Ühikukvaternioone

Ühikukvaternioone, also known as unit quaternions, are a mathematical construct used to represent rotations in three-dimensional space. They are a specific type of quaternion where the norm (or magnitude) of the quaternion is equal to one. A quaternion itself is an extension of complex numbers, 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 with specific multiplication rules (i² = j² = k² = ijk = -1).

For a unit quaternion, the condition ||q|| = √(a² + b² + c² + d²) = 1 must hold. This normalization is crucial because it ensures that the associated rotation is "proper," meaning it preserves distances and orientations. Unit quaternions offer several advantages over other rotation representations like Euler angles or rotation matrices. They avoid the problem of gimbal lock, a singularity that can occur with Euler angles, and are generally more computationally efficient and numerically stable than rotation matrices.

The multiplication of two unit quaternions corresponds to the composition of the rotations they represent. If q1 represents rotation R1 and q2 represents rotation R2, then the quaternion q1q2 represents the combined rotation R2 followed by R1 (or vice versa depending on convention). This associative property makes them powerful for combining multiple transformations. Unit quaternions are widely used in computer graphics, robotics, aerospace engineering, and physics for tasks involving 3D rotations.