ühikukvaternioonid

Ühikukvaternioonid, also known as unit quaternions, are a specific type of quaternion with a norm of 1. A quaternion is a number system that extends complex numbers, consisting of a real part and three imaginary parts. Mathematically, a quaternion q can be written as q = a + bi + cj + dk, where a, b, c, and d are real numbers, and i, j, and k are imaginary units satisfying the rules i^2 = j^2 = k^2 = ijk = -1.

For a quaternion to be a ühikukvaternioon, its norm must be equal to 1. The norm of a quaternion q = a + bi + cj + dk is defined as |q| = sqrt(a^2 + b^2 + c^2 + d^2). Therefore, for a ühikukvaternioon, a^2 + b^2 + c^2 + d^2 = 1.

Ühikukvaternioonid are particularly important in three-dimensional geometry and computer graphics for representing rotations. Unlike Euler angles, which can suffer from gimbal lock, or rotation matrices, which can be computationally expensive and prone to accumulating errors, ühikukvaternioonid offer a more compact and stable representation. A rotation in 3D space can be uniquely represented by a ühikukvaternioon. Multiplying a vector by a ühikukvaternioon results in the rotated vector. This makes them a preferred choice for animation, robotics, and virtual reality applications.