Rodrigues' rotation formula: Information from Answers.com

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, the Rodrigues rotation formula (named after Olinde Rodrigues) a formula for rotating a vector in space, given an axis and angle of rotation. If v is a vector in $R 3$ and z 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:

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

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 is an algorithm to compute the exponential map from so(3) to SO(3).

1 Derivation
- 1.1 Computing a rotation matrix
2 See also
3 References
4 External links

Derivation

The Rodrigues rotation formula rotates v around z by decomposing it into its component parallel to z and perpendicular (x) to z, rotating the perpendicular part in the plane orthogonal to z and adding back the parallel part to get v_rot.

Given a unit vector, z and a vector v that we wish to rotate about z, the vector

$\mathbf{x}=\mathbf{v} - \mathbf{z} (\mathbf{z}\cdot\mathbf{v})$

is the projection of v onto the plane orthogonal to z. Next let

$\mathbf{y}=\mathbf{z}\times\mathbf{v}$ .

Using trigonometry, we can rotate x by θ around z to obtain the projection of the rotated vector v_rot:

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

Notice that both the vectors x and y have the same length. This follows from the following formulae for the scalar product and cross product:

$\mathbf{z} \cdot \mathbf{v} = |\mathbf{z}| \, |\mathbf{v}| \cos \phi \,$

$\mathbf{z} \times \mathbf{v} = |\mathbf{z}| \, |\mathbf{v}| \sin \phi \ \mathbf{n}$

where Φ denotes the angle between z and v and n is a to both orthogonal unit vector. Since z has unit length it immediately follows from the second formula that

$|\mathbf{y}| = |\mathbf{z} \times \mathbf{v}| = |\mathbf{v}| \sin \phi.$

The first formula gives the length of z(z•v) and because this forms a right triangle with v and x it becomes clear that this is also the length of x.

To get the rotated vector v, we have to add back the component of v parallel to z:

$(\mathbf{z} \cdot \mathbf{v}) \mathbf{z}$ .

So the result is

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

as required. Or in matrix notation:

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

Computing a rotation matrix

Denoting by $[\mathbf{z}]_\times$ the "cross-product matrix" for $\mathbf{z}$ , i.e.,

$[\mathbf{z}]_\times \mathbf{v} = \mathbf{z}\times\mathbf{v} = \left[\begin{array}{ccc} 0 & -z_3 & z_2 \\ z_3 & 0 & -z_1 \\ -z_2 & z_1 & 0 \end{array}\right]\mathbf{v}$ ,

the equation can be written as

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

where I is the 3×3 identity matrix. Thus we have a formula for the rotation matrix, R, corresponding to an axis angle vector, z:

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

Noting that $\mathbf{z} \mathbf{z}^\top = [\mathbf{z}]_\times^2 + I$ , we have

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

or, equivalently,

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

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

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

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Rodrigues' rotation formula

Contents

Derivation

Computing a rotation matrix

See also

References

External links