3},{y,-3,3},PlotRange->All,AxesLabel->{"x","y","P"}] ↩︎
Equality holds only at $x=1$. ↩︎
Mathematica code omitted for brevity. ↩︎
The Jacobian matrix is $J=\begin{pmatrix}x^{-2}&0&0\\-zx^{-2}&0&-x^{-1}\\yx^{-2}&x^{-1}&0\end{pmatrix}$, with determinant $x^{-4}$. ↩︎
The Jacobian matrix is $J=\begin{pmatrix}\frac{1+\tan^2\theta_k}{(1-x\tan\theta_k)^2}&0&0\\ \cdots&\frac{1}{1-x\tan\theta_k}&\frac{-\tan\theta_k}{1-x\tan\theta_k}\\ \cdots&\frac{\tan\theta_k}{1-x\tan\theta_k}&\frac{1}{1-x\tan\theta_k}\end{pmatrix}$. Taking the cross product of columns 2 and 3, then dotting with column 1, gives the result. ↩︎