Rodrigues' rotation formula

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 heta + mathbf{u} imes mathbf{v} cdot sin heta + langle mathbf{u}, mathbf{v} angle mathbf{u} cdot (1 - cos heta).

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&times;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:

$$egin{align} mathbf{w}_{rot} &= mathbf{w} cdot cos heta + mathbf{z} cdot sin heta \ &= (mathbf{v} - langle mathbf{u}, mathbf{v} angle mathbf{u}) cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta.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

$$egin{align} mathbf{v}_{rot} &= (mathbf{v} - langle mathbf{u}, mathbf{v} angle mathbf{u}) cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta + langle mathbf{u}, mathbf{v} angle mathbf{u} \ &= mathbf{v} cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta + langle mathbf{u}, mathbf{v} angle mathbf{u} cdot (1 - cos heta),end{align}

which is exactly what we were looking for.

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

$$egin{align} mathbf{v}_{rot} &= cos heta cdot mathbf{v} + sin heta cdot mathbf{u} imes mathbf{v} + (1 - cos heta) cdot mathbf{u} cdot mathbf{u}^T mathbf{v}end{align}

Substituting the cross product with matrix multiplication:

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

and multiplying with the identity matrix "I", we get

$$egin{align} mathbf{v}_{rot} &= cos heta cdot I mathbf{v} + sin heta cdot left [ mathbf{u} ight] _ imes mathbf{v} + (1 - cos heta) cdot mathbf{u} cdot mathbf{u}^T mathbf{v} \ &= left( cos heta I + sin heta left [ mathbf{u} ight] _ imes + (1 - cos heta) mathbf{u} mathbf{u}^T ight) mathbf{v} end{align}

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

$$egin{align} R = cos heta I + sin heta left [ mathbf{u} ight] _ imes + (1 - cos heta) mathbf{u} mathbf{u}^Tend{align}

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