Kvaternioina

Kvaternioina, commonly known as quaternions, are a number system that extends the complex numbers. They consist of a real part and three imaginary units, usually written as a + bi + cj + dk, with i^2 = j^2 = k^2 = ijk = -1. Multiplication in this system is non-commutative, meaning that generally pq ≠ qp for quaternions p and q. The set of quaternions forms a four-dimensional real division algebra, commonly denoted H.

As an algebra, a quaternion q = a + bi + cj + dk has a conjugate q* = a - bi - cj - dk and a norm N(q) = a^2 + b^2 + c^2 + d^2. If q ≠ 0, its inverse is q^{-1} = q*/N(q). The unit quaternions, those with N(q) = 1, form a group under multiplication and encode rotations in three-dimensional space. There is a close relationship between unit quaternions and the special orthogonal group SO(3): each unit quaternion corresponds to a rotation, and every rotation can be represented by two opposite unit quaternions.

Kvaternioina were introduced by William Rowan Hamilton in 1843 as a means to generalize complex numbers and to model spatial rotations more robustly. They provided a toolset that avoided some complications of Euler angles, such as gimbal lock, and inspired developments in linear algebra and geometric algebra. In modern applications, quaternions are widely used in computer graphics, robotics, aerospace engineering, and animation, particularly for stable rotation representations and smooth interpolation between orientations via spherical linear interpolation (SLERP).