yksikkökvaterniolla

Yksikkökvaternioni refers to a quaternion whose norm is equal to one. A quaternion is a number system that extends complex numbers, often represented as $a + bi + cj + dk$, where $a$, $b$, $c$, and $d$ are real numbers, and $i$, $j$, and $k$ are imaginary units with specific multiplication rules: $i^2 = j^2 = k^2 = ijk = -1$. The norm of a quaternion $q = a + bi + cj + dk$ is defined as $||q|| = \sqrt{a^2 + b^2 + c^2 + d^2}$. Therefore, a unit quaternion satisfies the condition $a^2 + b^2 + c^2 + d^2 = 1$.

Unit quaternions are particularly important in three-dimensional rotations. They provide a more efficient and numerically stable way to represent rotations compared to Euler angles or rotation matrices. A unit quaternion can uniquely represent any rotation in 3D space. Specifically, a rotation of an angle $\theta$ around an axis defined by a unit vector $(x, y, z)$ can be represented by the unit quaternion $q = \cos(\theta/2) + \sin(\theta/2)(xi + yj + zk)$. Applying this rotation to a vector $v$ can be achieved by the quaternion multiplication $v' = qvq^{-1}$, where $v$ is treated as a pure quaternion $(0, v_x, v_y, v_z)$ and $q^{-1}$ is the conjugate of $q$.

The set of all unit quaternions forms a Lie group called SU(2), which is also a double cover of the special orthogonal group SO(3), the group of all rotations in three-dimensional Euclidean space. This connection is fundamental in fields such as computer graphics, robotics, and physics for representing and manipulating orientations and rotations.