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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:
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]
Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by a unit vector k → = (k x, k y, k z)) and the rotation angle φ.The Euler parameters for this rotation are calculated as follows:
The quaternion can be related to the rotation vector form of the axis angle rotation by the exponential map over the quaternions, = /, where v is the rotation vector treated as a quaternion. A single multiplication by a versor, either left or right, is itself a rotation, but in four dimensions.
Extracting the angle and axis of rotation is simpler. Interpolation is more straightforward. See for example slerp. Quaternions do not suffer from gimbal lock as Euler angles do. Regardless, the rotation matrix calculation is the first step for obtaining the other two representations.
Other widely used methods include rotation quaternions, rotors, Euler angles, or rotation matrices. More specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. A unit vector may also be used to represent an object's normal vector direction or the relative direction between two points.
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...