Euler angels used for describing 3D rotations to easily understandable to readable way, and its commonly seen in user interfaces and applications. If you are dealing with 3D rotations, you need to be have understanding and familiar with both Euler angles and rotation matrices. Rotation matrices are helps to implement efficient rotations in software to define Euler angles.

Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about x by an angle α, a second rotation about y by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.

Euler Angle Conventions

Euler angles are a set of three angles – Raw, Pitch and Yaw used to specify the orientation—or change in orientation—of an object in 3D space. Each of the three angles in a Euler angle triplet specifies an elemental rotation around one of the axes in a 3D Cartesian coordinate system (see Figure 1).

Airplane coordinate system

Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups:

  1. Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)
  2. Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z).


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