It's easy to get muddled up over quaternions, a fashion said to have been started by Hamilton himself. There are two traps: firstly, quaternions are non-commutative. As a result, rotation of a 3-space point V via V -> (1/X) V X --- where X itself represents only a half-angle rotation --- does not collapse into a single product as happens with complex numbers in 2-space. [By the way, I prefer traditional group-theoretic conjugation over the back-to-front matrix convention.] Secondly is the abiding confusion between axial or covariant vectors representing points, and polar or contravariant vectors representing directions (rotation axes): the two types transform and normalise quite differently. That is why the rotation of a rotation Y can be achieved by a single composition Y -> Y X , unlike rotation of a point. Re HB << x -> -y conj(x) y >> If you will insert a conjugation of your vector, you are indeed liable to find the result reflected! Fred Lunnon On 6/26/14, Henry Baker <hbaker1@pipeline.com> wrote:
Read Coxeter's paper.
Theorem 5.1. The reflection in the hyperplane sum(y_nu x_nu)=0 is represented by the transformation x -> -y conj(x) y. The product of two such reflections ... is a rotation.
Theorem 5.2. The general rotation through angle phi (about a plane [this is 4D, remember]) is x -> a x b, where N(a)=N(b)=1 and S(a)=S(b)=cos(phi/2). Conversely, [every] transformation x -> a x b (N(a)=N(b)=1) is a rotation whenever S(a)=S(b).
At 09:17 AM 6/26/2014, Dan Asimov wrote:
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.
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