Quaternion

Description

In mathematics, the quaternions form a number system similar to the complex numbers, with the usual arithmetical operations of addition, subtraction, multiplication, and division, but with four real-number components instead of two. Unlike with the complex numbers, quaternion multiplication is not commutative, meaning that the result of multiplying two quaternions depends on their order. Quaternions can be used to represent vectors in three-dimensional space, which provides a definition of the quotient of two vectors.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance imaging and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

As an abstract mathematical structure, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. Because of their non-commutative multiplication, they do not form a field.

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, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

Using quaternions as rotations

In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle $\theta$ about a fixed axis (called the Euler axis) that runs through the fixed point. The Euler axis is typically represented by a unit vector $\overrightarrow{u}$. Therefore, any rotation in three dimensions can be represented as a vector $\overrightarrow{u}$ and an angle $\theta$.

Quaternions give a simple way to encode this axis–angle representation using four real numbers, and can be used to apply (calculate) the corresponding rotation to a position vector (x,y,z), representing a point relative to the origin in $R^3$.