What is a Quaternion?
A quaternion is a number system that extends complex numbers. Introduced by William Rowan Hamilton in 1843, a quaternion consists of one real part and three imaginary parts, typically written in the form q = w + xi + yj + zk, where w, x, y, z are real numbers, and i, j, k are imaginary units that follow specific multiplication rules (e.g., i² = j² = k² = ijk = -1). Quaternions are particularly powerful for representing rotations in three-dimensional space.
Key Principles of Quaternion Operations
Unlike vectors or rotation matrices, quaternions offer a more elegant and computationally efficient way to perform 3D rotations without encountering issues like 'Gimbal lock'. Key operations include addition (adding corresponding components), multiplication (which is non-commutative, meaning the order matters, similar to matrix multiplication), and conjugation (changing the sign of the imaginary parts). A unit quaternion, where its magnitude is one, is specifically used for rotations.
Practical Application: 3D Graphics and Robotics
In computer graphics, quaternions are extensively used to calculate and interpolate rotations for objects like cameras, characters, or vehicles. For instance, when animating a character's arm swinging, a sequence of quaternion rotations can smoothly transition the arm through different orientations, providing realistic movement. Similarly, in robotics, they help control the orientation of robotic arms or drones, avoiding abrupt changes in movement.
Advantages of Using Quaternions for Rotations
Quaternions offer several advantages over other rotation representations, such as Euler angles or rotation matrices. They are more compact than rotation matrices, requiring only four numbers instead of nine. Crucially, they prevent Gimbal lock, a phenomenon where two axes of rotation become aligned, causing a loss of a degree of freedom and making further rotations unpredictable. Quaternions also allow for straightforward spherical linear interpolation (Slerp), which produces smooth and uniform rotation animations.