quaternion

A quaternion is a number system that extends the complex numbers to four dimensions. An element, called a quaternion, has a real part and three imaginary parts: q = a + b i + c j + d k, where a, b, c, d are real numbers and i, j, k satisfy i^2 = j^2 = k^2 = ijk = -1.

Quaternions form a noncommutative algebra. The multiplication rules are defined by i j = k, j k = i, k i = j and their reverses j i = -k, k j = -i, i k = -j. Conjugation is q* = a - b i - c j - d k, and the norm is N(q) = q q* = a^2 + b^2 + c^2 + d^2. If N(q) ≠ 0, the inverse is q^{-1} = q* / N(q). Unit quaternions satisfy N(q) = 1.

Quaternions provide a compact way to represent rotations in three dimensions. A rotation by angle θ about a unit vector u = (u_x, u_y, u_z) is described by the unit quaternion q = cos(θ/2) + (u_x i + u_y j + u_z k) sin(θ/2). To rotate a vector v, one treats v as a pure quaternion (0 + v_x i + v_y j + v_z k) and computes v' = q v q^{-1}. Unit quaternions correspond to orientation changes, and every spatial rotation has a corresponding quaternion.

Advantages include compact storage, smooth interpolation via slerp (spherical linear interpolation), and avoidance of gimbal lock common with Euler angles. Quaternions are widely used in computer graphics, robotics, aerospace, and physics for efficient and robust rotation computations. The concept was introduced by William Rowan Hamilton in 1843.