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 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:

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:

Failed to parse (unknown function\begin): \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

Failed to parse (unknown function\begin): \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.

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

Donate to Wikimedia