quaternion multiplication order changes rotation direction?

temp_ie_cant_thinkof_name · 2004-05-26T13:49:07

I'm learning about quaternions and how they're used for rotations. In the proof that they represent a rotation b/v they're equivalent to the angle-axis formula, if you multiply this whole thing out (a lot easier using vectors): q*p*~q, where q = (cos t/2, w*sin t/2), ~q = conjugate of q, p = (0, v), |w|=1, N(q)=1, and v is the vector being rotated about w by theta (t). Well, I was doing the proof myself, but instead of starting by multiplying p times ~q and then q times the former, respectively, I did q times p and then (qp)*~q, which is the intuitive left-right approach. Well, it turns out that the order in which you do them, one or the other of the above, changes the direction of the angle of rotation. The first one mentioned becomes the axis-angle formula, which is: v' = v*cos t + (1-cos t)(w.v)w + (vxw)sin t ie, q(p*~q) = v' while the latter one mentioned becomes v'' = v*cos t + (1-cos t)(w.v)w + (wxv)sin t ie, (q*p)*~q = v'', note the bold part. Basically, instead of adding the final vector offset in the "right" direction derived by the axis-angle method using the righ-hand rule for everything, it rotates in the -t direction, hmm... Does this become confusing when you actually implement it and expect to rotate in the same direction, or is it the de facto practice just to multiply quaternions in the "backwards" order (the first mehtod, q(p*~q). I'll post the proof if you want it, or you can just believe me. edit: if i'm a little sketchy, say so. I forgot to mention, but you probably know, wxv = -vxw. [edited by - temp_ie_cant_thinkof_name on May 25, 2004 1:45:23 PM]

Math and Physics Programming

Started by temp_ie_cant_thinkof_name May 25, 2004 12:43 PM

12 comments, last by temp_ie_cant_thinkof_name 19 years, 11 months ago

szinkopa

198

May 26, 2004 09:02 AM

Your mistake was the following:

You thought that (a * b) * c == a * (b * c) (vector dot product), but it''s not true.

It isn''t associative ! But it''s commutative.

And quaternion mult. is associative, but not commutative

Cross product is neither associative, neithaer commutative.

                |Commutative|Associative|Distibutive for addition----------------+-----------+-----------+------------------------dot product     |    Yes    |     No    |     Yes----------------+-----------+-----------+------------------------cross product   |     No    |     No    |     Yes----------------+-----------+-----------+------------------------quaternion mult.|     No    |    Yes    |     Yes----------------+-----------+-----------+------------------------matrix mult.    |     No    |    Yes    |     Yes

capn_midnight | Captain Midnight | deviantArt
ACM | SIGGRAPH | Generation5

temp_ie_cant_thinkof_name

100

Author

May 26, 2004 12:30 PM

quote:
Now comparing:The scalar component is the same because v1 * (v2 x v3)=(v1 x v2) * v3

Do you have a proof or reference for this?

Thanks for showing that to me, btw, but I won''t be sure until i conduct a proof myself

I''ll also post the work I had trouble with with the rotation quaternions, you could probably see where I made my mistake.

"I study differential and integral calculus in my spare time." -- Karl Marx

szinkopa

198

May 26, 2004 01:16 PM

OK.

v1 * (v2 x v3)  =?=  (v1 x v2) * v3 v1(x1,y1,z1), v2(x2,...They are a scalar value, I will calculate both: v1 * (v2 x v3) = x1 * (y2 * z3 - z2 * y3) + y1 * (z2 * x3 - x2 * z3) + z1 * (x2 * y3 - y2 * x3) = x1 * y2 * z3 + y1 * z2 * x3 + z1 * x2 * y3 - z1 * y2 * x3 - y1 * x2 * z3 - x1 * z2 * y3 (v1 x v2) * v3=(y1 * z2 - z1 * y2) * x3 + (z1 * x2 - x1 * z2) * y3 + (x1 * y2 - y1 * x2) * z3 =y1 * z2 * x3 +z1 * x2 * y3 +x1 * y2 * z3 -x1 * z2 * y3 -y1 * x2 * z3 -z1 * y2 * x3

They are the same.

capn_midnight | Captain Midnight | deviantArt
ACM | SIGGRAPH | Generation5

temp_ie_cant_thinkof_name

100

Author

May 26, 2004 01:45 PM

Thanks for proving that for me, szinkopa!

[edited by - temp_ie_cant_thinkof_name on May 26, 2004 2:49:07 PM]

"I study differential and integral calculus in my spare time." -- Karl Marx

quaternion multiplication order changes rotation direction?

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