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A direct formula for the conversion from a quaternion to Euler angles in any of the 12 possible sequences exists. [2] For the rest of this section, the formula for the sequence Body 3-2-1 will be shown. If the quaternion is properly normalized, the Euler angles can be obtained from the quaternions via the relations:
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. [ 1 ] They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in three dimensional linear algebra .
This explains why, for example, the Euler angles appear to give a variable in the 3-torus, and the unit quaternions in a 3-sphere. The uniqueness of the representation by Euler angles breaks down at some points (cf. gimbal lock), while the quaternion representation is always a double cover, with q and −q giving the same rotation.
The angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation. The eigenvector corresponding to the eigenvalue of 1 is the accompanying Euler axis, since the axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix.
Charts on SO(3) Chasles' theorem (kinematics) ... Conversion between quaternions and Euler angles; D. Davenport chained rotations; E. Euler angles; Euler–Rodrigues ...
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T 3 to the real projective space RP 3, which is the same as the space of rotations for three-dimensional rigid bodies, formally named SO(3)) is not a local homeomorphism at every point, and thus at some points the rank (degrees ...
The quaternion formulation of the composition of two rotations R B and R A also yields directly the rotation axis and angle of the composite rotation R C = R B R A. Let the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by,