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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:
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The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues' rotation formula (named after Olinde Rodrigues ), a method of calculating the position of a rotated point, is used in some software applications, such as flight simulators and computer games .
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 ...
They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. The most common orientation representations are the rotation matrices, the axis-angle and the quaternions, also known as Euler–Rodrigues parameters, which provide another mechanism for representing 3D rotations. This is equivalent to the special ...
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 Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to p = a, q = −b, r = −c, s = −d, or in quaternion representation: Q R = Q L ′ = Q L −1. The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations
Problems of this sort are inevitable, since SO(3) is diffeomorphic to real projective space P 3 (R), which is a quotient of S 3 by identifying antipodal points, and charts try to model a manifold using R 3. 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.