![]() Let's say p and q are defined as below, and q is normalized ( ) įor a typical usage, rotating a 3D vector, we can set the scalar component to 0 for p. In this section, we calculate from arbitrary Quaternions, p and q. Here explains why this double multiplication is required and the proof of the length conservation. This special double multiplication is called "conjugation by q". ![]() Therefore, the final equation for the rotation in Quaternion becomes. Since we multiply q twice, front and back, we only need a half rotation angle for q. However, the length of p is changed during multiplication, so we multiply q* ( conjugate of q) again at the back of p in order to cancel out the length changes. ![]() To rotate the quaternion p, we simply multiply the quaternion q. ( Note that we only uses half angle to define the rotation quaternion q because we are going to multiply q twice.) )Īnd, the Quaternion representation for the rotation axis vector and the half rotation angle can be written as The vector can be converted as Quaternion form ![]() Suppose there is a 3D vector, and rotate it along an arbitrary axis (unit vector) by angle. Qpq* means rotating P to P' along R axis by θ
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