quaternionte

Quaternions, sometimes referred to in older texts as quaternionte, are a number system that extends the complex numbers. A quaternion q can be written as 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 of quaternions is noncommutative: for example, ij = k, ji = -k, and so on.

Quaternions form a four-dimensional real algebra, with addition performed componentwise and multiplication defined by the above relations. Each nonzero quaternion has a conjugate q* = a - bi - cj - dk and a norm |q| = sqrt(a^2 + b^2 + c^2 + d^2). The inverse is q^{-1} = q* / |q|^2.

Unit quaternions (|q| = 1) are used to represent orientations in three-dimensional space. They provide a compact, robust way to compose rotations and to interpolate between orientations (slerp). A rotation can be applied to a vector v by the operation v' = q v q^{-1}, where v is treated as a pure quaternion (0 + xi + yj + zk). Unit quaternions form a group isomorphic to SU(2) and provide a double cover of the rotation group SO(3).

History and applications: quaternions were introduced by William Rowan Hamilton in 1843. They are widely used in computer graphics, robotics, aerospace, and virtual reality to represent and blend 3D rotations, avoiding singularities associated with Euler angles. They can be implemented with real-valued arithmetic and are compatible with standard linear algebra tools via matrix representations.

Quaternions are part of the broader family of hypercomplex numbers and have various extensions; they remain essential in modern 3D computation and attitude estimation methods.