Quaternions and spatial rotation

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,^[1] computer vision, robotics,^[2] navigation, molecular dynamics, flight dynamics,^[3] orbital mechanics of satellites,^[4] and crystallographic texture analysis.^[5]

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 $\theta$ radians about a unit axis $(X,Y,Z)$ that denotes the Euler axis is given by the quaternion $(C,X\,S,Y\,S,Z\,S)$ , where $C=\cos(\theta /2)$ and $S=\sin(\theta /2)$ .

Compared to rotation matrices, quaternions are more compact, efficient, and numerically stable. Compared to Euler angles, they are simpler to compose. However, they are not as intuitive and easy to understand and, due to the periodic nature of sine and cosine, rotation angles differing precisely by the natural period will be encoded into identical quaternions and recovered angles in radians will be limited to $[0,2\pi ]$ .

^ Shoemake, Ken (1985). "Animating Rotation with Quaternion Curves" (PDF). Computer Graphics. 19 (3): 245–254. doi:10.1145/325165.325242. Presented at SIGGRAPH '85.
^ J. M. McCarthy, 1990, Introduction to Theoretical Kinematics, MIT Press
^ Amnon Katz (1996) Computational Rigid Vehicle Dynamics, Krieger Publishing Co. ISBN 978-1575240169
^ J. B. Kuipers (1999) Quaternions and rotation Sequences: a Primer with Applications to Orbits, Aerospace, and Virtual Reality, Princeton University Press ISBN 978-0-691-10298-6
^ Karsten Kunze, Helmut Schaeben (November 2004). "The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis". Mathematical Geology. 36 (8): 917–943. doi:10.1023/B:MATG.0000048799.56445.59. S2CID 55009081.

[Shoemake-1] Shoemake, Ken (1985). "Animating Rotation with Quaternion Curves" (PDF). Computer Graphics. 19 (3): 245–254. doi:10.1145/325165.325242. Presented at SIGGRAPH '85.

[2] J. M. McCarthy, 1990, Introduction to Theoretical Kinematics, MIT Press

[3] Amnon Katz (1996) Computational Rigid Vehicle Dynamics, Krieger Publishing Co. ISBN 978-1575240169

[4] J. B. Kuipers (1999) Quaternions and rotation Sequences: a Primer with Applications to Orbits, Aerospace, and Virtual Reality, Princeton University Press ISBN 978-0-691-10298-6

[Lunze-5] Karsten Kunze, Helmut Schaeben (November 2004). "The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis". Mathematical Geology. 36 (8): 917–943. doi:10.1023/B:MATG.0000048799.56445.59. S2CID 55009081.

[1]

[2]

[3]

[4]

[5]