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

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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:

$r^{\prime }=Q_{L}r{Q}_L^{*}=[0,2ac+2bd,-2ab+2cd,a^2-b^2-c^2+d^2]$

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 Q_L and a, that $r^{\prime }=Q_L{a}^{*}{Q}_L^{*}a$ is a single rotation? Thanks for any help... PAR (talk) 05:31, 6 December 2014 (UTC)