Yksikköquaternion

Yksikköquaternion is the Finnish term for a unit quaternion. A quaternion is a number that extends the complex numbers, often used in mathematics and physics for representing rotations in three-dimensional space. A unit quaternion is a quaternion whose norm (or magnitude) is equal to one.

Mathematically, a quaternion is expressed in the form $a + bi + cj + dk$, where $a$, $b$, $c$, and $d$ are real numbers, and $i$, $j$, and $k$ are imaginary units. These units follow specific multiplication rules: $i^2 = j^2 = k^2 = ijk = -1$. The norm of a quaternion $q = a + bi + cj + dk$ is defined as $|q| = \sqrt{a^2 + b^2 + c^2 + d^2}$.

For a quaternion to be a unit quaternion, its norm must be 1, meaning $a^2 + b^2 + c^2 + d^2 = 1$. This property is crucial because unit quaternions can directly represent rotations. A rotation of an angle $\theta$ around an axis defined by a unit vector $(u_x, u_y, u_z)$ can be represented by a unit quaternion $q$ where $a = \cos(\theta/2)$ and $(b, c, d) = (u_x \sin(\theta/2), u_y \sin(\theta/2), u_z \sin(\theta/2))$.

Unit quaternions are highly valued in computer graphics, robotics, and aerospace engineering due to their ability to avoid gimbal lock, a phenomenon that can occur with other rotation representations like Euler angles. They provide a compact and efficient way to represent and interpolate rotations.