Quaternion algebra

Quaternions are a type of mathematical number system that extend complex numbers. They consist of one real part and three imaginary parts, denoted as \( a + bi + cj + dk \). Quaternions are particularly useful in three-dimensional space, especially in computer graphics and robotics, because they efficiently represent rotations without the limitations of other methods like Euler angles. They allow for smooth transitions and avoid issues such as gimbal lock. In essence, quaternions provide a robust framework for handling rotations and orientations in 3D space, making them vital in various fields of science and engineering.