Rotation in three dimensions

This application calculates the three-dimensional rotation of a point $p$ around an arbitrary axis. In three dimensions, the rotation axis can be determined from a directed line or a direction vector $\overrightarrow{v}$ and a (fixed) point $f$ which $\overrightarrow{v}$ goes through. The rotation is calculated using quaternion as follows. Let $P$ and $F$ be (pure imaginary) quaternions with $p$ and $f$ as vector parts, $\theta$ be the angle of rotation (positive for counterclockwise rotation), and unit quaternion of $\overrightarrow{v}$ defined as $$u=\cos \left(\frac{\theta }{2}\right)+\sin \left(\frac{\theta }{2}\right)\cdot \frac{\overrightarrow{v}}{\Vert \overrightarrow{v}\Vert }$$ then 3D rotation is calculated from the formula $$F+u(P-F)u^{-1}$$ Note that for unit quaternion $u^{-1}=\bar{u}$, the quaternion conjugate of $u$.

3D Rotation Calculator