# Math Dopers... Quaternions? (a primer, please?) **URL:** \ **Category:** Factual Questions\ **Created:** [May 10, 2005, 11:48am UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203 "2005-05-10T11:48:23Z")\ **Posts on this page:** 15\ **Page:** 4
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [May 12, 2005, 7:32pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/61 "2005-05-12T19:32:00Z")
> [@Mathochist](#): > > Just define f on _all_ quaternions so it’s constant on lines through the origin. Bingo: no more renormalizing. How to do that is not entirely obvious. ---
**Author:** ![Punoqllads](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Punoqllads](https://boards.straightdope.com/u/Punoqllads)\ **Post date:** [May 12, 2005, 7:40pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/62 "2005-05-12T19:40:25Z")
> [@Mathochist](#): > > He’s just testing nearby values of a function trying to find a minimum. Renormalizing each time adds dozens if not hundreds of multiplications and divisions. ugh! > > Just define f on _all_ quaternions so it’s constant on lines through the origin. Bingo: no more renormalizing. Most modern processors do multiplies and inverse square roots fairly quickly. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [May 12, 2005, 8:33pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/63 "2005-05-12T20:33:59Z")
> [@Punoqllads](#): > > Most modern processors do multiplies and inverse square roots fairly quickly. Not doing them is even quicker. ---
**Author:** ![Shalmanese](https://avatars.discourse-cdn.com/v4/letter/s/45deac/32.png) [@Shalmanese](https://boards.straightdope.com/u/Shalmanese)\ **Post date:** [May 12, 2005, 9:06pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/64 "2005-05-12T21:06:46Z")
Basically, my algorithm is: compute f®, compute f(R’) for all R’ “close” to R. If f® is the minimum, return Else for the minimum f(R’), recursively apply the formula. Now, this would be trivial in translation because the points “near” (x, y, z) would just be (x +or- delta, y +or- delta, z +or- delta) I’m trying to find the equivilant for rotations. ---
**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\ **Post date:** [May 12, 2005, 9:32pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/65 "2005-05-12T21:32:47Z")
> [@Shalmanese](#): > > Basically, my algorithm is: > compute f(R), > compute f(R’) for all R’ “close” to R. > If f(R) is the minimum, return > Else for the minimum f(R’), recursively apply the formula. You may find it useful to precompute a set of near-identity rotations (you can do this with quaternions by creating a list L of elements close to 1; with rotation matrices, you can choose elements in the Lie algebra so(3) and find corresponding SO(3) elements, or just choose three Euler angles close to zero). This avoids having to do the time-consuming normalizations at each stage; for any rotation R, just evaluate f(Rx) for each x in L. You still have one matrix multiplication per evaluation; I’m not sure if that’s acceptable for you. This also uses a constant set of rotations (probably scaled with time, as for an annealing search: that is, you start with rotations of magnitude d; after a while, you further optimize with rotations of magnitude (say) d/2, using a new list L’; etc.), which may affect the convergence speed if f is sufficiently annoying. If this is a problem, you can occasionally rotate all of the elements of L by some random matrix to randomize the directions of your small-rotation matrices. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [May 12, 2005, 9:37pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/66 "2005-05-12T21:37:54Z")
Depending on what language you’re using, multiplication by a compile-time constant matrix can be made very fast, so that may not be a horrible issue. ---
**Author:** ![Punoqllads](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Punoqllads](https://boards.straightdope.com/u/Punoqllads)\ **Post date:** [May 12, 2005, 9:53pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/67 "2005-05-12T21:53:23Z")
I don’t remember if it’s already been stated, but quaternions representing orientations can be thought of as an axis (x, y, z) and a rotation about that axis, r, in the form: cos(r/2) + i \* sin(r/2) \* x + j \* sin(r/2) \* y + k \* sin(r/2) where the starting orientation to be pointing along the x axis with the y axis to its left and the z axis to its top. If you can multiply two unit quaternions together and will get another unit quaternion representing the rotation from the starting point to the orientation you would get to if you did those two rotations consecutively. So, if you gave some small number, s, you can use a rotation about the y and z axis by that amount. So say that some number s is small enough. you would come up with another number t = sqrt(1 - s[sup]2[/sup]). Suppose at the current iteration you have the quaternion a + bi + cj + dk you would multiply it by each of the four quaternions: t + sj t - sj t + sk t - sk you would then get the four quaternions: at + bti + ctj + dtk + asj + bsij + csj[sup]2[/sup] + dskj at + bti + ctj + dtk - asj - bsij - csj[sup]2[/sup] - dskj at + bti + ctj + dtk + asj + bsik + csjk + dsk[sup]2[/sup] at + bti + ctj + dtk - asj - bsik - csjk - dsk[sup]2[/sup] Remember, the ordering of the multiplications matters, and ij = -ji = k, jk = -kj = i, ki = -ik = j, and i[sup]2[/sup] + j[sup]2[/sup] + k[sup]2[/sup] = -1. Multiplying it all out: at - cs + (bt - ds)i + ctj + (dt + bs)k at + cs + (bt + ds)i + ctj + (dt - bs)k at - ds + (bt + cs)i + (ct - bs)j + dtk at + ds + (bt - cs)i + (ct + bs)j + dtk would be four small steps to the up, down, left, and right directions. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [May 12, 2005, 11:52pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/68 "2005-05-12T23:52:56Z")
> [@ultrafilter](#): > > How to do that is not entirely obvious. Well, not without a better notion of what this function is. Still, if I were writing the program I’d at least try to see if there’s a natural formula before coding up the ugly algorithm which renormalizes each time. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [May 12, 2005, 11:56pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/69 "2005-05-12T23:56:14Z")
> [@Punoqllads](#): > > at - cs + (bt - ds)i + ctj + (dt + bs)k > at + cs + (bt + ds)i + ctj + (dt - bs)k > at - ds + (bt + cs)i + (ct - bs)j + dtk > at + ds + (bt - cs)i + (ct + bs)j + dtk > > would be four small steps to the up, down, left, and right directions. This gives a slight displacement either way in two dimensions. As we’ve said before, the Lie group SO(3) is three-dimensional. I’m not saying your answer is wrong, but rather that it’s incomplete since there should be two more displacements to find. ---
**Author:** ![Punoqllads](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Punoqllads](https://boards.straightdope.com/u/Punoqllads)\ **Post date:** [May 13, 2005, 12:51am UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/70 "2005-05-13T00:51:26Z")
> [@Mathochist](#): > > This gives a slight displacement either way in two dimensions. As we’ve said before, the Lie group SO(3) is three-dimensional. I’m not saying your answer is wrong, but rather that it’s incomplete since there should be two more displacements to find. There’s also the alteration of twist, around the x axis, but **Shalmanese** implies that it’s only the direction that is important. If the complete orientation was important, you’d then need: at - bs + bti + (ct + ds)j + (dt - cs)k at + bs + bti + (ct - ds)j + (dt + cs)k too. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [May 13, 2005, 7:19am UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/71 "2005-05-13T07:19:38Z")
> [@Punoqllads](#): > > There’s also the alteration of twist, around the x axis, but **Shalmanese** implies that it’s only the direction that is important. If the complete orientation was important, you’d then need: > > at - bs + bti + (ct + ds)j + (dt - cs)k > at + bs + bti + (ct - ds)j + (dt + cs)k > > too. I don’t get that implication at all. what **Shalmanese** asked for was a way to find nearby rotations, not nearby directions. If the function depends on a rotation (as stated) rather than on only the direction into which the x-axis is rotated then it can depend on that third coordinate and leaving it out of the calculations will lead to a bad algorithm. ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 13, 2005, 12:11pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/72 "2005-05-13T12:11:44Z")
> [@tim314](#): > > I think you misunderstand me. I’m not denigrating Hamilton’s contributions to physics and mathematics, which were extremely significant. OK, no sweat. > [@tim314](#): > > But you seemed to be suggesting (and perhaps _I_ misunderstood _you_) that the reason quaternions turned out to be useful in quantum mechanics (which is what **Loopydude** was commenting on) is because all of modern physics was built on vector calculus, which was in turn based on quaternions. Nah, my only point was that quaternions didn’t get ignored for a hundred years, and that the guy who didn’t ignore them was Maxwell, whose _own_ work led inexorably to… ---
**Author:** ![Loopydude](https://avatars.discourse-cdn.com/v4/letter/l/e56c9b/32.png) [@Loopydude](https://boards.straightdope.com/u/Loopydude)\ **Post date:** [May 13, 2005, 2:49pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/73 "2005-05-13T14:49:07Z")
> [@Desmostylus](#): > > OK, no sweat.Nah, my only point was that quaternions didn’t get ignored for a hundred years… Well, jeez, nobody was saying _that_. It just seems to me that the discovery of abstractions like quaternions can easily predate the idea that they might have any connection whatsoever to the physical world; and I brought up the interesting anecdote of Heisenberg’s invention of matrix mechanics to show that while these abstractions don’t sit around collecting dust in some circles, in others, they remain completely unknown, and can be, in fact, reinvented for the specific purpose of describing physical phenomena. It’s that (seemingly) deep and unanticipated connection between these abstractions and unforseen observations that gives me a bit of a tingle. I readily admit, and am aware, that there’s a certain amount of mytholologizing going on when describing these Great Men, in no small part because the reality of the history is so messy and convoluted it makes a very difficult historical narrative. If I’m completely off base about the myths, even, then fine, call me a dunderhead. The received wisdom from the Greats to the dunderheads is sometimes there are these uncanny connections, an I find the idea fascinating. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [May 13, 2005, 6:03pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/74 "2005-05-13T18:03:28Z")
An earlier thread on the subject: [http://boards.straightdope.com/sdmb/showthread.php?t=260773&highlight=quarternions](http://boards.straightdope.com/sdmb/showthread.php?t=260773&highlight=quarternions) ---
**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\ **Post date:** [May 13, 2005, 6:36pm UTC](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203/75 "2005-05-13T18:36:39Z")
> [@Desmostylus](#): > > Nah, my only point was that quaternions didn’t get ignored for a hundred years, and that the guy who didn’t ignore them was Maxwell, whose _own_ work led inexorably to… OK, I gotcha. [Previous page](https://boards.straightdope.com/t/math-dopers-quaternions-a-primer-please/303203.md?page=3)