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Quaternions and spatial rotation 131
Describing rotations with quaternions
Let (w,x,y,z) be the coordinates of a rotation by around the axis as previously described. Define the quaternion
where is a unit vector. Let also be an ordinary vector in 3 dimensional space, considered as a quaternion with a
real coordinate equal to zero. Then it can be shown (see next section) that the quaternion product
yields the vector upon rotation of the original vector by an angle around the axis . The rotation is
clockwise if our line of sight points in the direction pointed by . This operation is known as conjugation by q.
It follows that quaternion multiplication is composition of rotations, for if p and q are quaternions representing
rotations, then rotation (conjugation) by pq is
which is the same as rotating (conjugating) by q and then by p.
,
The quaternion inverse of a rotation is the opposite rotation, since . The square of a quaternion
rotation is a rotation by twice the angle around the same axis. More generally q n is a rotation by n times the angle
around the same axis as q. This can be extended to arbitrary real n, allowing for smooth interpolation between spatial
orientations; see Slerp.
Proof of the quaternion rotation identity
Let be a unit vector (the rotation axis) and let . Our goal is to show that
yields the vector rotated by an angle around the axis . Expanding out, we have
where and are the components of perpendicular and parallel to respectively. This is the formula of a
rotation by around the axis.