Euler angles

Euler angles are a set of three angles — traditionally called the precession angle φ, the nutation angle θ, and the spin angle ψ — that parameterise the orientation of a rigid body in three-dimensional space relative to a fixed inertial reference frame. They arise naturally in the analysis of rotations and are central to rigid-body mechanics, orbital mechanics, and aerospace engineering.

The conventional interpretation: starting from the fixed reference frame, you first rotate by φ about the z-axis, then by θ about the new x-axis, then by ψ about the new z-axis. The three rotations, composed, produce the final orientation. The specific ordering is a convention — other conventions (roll-pitch-yaw, Tait-Bryan angles) split the three rotations across different axes — and different fields use different conventions. For rigid-body mechanics, the ZXZ convention due to Euler himself is traditional; for aerospace, ZYX (yaw-pitch-roll) is more common.

Euler angles have two shortcomings: they suffer from gimbal lock (two of the three rotations degenerate when θ = 0 or π, losing a degree of freedom), and they are awkward for composing successive rotations. Modern orientation-tracking systems often use quaternions internally and convert to Euler angles only for display. But for analytical work on spinning tops, gyroscopes, and Earth-rotation problems, Euler's three angles remain the standard description.