Euler angles

lastmod 2018-08-17created 2014-05-032 edits

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. Euler angles are three angles introduced by Leonhard Euler to represent, in three-dimensional space, the orientation in which a rigid body is placed.

The angles of the rotated coordinate system are defined as follows. Roll, Pitch, Yaw

As in the figure beside, if the given three-dimensional coordinate system is (x, y, z) and the rotated coordinate system is (X,Y,Z), the orientation of the rigid body can be expressed by the following three angles.

1. α (or ψ): the angle of the x-y axes rotated about the z-axis (blue) as the rotation axis.
2. β (or θ): the angle of the z-y axes rotated about the rotated x-axis (i.e., the N-axis, green) as the rotation axis.
3. γ (or ϕ): the angle of the x-y axes rotated about the z-axis rotated above (i.e., the Z-axis, red) as the rotation axis.

In this way, the orientation of a rigid body can be expressed with three angles. Besides Euler angles, another way to express the attitude of a rigid body is the yaw, pitch, roll (RPY; Roll-Pitch-Yaw) method, X-Y-Z coordinates in which, for the coordinate system (X, Y, Z), rotation about the X-axis is called Roll, rotation about the Y-axis Pitch, and rotation about the Z-axis Yaw.

Favorite site

• Wikipedia (en) description of Euler angles ¹

References