realquaternion

Realquaternion is a term used to describe the quaternion algebra over the real numbers. In this context, a realquaternion is an element q = a + bi + cj + dk, where a, b, c, and d are real numbers and the units i, j, k satisfy the relations i^2 = j^2 = k^2 = ijk = -1, with ij = k, ji = -k, and cyclic permutations. This establishes a four-dimensional real vector space with a non-commutative multiplication.

The realquaternions form a real associative division algebra, often denoted H. Key operations include the conjugate q* = a - bi - cj - dk and the norm N(q) = q q* = a^2 + b^2 + c^2 + d^2. For q ≠ 0, the inverse is q^{-1} = q* / N(q). These properties make the realquaternions a robust setting for representing and manipulating orientations in three-dimensional space.

Unit realquaternions, those with N(q) = 1, correspond to rotations in 3D. They can be used to encode a rotation about a unit axis by a certain angle, and a rotation of a vector v is performed as q v q^{-1} in suitable representations. Unit realquaternions avoid some issues of other rotation representations, such as gimbal lock, and support smooth interpolation through methods like spherical linear interpolation (slerp).

Applications of realquaternions are widespread in computer graphics, robotics, aerospace, and simulation, where robust handling of 3D orientations and efficient composition of rotations are important. In practice, they provide a compact, numerically stable alternative to rotation matrices for many tasks, while offering convenient algebraic properties for composition and interpolation.

See also: Quaternion, rotation, slerp, axis-angle.