Rodrigues' rotation formula

This article is about the Rodrigues' rotation formula, which is distinct from Euler–Rodrigues parameters and The Euler–Rodrigues formula for 3D rotation.

In the theory of three-dimensional rotation, Rodrigues' rotation formula (named after Olinde Rodrigues) is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix from an axis-angle representation. In other words, the Rodrigues formula provides an algorithm to compute the exponential map from so(3) to SO(3) without computing the full matrix exponent.

If v is a vector in $\mathbb{R}^3$ and k is a unit vector describing an axis of rotation about which we want to rotate v by an angle θ (in a right-handed sense), the Rodrigues formula is:

$\mathbf{v}_\mathrm{rot} = \mathbf{v} \cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) (1 - \cos\theta).$

Derivation

Rodrigues' rotation formula rotates v by an angle θ around an axis z by decomposing it into its components parallel and perpendicular to z, and rotating only the perpendicular component.

Given a rotation axis z represented by a unit vector k = (k_X, k_Y, k_Z), and a vector v = (v_X, v_Y, v_Z) that we wish to rotate about z, the vector

$\mathbf{v}_z = (\mathbf{k} \cdot \mathbf{v}) \mathbf{k}$

is the component of v parallel to z, also called the vector projection of v on k, and the vector

$\mathbf{v}_x = \mathbf{v} - \mathbf{v}_z = \mathbf{v} - (\mathbf{k} \cdot \mathbf{v}) \mathbf{k}$

is the projection of v onto the xy plane orthogonal to z, also called the vector rejection of v from k.

Notice that we chose to define a reference frame xyz in which the z axis is aligned with the rotation axis, and the x axis with the rejection of v from k. This simplifies the demonstration, as it implies that v lies on the xz plane, and its component v_y is null (see figure). However, xyz does not coincide with the reference frame XYZ in which vectors v, k, v_x, and v_z are actually represented. For instance, v = (v_X, v_Y, v_Z) ≠ (v_x, v_y, v_z). In other words, Rodrigues' formula is independent of the orientation in space of the reference frame XYZ in which v and k are represented.

Next let

$\mathbf{w} = \mathbf{k}\times\mathbf{v}$.

Notice that v_x and w have the same length. By definition of the cross product, the length of w is:

$|\mathbf{w}| = |\mathbf{k} \times \mathbf{v}| = |\mathbf{k}| \, |\mathbf{v}| \sin \phi \$

where Φ denotes the angle between z and v. Since k has unit length,

$|\mathbf{w}| = |\mathbf{k} \times \mathbf{v}| = |\mathbf{v}| \sin \phi.$

This coincides with the length of v_x, computed trigonometrically as follows:

$|\mathbf{v}_x| = |\mathbf{v}| \cos(\pi/2-\phi) = |\mathbf{v}| \sin \phi.$

So w can be viewed as a copy of v_x rotated by 90° about z. Using trigonometry, we can now rotate v_x by θ around z to obtain v_{x rot}. Its two components with respect of x and y are v_xcosθ and wsinθ, respectively. Thus,

$\begin{align} \mathbf{v}_{x\ \mathrm{rot}} &= \mathbf{v}_x\cos\theta + \mathbf{w}\sin\theta\\ &= (\mathbf{v} - (\mathbf{k} \cdot \mathbf{v}) \mathbf{k})\cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta. \end{align}$

v_{x rot} can be also described as the projection on xy (or rejection from z) of the rotated vector v_rot (see figure). Since v_z is obviously not affected by a rotation about z, the other component of v_rot (i.e. its projection on z) coincides with v_z. Thus,

$\begin{align} \mathbf{v}_{\mathrm{rot}} &= \mathbf{v}_{x\ \mathrm{rot}} + \mathbf{v}_{z\ \mathrm{rot}} \\ &= \mathbf{v}_{x\ \mathrm{rot}} + \mathbf{v}_z \\ &= (\mathbf{v} - (\mathbf{k} \cdot \mathbf{v}) \mathbf{k}) \cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta + (\mathbf{k} \cdot \mathbf{v}) \mathbf{k} \\ &= \mathbf{v} \cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) (1 - \cos\theta), \end{align}$

as required.

Matrix notation

By representing v and k as column matrices, and defining a matrix $[\mathbf{k}]_\times$ as the "cross-product matrix" for $\mathbf{k}$, i.e.,

$[\mathbf{k}]_\times \mathbf{v} = \mathbf{k}\times\mathbf{v} = \left[\begin{array}{ccc} 0 & -k_3 & k_2 \\ k_3 & 0 & -k_1 \\ -k_2 & k_1 & 0 \end{array}\right]\mathbf{v}$,

Rodrigues' formula can be written in matrix notation:

$\begin{align} \mathbf{v}_{\mathrm{rot}} &= \mathbf{v} \cos\theta + ([\mathbf{k}]_\times \mathbf{v}) \sin\theta + \mathbf{k} (\mathbf{k}^\top \mathbf{v}) (1 - \cos\theta) \\ &= \mathbf{v} \cos\theta + [\mathbf{k}]_\times \mathbf{v} \sin\theta + \mathbf{k} \mathbf{k}^\top \mathbf{v} (1 - \cos\theta). \end{align}$

Using the triple product expansion in can be written as:

$\begin{align} \mathbf{v}_{\mathrm{rot}}&=([\mathbf{k}]_{\times}\mathbf{v})\sin\theta +(\mathbf{k}(\mathbf{k}^\top \mathbf{v})-\mathbf{v}(\mathbf{k}\cdot\mathbf{k})) (1-\cos\theta)+\mathbf{v}(\mathbf{k}^\top \mathbf{k}) \\ &=\mathbf{v}+([\mathbf{k}]_{\times}\mathbf{v})\sin\theta +([\mathbf{k}]_{\times}[\mathbf{k}]_{\times}\mathbf{v})(1-\cos\theta). \end{align}$

since $\mathbf{k}^\top \mathbf{k}=1$ for a normalized vector.

Conversion to rotation matrix

The equation can be also written as

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

where I is the 3×3 identity matrix. Thus we have a formula for the rotation matrix R corresponding to an axis angle representation [k θ]:

$R = I \cos\theta + [\mathbf{k}]_\times \sin\theta + (1 - \cos\theta) \mathbf{k} \mathbf{k}^\top$.

Noting that, using the outer product $\mathbf{k} \mathbf{k}^\top = [\mathbf{k}]_\times^2 + I$, we have

$R = I + [\mathbf{k}]_\times \sin\theta + (1 - \cos\theta) [\mathbf{k}]_\times^2$

or, equivalently,

$R = I + \sin\theta [\mathbf{k}]_\times + (1 - \cos\theta) (\mathbf{k} \mathbf{k}^\top-I)$.

For the inverse mapping, see Log map from SO(3) to so(3).

See also

• Axis angle
• Rotation (mathematics)
• SO(3) and SO(4)
• History of quaternions

References

• Don Koks, (2006) Explorations in Mathematical Physics, Springer Science+Business Media,LLC. ISBN 0-387-30943-8. Ch.4, pps 147 et seq. A Roundabout Route to Geometric Algebra'

External links

• Weisstein, Eric W., "Rodrigues' Rotation Formula" from MathWorld.
• Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007.
• For a proof that begins with the xyz frame see http://myyn.org/m/article/proof-of-rodrigues-rotation-formula/
• For another descriptive example see http://chrishecker.com/Rigid_Body_Dynamics#Physics_Articles, Chris Hecker, physics section, part 4. "The Third Dimension" -- on page 3, section ``Axis and Angle, http://chrishecker.com/images/b/bb/Gdmphys4.pdf
• Matthew T. Mason, Representing rotation.