quaternioni

Quaternions are a number system that extends complex numbers. They form a four-dimensional associative algebra over the real numbers, invented by William Rowan Hamilton in 1843. A quaternion is commonly written q = a + bi + cj + dk, where a, b, c, d are real numbers and i, j, k are imaginary units satisfying i^2 = j^2 = k^2 = ijk = -1. The multiplication is noncommutative, with ij = k, jk = i, ki = j and ji = -k, kj = -i, ik = -j.

Conjugation and norm are basic operations. The conjugate of q is q* = a - bi - cj - dk, and the norm is |q| = sqrt(a^2 + b^2 + c^2 + d^2). If q ≠ 0, its inverse is q^{-1} = q*/|q|^2. The set of quaternions with unit norm forms a group under multiplication and is isomorphic to SU(2).

Geometrically, unit quaternions encode 3D rotations. A vector v = (x, y, z) is represented as a pure imaginary quaternion v = xi + yj + zk. A rotation by angle θ about a unit axis u is given by q = cos(θ/2) + u sin(θ/2). The rotated vector is obtained by v' = q v q^{-1}. This representation avoids some pitfalls of other parameterizations and supports smooth interpolation.

Quaternions have widespread applications in computer graphics, robotics, aerospace, and physics for orientation, rotation, and interpolation (notably spherical linear interpolation, slerp). They relate to the double cover of SO(3) through SU(2) and are part of broader algebraic structures, including normed division algebras (per Hurwitz’s theorem) and related extensions such as split-quaternions.