ühikukvaterniooni

An ühikukvaternioon, or unit quaternion, is a quaternion with a magnitude of one. Quaternions are an extension of complex numbers, with the form a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary units satisfying i^2 = j^2 = k^2 = ijk = -1. The magnitude of a quaternion q = a + bi + cj + dk is defined as |q| = sqrt(a^2 + b^2 + c^2 + d^2). Therefore, an ühikukvaternioon satisfies the condition a^2 + b^2 + c^2 + d^2 = 1.

Ühikukvaternioonid are particularly useful in three-dimensional space for representing rotations. A unit quaternion can uniquely represent any rotation in 3D space. Specifically, a rotation by an angle theta around an axis defined by a unit vector (x, y, z) can be represented by the quaternion q = cos(theta/2) + (xi + yj + zk)sin(theta/2). When applied to a vector, this quaternion performs the specified rotation. This application makes them essential in fields like computer graphics, robotics, and aerospace engineering, as they avoid issues like gimbal lock that can occur with other rotation representations.