Without considering the possibility of using two different convertions for the definition of the rotation axes , there exists twelve possible sequences of rotation axes, divided into two groups:
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 after each elemental rotation).
Extrinsic and intrinsic rotations are relevant.
The definition of the Euler angles is as following,
represents the first rotation angle,
represents the second rotation angle,
represents the third rotation angle.
For intrinsic rotations in the order of X-Y-Z, the rotation matrix R can be calculated by:
For extrinsic rotations in the order of X-Y-Z, the rotation matrix R can be calculated by:
where
The function is designed according to this set of conventions:
Each matrix is meant to represent an active rotation (the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed reference frame and give as a result the coordinates of a rotated vector defined in the same reference frame).
For and , the valid range is (−π, π].
For , the valid range is [−π/2, π/2] or [0, π].
For Tait–Bryan angles, the valid range of is [−π/2, π/2]. When transforming a quaternion to Euler angles, the solution of Euler angles is unique in condition of . If or , there are infinite solutions. The common name for this situation is gimbal lock. For Proper Euler angles,the valid range of is in [0, π]. The solutions of Euler angles are unique in condition of . If or , there are infinite solutions and gimbal lock will occur.
Enumerator
INT_XYZ
Intrinsic rotations with the Euler angles type X-Y-Z.
INT_XZY
Intrinsic rotations with the Euler angles type X-Z-Y.
INT_YXZ
Intrinsic rotations with the Euler angles type Y-X-Z.
INT_YZX
Intrinsic rotations with the Euler angles type Y-Z-X.
INT_ZXY
Intrinsic rotations with the Euler angles type Z-X-Y.
INT_ZYX
Intrinsic rotations with the Euler angles type Z-Y-X.
INT_XYX
Intrinsic rotations with the Euler angles type X-Y-X.
INT_XZX
Intrinsic rotations with the Euler angles type X-Z-X.
INT_YXY
Intrinsic rotations with the Euler angles type Y-X-Y.
INT_YZY
Intrinsic rotations with the Euler angles type Y-Z-Y.
INT_ZXZ
Intrinsic rotations with the Euler angles type Z-X-Z.
INT_ZYZ
Intrinsic rotations with the Euler angles type Z-Y-Z.
EXT_XYZ
Extrinsic rotations with the Euler angles type X-Y-Z.
EXT_XZY
Extrinsic rotations with the Euler angles type X-Z-Y.
EXT_YXZ
Extrinsic rotations with the Euler angles type Y-X-Z.
EXT_YZX
Extrinsic rotations with the Euler angles type Y-Z-X.
EXT_ZXY
Extrinsic rotations with the Euler angles type Z-X-Y.
EXT_ZYX
Extrinsic rotations with the Euler angles type Z-Y-X.
EXT_XYX
Extrinsic rotations with the Euler angles type X-Y-X.
EXT_XZX
Extrinsic rotations with the Euler angles type X-Z-X.
EXT_YXY
Extrinsic rotations with the Euler angles type Y-X-Y.
EXT_YZY
Extrinsic rotations with the Euler angles type Y-Z-Y.
EXT_ZXZ
Extrinsic rotations with the Euler angles type Z-X-Z.
EXT_ZYZ
Extrinsic rotations with the Euler angles type Z-Y-Z.
EULER_ANGLES_MAX_VALUE
The documentation for this class was generated from the following file: