kvaternäärisestä

Kvaternääriset, more commonly known as quaternions, are a number system that extends complex numbers. They were first described by the Irish mathematician William Rowan Hamilton in 1843. A quaternion is typically represented 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: i² = j² = k² = ijk = -1, and ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j.

The primary application of quaternions is in representing rotations in three-dimensional space. They offer a more compact and computationally efficient way to perform rotations compared to other methods like Euler angles or rotation matrices. This makes them invaluable in fields such as computer graphics, robotics, and aerospace engineering where precise and smooth rotations are crucial. Unlike Euler angles, quaternions do not suffer from gimbal lock, a phenomenon that can cause loss of a degree of freedom in certain orientations.

The algebra of quaternions is non-commutative, meaning that the order of multiplication matters (e.g., ij ≠ ji). This property is fundamental to their structure and their effectiveness in describing rotations. While they have a rich mathematical theory, their practical utility is largely centered on their ability to represent and manipulate 3D orientations and rotations.