α ) is the signed angle between the N axis and the X axis ( x-convention).Įuler angles between two reference frames are defined only if both frames have the same handedness.Ĭonventions by intrinsic rotations.Using it, the three Euler angles can be defined as follows:

The geometrical definition (sometimes referred to as static) begins by defining the line of nodes (N) as the intersection of the planes xy and XY (it can also be defined as the common perpendicular to the axes z and Z and then written as the vector product N = z × Z). The axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z. Right: A simple diagram showing similar Euler angles. Left: A gimbal set, showing a z- x- z rotation sequence. In that case, the sequences of the first group are called proper or classic Euler angles. Sometimes, both kinds of sequences are called 'Euler angles'. Tait–Bryan angles are also called Cardan angles nautical angles heading, elevation, and bank or yaw, pitch, and roll. Tait–Bryan angles ( x- y- z, y- z- x, z- x- y, x- z- y, z- y- x, y- x- z).Proper Euler angles ( z- x- z, x- y- x, y- z- y, z- y- z, x- z- x, y- x- y).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: Therefore, any discussion employing Euler angles should always be preceded by their definition. Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. In the sections below, an axis designation with a prime mark superscript (e.g., z″) denotes the new axis after an elemental rotation.Įuler angles are typically denoted as α, β, γ, or ψ, θ, φ. The three elemental rotations may be extrinsic (rotations about the axes xyz of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system XYZ, solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation). The geometrical definition demonstrates that three composed elemental rotations (rotations about the axes of a coordinate system) are always sufficient to reach any target frame. Euler angles can be defined by elemental geometry or by composition of rotations.

90 rotation rule for geometry