Euler Angles

Ask a pilot how their aircraft is oriented and they won't hand you a rotation matrix — they'll say "nose up ten degrees, banked thirty to the left, heading north-east". Those three plain-English numbers are Euler angles: yaw, pitch and roll. They are the most human-readable way to name an orientation, and — for exactly the same reason — one of the most persistent sources of bugs in a game engine.

Three turns about three axes

The idea is to reach any orientation by three successive axis rotations, each with its own intuitive name.

Step 1 — yaw: turn about the up axis. Like shaking your head "no", this swings the heading left and right. Taking up as z, yaw by \psi is

R_z(\psi) = \begin{bmatrix} \cos\psi & -\sin\psi & 0 \\ \sin\psi & \cos\psi & 0 \\ 0 & 0 & 1 \end{bmatrix}.

Step 2 — pitch: tip about the right axis. Like nodding "yes", this raises and lowers the nose. Taking right as y, pitch by \theta is

R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{bmatrix}.

Step 3 — roll: bank about the forward axis. Like tilting your head toward a shoulder, this banks left and right. Taking forward as x, roll by \phi is

R_x(\phi) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi & \cos\phi \end{bmatrix}.

Multiply them into one orientation

Step 4 — compose the three rotations. The full orientation is the matrix product of the three elementary spins:

R = R_z(\psi)\,R_y(\theta)\,R_x(\phi).

Step 5 — read the order right-to-left. Applied to a vector \mathbf{v}, the rightmost matrix acts first:

R\,\mathbf{v} = R_z(\psi)\,\Big(R_y(\theta)\,\big(R_x(\phi)\,\mathbf{v}\big)\Big),

so the object is rolled, then pitched, then yawed. Order and axis convention matter. Because 3-D rotations don't commute (see composing transformations), R_z R_y R_x and R_x R_y R_z give different orientations from the very same three angles. There is no universal convention: one engine yaws first, another rolls first; one calls up z, another calls it y. Reading another team's Euler angles without knowing their convention is a reliable way to fly a spaceship into a wall.

An orientation can be specified by three angles applied as successive axis rotations:

Euler angles are a joy to author. A designer scrubbing three sliders in an editor — yaw, pitch, roll — gets instant, legible control over an orientation, and a keyframe of three numbers is trivial to store and to interpolate by eye. For posing a streetlamp or animating a turret, nothing beats them.

Engines, though, treat them warily, and not only for the ordering trap above. Stack the rotations a certain way and two of the three axes can line up, collapsing three independent dials into two — the orientation loses a whole degree of freedom and the controls fight each other. That failure has a name, gimbal lock, and it is the reason engines convert Euler angles to a matrix or a quaternion the moment the artist looks away. Read them in, compute with something better.

Euler angles are a fine format for a human to type in — but a poor format to animate between. It's tempting to smoothly blend from one pose to another by lerping each of yaw, pitch and roll independently, since they're "just three numbers". Don't: turning three angles independently doesn't turn the object smoothly. Each angle wraps and jumps on its own schedule, so the blended path can twist the long way around, wobble, or even sail straight through a gimbal lock pose in the middle of the blend — even when the start and end orientations look perfectly ordinary.

The robust fix is to interpolate the orientation itself, not its three Euler coordinates: convert both poses to unit quaternions and slerp between them. Slerp always takes the shortest arc on the rotation "sphere", so it can't overshoot into gimbal lock or take the long way round. Keep Euler angles for the inspector panel; keep quaternions for the motion.

Pose an object

Drag the three sliders and watch the little frame take its orientation, shown in isometric projection. The rotation applied is R_z(\psi)\,R_y(\theta)\,R_x(\phi): roll first, then pitch, then yaw. Notice how the same three numbers, combined in this fixed order, name one orientation — and imagine the confusion if another engine multiplied them the other way around.