Yksikkökvaternionen

Yksikkökvaternionen, often referred to as a unit quaternion, is a type of quaternion where the norm (or magnitude) is equal to one. A quaternion is a number system that extends complex numbers, with four components: one real part and three imaginary parts, typically represented as $a + bi + cj + dk$, where $i, j, k$ are the imaginary units satisfying $i^2 = j^2 = k^2 = ijk = -1$. For a unit quaternion $q = a + bi + cj + dk$, its norm is defined as $||q|| = \sqrt{a^2 + b^2 + c^2 + d^2}$. Therefore, for a unit quaternion, $a^2 + b^2 + c^2 + d^2 = 1$.

Unit quaternions are particularly important in three-dimensional geometry and computer graphics. They provide a computationally efficient and mathematically elegant way to represent rotations in 3D space. Unlike Euler angles, which can suffer from gimbal lock, unit quaternions avoid this singularity. A rotation by an angle $\theta$ around an axis represented by a unit vector $(x, y, z)$ can be expressed by the unit quaternion $q = \cos(\theta/2) + (x\sin(\theta/2))i + (y\sin(\theta/2))j + (z\sin(\theta/2))k$. The real part of this quaternion corresponds to the scalar component, and the imaginary part corresponds to the vector component representing the axis of rotation scaled by half the sine of the rotation angle. Multiplication of two unit quaternions corresponds to the composition of their respective rotations.