Euler Angles

A way of describing an object's orientation using three angles of rotation applied one after another about coordinate axes.

Euler angles, introduced by the Swiss mathematician Leonhard Euler, describe a rigid body's orientation through three successive rotations about coordinate axes, commonly written α, β, γ or φ, θ, ψ. There are 12 possible rotation-order sequences: ones where the first and last axis are the same (like Z-X-Z) are the classical Euler angles, and ones where all three axes differ (like Z-Y-X) are called Tait-Bryan angles — the familiar roll, pitch, and yaw fall into this second category. Rotating about axes that move with the object is called intrinsic rotation; rotating about fixed axes is called extrinsic. Euler angles are intuitive and easy to read, but the same three numbers describe a different orientation depending on the rotation order and the intrinsic/extrinsic convention used, and at certain angles two of the rotation axes line up and a degree of freedom is lost — a problem called gimbal lock. For this reason, code typically represents rotation internally with quaternions or rotation matrices, and neural networks predicting rotation often use a 6D representation instead.

ExampleAn aircraft that turns 30° to the right, then pitches its nose up 10°, with no roll, can be written as a Z-Y-X sequence of Euler angles: yaw 30°, pitch 10°, roll 0°.

Related

Roll-Pitch-Yaw (RPY) · Gimbal Lock · Quaternion · Rotation Matrix · Intrinsic vs. Extrinsic Rotations · 6D Rotation Representation

Sources

Wikipedia: Euler angles
Wikipedia: Gimbal lock
On the Continuity of Rotation Representations in Neural Networks（Zhou et al.）