Rodrigues' rotation formula

From Wikipedia, the free encyclopedia

This article may require cleanup to meet Wikipedia's quality standards.
Please improve this article if you can. (March 2008)

In geometry, Rodrigues' rotation formula (named after Olinde Rodrigues) is a vector formula for a rotation in space, given its axis and angle of rotation.

Say u,v $\in$ R³ and we want to obtain a representation for the rotation v_rot of the vector v around the vector u (which is assumed to have unit length) by an angle θ in the counterclockwise (i.e. positive) direction. Rodrigues' formula reads as follows:

$\mathbf{v}_{rot} = \mathbf{v} \cdot \cos\theta + \mathbf{u} \times \mathbf{v} \cdot \sin\theta + \langle \mathbf{u}, \mathbf{v} \rangle \mathbf{u} \cdot (1 - \cos\theta).$

[edit] Proof of the formula

Take the vector w = v − <u,v>u, which is the projection of v on the plane orthogonal to u, and the cross product of the vectors u and v: z = u×v. Turn the vector w by the angle θ around the base of the vector u to obtain the projection of the rotated vector v_rot:

$\begin{align} \mathbf{w}_{rot} &= \mathbf{w} \cdot \cos\theta + \mathbf{z} \cdot \sin\theta \\ &= (\mathbf{v} - \langle \mathbf{u}, \mathbf{v} \rangle \mathbf{u}) \cdot \cos\theta + \mathbf{u} \times \mathbf{v} \cdot \sin\theta. \end{align}$

Notice that both the vectors w and z have the same length: |w|,|z| = |v - <u,v>u|, because the vector u is of unit length. To get the rotated vector v, we have to add back the adjustment <u,v>u. Hence

$\begin{align} \mathbf{v}_{rot} &= (\mathbf{v} - \langle \mathbf{u}, \mathbf{v} \rangle \mathbf{u}) \cdot \cos\theta + \mathbf{u} \times \mathbf{v} \cdot \sin\theta + \langle \mathbf{u}, \mathbf{v} \rangle \mathbf{u} \\ &= \mathbf{v} \cdot \cos\theta + \mathbf{u} \times \mathbf{v} \cdot \sin\theta + \langle \mathbf{u}, \mathbf{v} \rangle \mathbf{u} \cdot (1 - \cos\theta), \end{align}$

which is exactly what we were looking for.

Using the annotation $\mathbf{u}^T \mathbf{v}$ for the scalar product, we get:

$\begin{align} \mathbf{v}_{rot} &= \cos\theta \cdot \mathbf{v} + \sin\theta \cdot \mathbf{u} \times \mathbf{v} + (1 - \cos\theta) \cdot \mathbf{u} \cdot \mathbf{u}^T \mathbf{v} \end{align}$

Substituting the cross product with matrix multiplication:

$\begin{align} \mathbf{u} \times \mathbf{v} &= \left(\begin{array}{ccc} 0 & -u_3 & u_2 \\ u_3 & 0 & -u_1 \\ -u_2 & u_1 & 0 \end{array}\right) \mathbf{v} = \left[ \mathbf{u} \right]_\times \mathbf{v} \end{align}$

and multiplying with the identity matrix I, we get

$\begin{align} \mathbf{v}_{rot} &= \cos\theta \cdot I \mathbf{v} + \sin\theta \cdot \left[ \mathbf{u} \right]_\times \mathbf{v} + (1 - \cos\theta) \cdot \mathbf{u} \cdot \mathbf{u}^T \mathbf{v} \\ &= \left( \cos\theta I + \sin\theta \left[ \mathbf{u} \right]_\times + (1 - \cos\theta) \mathbf{u} \mathbf{u}^T \right) \mathbf{v} \end{align}$

where the expression in the paranthesis can be identified as the rotation matrix R:

$\begin{align} R = \cos\theta I + \sin\theta \left[ \mathbf{u} \right]_\times + (1 - \cos\theta) \mathbf{u} \mathbf{u}^T \end{align}$

[edit] External links

For another descriptive example see www.d6.com, Chris Hecker, physics section, part 4. "The Third Dimension" -- on page 3, section ``Axis and Angle, http://www.d6.com/users/checker/pdfs/gdmphys4.pdf