Quaternions and spatial rotation in the context of "Axis–angle representation"

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

When used to represent rotation, unit quaternions are also called rotation quaternions as they represent the 3D rotation group. When used to represent an orientation (rotation relative to a reference coordinate system), they are called orientation quaternions or attitude quaternions. A spatial rotation around a fixed point of ${\displaystyle \theta }$ radians about a unit axis ${\displaystyle (X,Y,Z)}$ that denotes the Euler axis is given by the quaternion ${\displaystyle (C,X\,S,Y\,S,Z\,S)}$, where ${\displaystyle C=\cos(\theta /2)}$ and ${\displaystyle S=\sin(\theta /2)}$.