Rodrigues' rotation formula from SO(3) comutator properties

In summary, Rodrigues' rotation formula can be obtained from the matrix exponential using SO(3) group commutator properties only, such as the commutator action and the adjoint action. This can be done by working with quaternions in scalar+3-vector format and utilizing the adjoint action and Euler's formula. Alternatively, the characteristic polynomial can also be used to compute the exponential.
  • #1
sergiokapone
302
17
Is any way to get Rodrigues' rotation formula from matrix exponential
\begin{equation}
e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ).
\end{equation}
using SO(3) groups comutators properties ONLY like
\begin{equation}\hat S_k\hat S_j-\hat S_j\hat S_k=i\varepsilon_{kjl}\hat S_l,\qquad \hat S^2=2\hat I ?\end{equation}
where ##\vec{n} = (n_x,n_y,n_z)^{\top}##, ##\vec{n}^2 = 1##, and
\begin{equation}\label{}
(\vec{n}\cdot\hat{\vec{S}}) = \star\vec{n} =
\begin{pmatrix}
0 & -n_z & n_y \\
n_z & 0 & -n_x \\
-n_y & n_x & 0 \\
\end{pmatrix},
\end{equation}

\begin{multline}\label{}
\hat{\vec{S}} =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & i \\
0 & -i & 0 \\
\end{pmatrix}
\vec{e}_x
+
\begin{pmatrix}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0 \\
\end{pmatrix}
\vec{e}_y
+
\begin{pmatrix}
0 & i & 0 \\
-i & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
\vec{e}_z =
\hat{S}_x \vec{e}_x +
\hat{S}_y \vec{e}_y +
\hat{S}_z \vec{e}_z
.
\end{multline}

References: https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_so(3)_to_SO(3)
 
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  • #2
The problem you will run into is that you will be expanding a commutator action [when you exponentiate] and that's not associative. [You want to work with an associative product when expanding an exponential.]

For a purely algebraic derivation you can work it through most easily (imnsho) utilizing quaternion algebra. You have to be aware that in that algebra you must act adjointly (left action yields the spinor representation).

I suggest working with quaternions in scalar+3-vector format. The quaternion product then manifests as:
[tex] (a+\mathbf{u})(b+\mathbf{v}) = ab+a\mathbf{v}+b\mathbf{u} -\mathbf{u}\bullet\mathbf{v}+\mathbf{u}\times\mathbf{v}[/tex] where ##\times## is the cross product and ##\bullet## the dot product.

The adjoint action is:
[tex]R_{\theta,\mathbf{u}} \mathbf{v} = e^{\theta\mathbf{u}/2} \mathbf{v}e^{-\theta\mathbf{u}/2}[/tex]
where ##\mathbf{u}## is a unit vector in the direction about which you rotate ( 👎 right hand rule). The exponential of a pure quaternion is then a version of the Euler formula as ##\mathbf{u}^2 = -1##:
[tex] e^{\theta\mathbf{u}/2} = \cos(\theta/2)+\mathbf{u}\sin(\theta/2)[/tex] Work out the adjoint action of this on ##\mathbf{v}## and play with your vector product identities and half-angle trig. identities and the Rodigues' formula will pop right out.
 
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  • #3
There is an analog of the binomial theorem for noncommutative associative algebras. It is quite useful when the commutator assumes a special form. However, it’s not strictly necessary in this case - for so(3), it’s simple enough just to use the characteristic polynomial ##\mathbf{[x]_\times^3}+\mathbf{|x\|^2[x]_\times}=0## to compute the exponential.
 
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FAQ: Rodrigues' rotation formula from SO(3) comutator properties

1. What is Rodrigues' rotation formula from SO(3) comutator properties?

Rodrigues' rotation formula is a mathematical formula used to describe the rotation of a rigid body in three-dimensional space. It is based on the properties of the special orthogonal group SO(3) and is commonly used in computer graphics and robotics.

2. How is Rodrigues' rotation formula derived?

Rodrigues' rotation formula is derived from the properties of the commutator of two elements in the special orthogonal group SO(3). This commutator is a vector that represents the axis of rotation and the angle of rotation of a rigid body in three-dimensional space.

3. What are the applications of Rodrigues' rotation formula?

Rodrigues' rotation formula has numerous applications in computer graphics and robotics, such as in 3D animation, virtual reality, and robot kinematics. It is also used in physics and engineering to describe the motion of rigid bodies.

4. Can Rodrigues' rotation formula be applied to non-rigid bodies?

No, Rodrigues' rotation formula is only applicable to rigid bodies, which are objects that maintain their shape and size even when subjected to external forces. It cannot be used for non-rigid bodies, such as fluids or gases.

5. Is Rodrigues' rotation formula the only way to describe rotations in three-dimensional space?

No, there are other mathematical methods for describing rotations in three-dimensional space, such as Euler angles and quaternions. However, Rodrigues' rotation formula is often preferred for its simplicity and intuitive geometric interpretation.

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