Rotation Matrix - in Three Dimensions - Conversion From and To Axis-angle - Rotation Matrix From Axis and Angle

Rotation Matrix From Axis and Angle

For some applications, it is helpful to be able to make a rotation with a given axis. Given a unit vector u = (ux, uy, uz), where ux2 + uy2 + uz2 = 1, the matrix for a rotation by an angle of θ about an axis in the direction of u is

R = \begin{bmatrix} \cos \theta +u_x^2 \left(1-\cos \theta\right) & u_x u_y \left(1-\cos \theta\right) - u_z \sin \theta & u_x u_z \left(1-\cos \theta\right) + u_y \sin \theta \\ u_y u_x \left(1-\cos \theta\right) + u_z \sin \theta & \cos \theta + u_y^2\left(1-\cos \theta\right) & u_y u_z \left(1-\cos \theta\right) - u_x \sin \theta \\ u_z u_x \left(1-\cos \theta\right) - u_y \sin \theta & u_z u_y \left(1-\cos \theta\right) + u_x \sin \theta & \cos \theta + u_z^2\left(1-\cos \theta\right) \end{bmatrix}. {{ }}

This can be written more concisely as

where is the cross product matrix of u, ⊗ is the tensor product and I is the Identity matrix. This is a matrix form of Rodrigues' rotation formula, with

\mathbf{u}\otimes\mathbf{u} = \begin{bmatrix} u_x^2 & u_x u_y & u_x u_z \\ u_x u_y & u_y^2 & u_y u_z \\ u_x u_z & u_y u_z & u_z^2 \end{bmatrix},\qquad _{\times} = \begin{bmatrix} 0 & -u_z & u_y \\ u_z & 0 & -u_x \\ -u_y & u_x & 0 \end{bmatrix}.

If the 3D space is right-handed, this rotation will be counterclockwise for an observer placed so that the axis u goes in his direction (Right-hand rule).

Read more about this topic:  Rotation Matrix, In Three Dimensions, Conversion From and To Axis-angle

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