Perhaps I'm not understanding this, but left multiplication of points in R^4 (identified with H, the ring of quaternions) by any fixed unit quaternion q L_q: R^4 -> R^4 via L_q(x) := qx results in a rotation of R^4. (Same for right multiplication R_q(x) := xq.) If instead we're talking about a pure unit quaternion u (Re(u) = 0), then identifying R^3 with the pure quaternions H_0 := {x in H | Re(x) = 0} results in left multiplication by q f_q: R^3 -> R^3 via f_q(x) := qx yielding the cross product of q with x, which of course is a projection of R^3 onto the 2-plane perpendicular to q. So, I'm not sure in what sense a quaternion multiply computes a reflection. --Dan On Jun 26, 2014, at 6:07 AM, Henry Baker <hbaker1@pipeline.com> wrote:
. . . each quaternion multiply only computes a _reflection_, and you need 2 reflections to make a rotation.