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The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle.
Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the ...
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,
Therefore, a = 0 and b 2 + c 2 + d 2 = 1. In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere. Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0). [citation needed] [d]
The planar quaternions can then be seen to represent a subset of those rigid-body motions. This requires some familiarity with the way the dual quaternions act on Euclidean space. We will not describe this approach here as it is adequately done elsewhere. The dual quaternions can be understood as an "infinitesimal thickening" of 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. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two different unit quaternions.
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.
A spatial rotation is a linear map in one-to-one correspondence with a 3 × 3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R , there is a nonzero vector n for which Rn = n ; this is exactly the claim that n is an eigenvector of R ...