Euler

Euler Angels

The Euler angles are three angles that can be used to describe the orientation of a rigid body with respect to a fixed coordinate system (such as the Cartesian or Polar )

The Euler angles are so frequently used that they have been used as the definition for the terms “Role”, “Pitch”, and “Yaw” by engineers that wish to refer the the orientation of a rigid body such as an aircraft, vessel, structure, or vehicle.

Geometrical Definitions

Let the three Euler angles be represented as $\psi,\ \phi,\ and\ \theta$ and the three dimensional axes of the original frame before applied rotation be denoted as $x,\ y,\ and\ z$ and such after rotation denoted as $X,\ Y,\ and\ Z$

Conventions for Achieving Orientation by Intrinsic Rotations

In order to achieve the desired orientation of the rigid body, we break down the rotation into three separate rotations and then describe the original rotation as a composition of these three rotations. At each of these rotations, we apply one of the Euler angles. For consistency we will set the definitions of the angles.

• Let $\psi$ represent the angle of rotation about the x-axis.
• Let $\phi$ represent the angle of rotation about the newly arrived y-axis (call it y-prime ($y'$))
• Let $\theta$ represent the angle of rotation about the newly arrived z-axis. (call it z-double-prime, as its the third z-axis we have see. ($z''$))