Wikipedia:Reference desk/Archives/Mathematics/2014 December 5

= December 5 =

Problem with single rotation in SO(4) using quaternions
Re. the article Rotations in 4-dimensional Euclidean space, I'm having a problem with a single rotation. Suppose I have a unit quaternion QL=[a,b,c,d]. In the article it states "In quaternion notation, a rotation in SO(4) is a single rotation if and only if QL and QR are conjugate elements of the group of unit quaternions." So if I want to rotate a point at r=[0,0,0,1] (i.e. a point displaced 1 unit along the z axis), the rotated point will be:


 * $$r'=Q_L\, r\, Q_L^* = [0, 2 a c + 2 b d, -2 a b + 2 c d, a^2 - b^2 - c^2 + d^2]$$

which means that the point can never be rotated into the u dimension, it must be constrained to the xyz space. This seems wrong, one should be able to rotate [0,0,0,1] into the u dimension.PAR (talk) 16:50, 5 December 2014 (UTC)


 * I concur that it's wrong as written. The correct condition is that there is a unit quaternion a such that QL^* a = a QR.  Sławomir Biały  (talk) 17:32, 5 December 2014 (UTC)


 * Is there a short explanation for your statement? Does that mean (since a*=a-1) that for any two unit quaternions QL and a, that $$r'=Q_L\,a^*Q_L^*a$$ is a single rotation? Thanks for any help... PAR (talk) 05:31, 6 December 2014 (UTC)