Kvaternione

Kvaternione are a number system that extends the complex numbers by adding three imaginary units. A quaternion is written as q = a + bi + cj + dk, with a, b, c, d real and i^2 = j^2 = k^2 = ijk = -1. The set H of quaternions forms a non-commutative, associative division algebra over the reals.

Quaternions can be represented as q = (a, u) where a ∈ R and u ∈ R^3. For p = (b, v) the product is qp = (ab - u·v) + (a v + b u + u × v).

Conjugate q* = a - u and norm N(q) = a^2 + ||u||^2. If q ≠ 0, the inverse is q^{-1} = q*/N(q). Unit quaternions (N(q) = 1) form a 3-sphere S^3.

Unit quaternions provide a representation of 3D rotations. A rotation by q maps a vector v to q v q^{-1} (treating v as a pure imaginary quaternion). This avoids gimbal lock and supports smooth interpolation (slerp).

Historically, quaternions were invented by William Rowan Hamilton in 1843 as a four-dimensional generalization of complex numbers. They gained practical importance in computer graphics, robotics, aerospace, and physics for describing orientations and rotations.

Quaternions also have matrix representations, linking them to 4×4 real or 2×2 complex matrices. This dual view underpins many numerical and geometric algorithms for rotations and orientation estimation.