📐 Quaternion Rotation — Axis-Angle & Slerp Explorer

A quaternion is a four-component number of the form q = w + xi + yj + zk, where w is the scalar (real) part and x, y, z form the vector (imaginary) part. A unit quaternion, where w²+x²+y²+z²=1, can represent any 3D rotation: the vector part encodes the rotation axis and the scalar part encodes half the rotation angle.

Given a unit rotation axis (x, y, z) and an angle θ, the rotation quaternion is w = cos(θ/2) and the vector part is sin(θ/2) multiplied by the axis. This axis-angle to quaternion conversion is exactly what the Axis X/Y/Z and Angle sliders in this simulation compute in real time.

Gimbal lock happens when using Euler angles (pitch, yaw, roll) and one rotation axis lines up with another, so two of the three rotational degrees of freedom collapse into one, permanently losing the ability to rotate independently around one axis. Quaternions avoid this because they represent orientation as a single point on the surface of a 4D unit sphere, with no separate axes that can ever become aligned or degenerate.

Slerp (spherical linear interpolation) blends two orientation quaternions along the shortest arc on the 4D unit sphere, producing rotation at a constant angular speed with no jumps or distortion. Linearly interpolating the raw w, x, y, z numbers instead would leave the sphere’s surface, requiring renormalisation and producing uneven, non-constant-speed rotation.

Quaternions are the standard way to track orientation in 3D graphics engines, video games, robotics, drones, spacecraft attitude control and virtual reality headsets, because they are compact, numerically stable, free of gimbal lock, and interpolate smoothly — all properties that make them preferable to rotation matrices or Euler angles for real-time orientation tracking.