the Euler angles
/ OY-ler /
The Euler angles are three angles that pin down the orientation of a rigid body -- the rotational analog of giving three coordinates for position. The question they answer is: given a reference orientation, exactly how is the body turned? Any orientation, no matter how twisted, can be reached by three successive rotations about well-chosen axes, and the three angles are how much you turn at each step.
In the common z-x-z convention of physics (Goldstein), you first rotate by phi about the space z-axis, then by theta about the new x-axis (the 'line of nodes'), then by psi about the new z-axis. The three angles carry names tied to the spinning top: phi is the precession angle, theta the nutation (tilt) angle, and psi the spin angle. Their rates of change combine to give the angular velocity; for a symmetric top in body coordinates, omega_3 = psi_dot + phi_dot cos(theta) along the symmetry axis, with the other components built from phi_dot, theta_dot, and psi_dot.
Euler angles are the standard coordinates for orientation in classical mechanics, spacecraft attitude, robotics, and computer graphics. Their one serious flaw is a coordinate singularity called gimbal lock: when theta = 0 the phi and psi rotations act about the same axis and become indistinguishable, so the parameterization degenerates. This is why navigation and graphics systems often switch to quaternions, which have no such singularity.
An aircraft's orientation is often reported as yaw, pitch, and roll -- an everyday set of Euler-like angles. As a fighter jet points its nose straight up (pitch = 90 degrees) the yaw and roll axes align, and the instruments hit gimbal lock, the same degeneracy Apollo navigators had to design around.
Three angles fix any orientation, but the parameterization degenerates at gimbal lock.
There is no single 'the' Euler-angle convention -- at least a dozen orderings are in use (z-x-z, z-y-z, x-y-z 'Tait-Bryan', and so on). Always state the convention, or formulas from two sources will silently disagree.