# Haar measure from axis-angle representation of $SO(3)$

My aim is to study some special representations of $$SO(3)$$ using characters, and to do this I need an explicit expression of the Haar measure on $$SO(3)$$. I've found some "versions" of the Haar measure on $$SO(3)$$, and the most common one seem to be the acquired by parameterizing $$SO(3)$$ using Euler angles (can be found here).

Since the characters of $$SO(3)$$ only depend on the angle from the axis-angle representation, it seems as if it would be suitable to find a Haar measure based on this parameterization. However, I don't really know where to start, and I've searched the web like a maniac.

Could someone provide some help?

The Haar measure can be explicitly calculated as follows in terms of a given parametrization $$R(\phi,\theta,\psi)$$ of $$SO(3)$$.

The first step is to define a metric thereon, which can be done by means of the Killing form: denoting by $$X$$, $$Y$$ and $$Z$$ the generators of the $$so(3)$$ algebra, $$[X,Y]=Z\,,\qquad [Y,Z]=X\,,\qquad [Z,X]=Y\,,$$ the Killing form on $$so(3)$$ is defined as the symmetric, bilinear form $$(\cdot\,,\cdot)$$ on $$so(3)$$ satisfying $$(X,X)=(Y,Y)=(Z,Z)=1\,,\qquad (X,Y)=(Y,Z)=(Z,X)=0\,.$$ In fact, one can check that this Killing form is indeed invariant under the Lie product, $$([A,B],C)+(B,[A,C])=0$$.

Now that we have a metric on the tangent space to the identity, we can define a metric at any point as follows: let $$\dot R=\frac{d}{dt}R(t)$$ be some vector, defined at the point $$R=R(t)$$; we may transport it back to the tangent space to the identity by means of the map induced by left multiplication. In this case, this simply means taking $$\Omega=R^T\dot R$$, which is indeed an element of the Lie algebra. Then, we may define a metric $$ds^2$$ by means of $$ds^2=(\Omega, \Omega)\,dt^2$$, the so-called Killing metric.

More explicitly, in the fundamental representation, we have $$(A,B)=\frac{1}{2}\mathrm{tr}(A^TB)\,$$ and $$ds^2=\frac{1}{2}\mathrm{tr}(\Omega^T\Omega)\,dt^2\,.$$ For instance, employing the Euler parametrization $$R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_z(\psi)$$ with $$\phi,\psi\in[0,2\pi)$$ and $$\theta\in[0,\pi)$$, $$R_z(\phi)=\left( \begin{matrix} \cos\phi & -\sin\phi & 0\\ \sin\phi & \cos\phi &0\\ 0 & 0 & 1 \end{matrix} \right) \,,\qquad R_y(\theta)=\left( \begin{matrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta &0 & \cos\theta \\ \end{matrix} \right)$$ you should obtain (after a somewhat lengthy calculation) $$ds^2= d\phi^2 + 2\cos\theta\, d\phi\, d\psi + d\theta^2 + d\psi^2\,.$$ The determinant of this metric, $$(\sin\theta)^2$$, provides the Haar measure/volume form by means of $$\omega = \sqrt{\mathrm{det}g}\,d\phi\wedge d\theta\wedge d\psi= \sin\theta \,d\phi \wedge d\theta\wedge d\psi.$$ Note that this measure is automatically right- and left-invariant, since so is the Killing metric.

I understand "angle-axis" representation as a parametrisation of the group manifold of $$SO(3)$$ in terms of an angle of rotation $$\theta$$ and an axis of rotation $$\hat{n}$$ of unit length. Explicitly, the group manifold of $$SO(3)$$ in terms of $$\theta$$ and $$\hat{n}(\beta_1,\beta_2)$$ is parametrised by coordinates $$(\theta,\beta_1,\beta_2)$$. In these coordinates, a general group element $$R(\theta,\beta_1,\beta_2) \in SO(3)$$ is given by $$R(\theta,\beta_1,\beta_2)=\mathbb{I}+\sin (\theta)K(\beta_1,\beta_2)+(1-\cos (\theta))K^2,$$ where $$\mathbb{I}$$ is the unit matrix in three dimenions. The matrix $$K(\beta_1,\beta_2)$$ is given in terms of the axis of rotation $$\hat{n}=(\hat{n}_1,\hat{n}_2,\hat{n}_3)$$ of unit length, $$K(\beta_1,\beta_2)=\left( \begin{matrix} 0&-\hat{n}_3&\hat{n}_2\\ \hat{n}_3&0&-\hat{n}_1\\ -\hat{n}_2&\hat{n}_1&0 \end{matrix} \right).$$ The axis of rotation $$\hat{n}(\beta_1,\beta_2)$$ is explicitly given by $$\hat{n}(\beta_1,\beta_2)=\left( \begin{matrix} \sin(\beta_1)\cos(\beta_2)\\ \sin(\beta_1)\sin(\beta_2)\\ \cos(\beta_1) \end{matrix} \right).$$ The rotation matrix $$R(\theta,\beta_1,\beta_2)$$ corresponds to a rotation of an angle $$\theta$$ around the axis $$\hat{n}(\beta_1,\beta_2)$$.
With the procedure described by Brightsun, using $$R(t)\equiv R(\theta(t),\beta_1(t),\beta_2(t))$$, the Haar measure $$\omega$$ of $$SO(3)$$ in the axis-angle representation is $$\omega=4\sin(\frac{\theta}{2})^2\sin(\beta_1)d\theta \ \wedge\beta_1\wedge\beta_2.$$ Note that the determinant of the Killing metric can be read of from $$\omega$$.
• Can you clarify how the Haar measure is normalized here? Your final expression does not seem to be unit-normalized, integrating to $8\pi^2$. What's the role of the $4$ there? Can you point to reliable references showing this Haar measure and its normalization? – E.P. Jun 21 at 18:24