ühikukvaternioon

An ühikukvaternioon, or unit quaternion, is a quaternion whose norm (or magnitude) is equal to one. Quaternions are an extension of complex numbers, 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 with specific multiplication rules. For a quaternion q = a + bi + cj + dk to be a unit quaternion, its norm, calculated as sqrt(a^2 + b^2 + c^2 + d^2), must equal 1.

Unit quaternions have significant applications in three-dimensional geometry, particularly in representing rotations. Unlike Euler angles, which can suffer from gimbal lock, unit quaternions provide a unique and unambiguous representation of rotations. A rotation in 3D space can be uniquely defined by a unit quaternion. The multiplication of quaternions corresponds to the composition of rotations, making them a powerful tool for computer graphics, robotics, and aerospace engineering.

The identity quaternion, 1 (or 1 + 0i + 0j + 0k), is a unit quaternion and represents no rotation. The inverse of a unit quaternion is equal to its conjugate. The conjugate of q = a + bi + cj + dk is q* = a - bi - cj - dk. For a unit quaternion, q * q* = q* * q = 1. This property is crucial for calculating inverse rotations. The set of all unit quaternions forms a group under quaternion multiplication, known as SU(2), which is topologically equivalent to the 3-sphere.