[Geometry]Rodrigues' formula - Rotation vector

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http://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula

Rodrigues' rotation formula

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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:

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

[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:

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

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

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

which is exactly what we were looking for.

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

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

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

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

the cross-product $\mathbf{u} \times \mathbf{v}$ can be represented with the matrix product $\left[ \mathbf{u} \right]_\times \mathbf{v}$ and we have

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

where I is the 3x3 identity matrix. The expression‎ in the parenthesis can be identified as the rotation matrix R:

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

Considering that $\mathbf{u} \mathbf{u}^T = \left[ \mathbf{u} \right]_\times^2 + I$ , the expression‎ for the rotation matrix R is sometimes written as

$\begin{align} R = I + \sin\theta \left[ \mathbf{u} \right]_\times + (1 - \cos\theta) \left[ \mathbf{u} \right]_\times^2 \end{align}$

or, equivalently,

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

[edit] See also

Axis angle

[edit] External links

Eric W. Weisstein, Rodrigues' Rotation Formula at MathWorld.

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

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Rodrigues' Rotation Formula

Rodrigues' rotation formula gives an efficient method for computing the rotation matrix corresponding to a rotation by an angle about a fixed axis specified by the unit vector . Then is given by