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Quaternions

Anonymous Tue 30 Nov 15:24:57 2021 No.13907215 View Reply Original Report

Quoted By: >>13907229 >>13908720 >>13911628 >>13911647 >>13911817 >>13911837 >>13911844 >>13912278 >>13912306 >>13915566 >>13915644 >>13915671

How the fuck do they work?

I just watched this video: https://www.youtube.com/watch?v=d4EgbgTm0Bg

And I only came back more confused.

What I know so far:
- Quaternions are a four dimensional representation of rotations in three dimensional space
- Quaternions avoid "gimbal lock" (I don't understand how gimbal lock is a problem).
- Quaternions make it easy to interpolate between rotations in 3d space and fast to compute (this makes perfect sense)
- They are somehow related to complex numbers???

Anonymous

Anonymous Tue 30 Nov 2021 15:30:00 No.13907229 Report

Quoted By:

>>13907215
Sounds like it could be a number in a higher dimension? Like a 4 with some parts you can't see?

Anonymous

Anonymous Tue 30 Nov 2021 15:33:46 No.13907238 Report

Quoted By: >>13908859

Do you play vidya, OP? This is a really easy way to see how quats and 'gimbal lock' happen. In most games when you look straight up/down you can then only rotate around the Z/vertical. However if the game uses quats for the camera, you should be able to continuously rotate on any axis. This is important for games where you fly around, so you don't get stuck in gimbal lock when facing along the Z axis. 6DOF games like Descent would be unplayable otherwise.

Anonymous

View Same Google iqdb SauceNAO quaternion_rotation.png, 106KiB, 1219x1312

Anonymous Tue 30 Nov 2021 22:34:02 No.13908674 Report

Quoted By: >>13915671

shh, no words, only math.
only talk if you have questions.

Anonymous

Anonymous Tue 30 Nov 2021 22:49:05 No.13908720 Report

Quoted By: >>13909852

>>13907215
>Quaternions avoid "gimbal lock"
when describing "discrete" rotations, gimbal lock isn't an issue. by discrete, i mean you are relating one thing to a rotated version of that thing, there is no "history" to the rotation.

when describing paths through rotation space, like when describing the rotation of a rigid-body like a plane, euler angles can get into configurations where there are discontinuities between the angles and the rotation they represent. for instance, when traveling across the poles of a sphere your longitude has a discontinuity but your motion doesn't. this can cause all sorts of problems.

the key point is that gimbal lock is an issue for representing continuous paths in rotation space. if you are just rotating something without needing to know the path, it's not an issue.

quaternion rotation is geometrically understood to be an axis-angle representation for rotation, which can be encoded in a single 3d vector whose direction is the axis and magnitude the rotation angle. such a representation doesn't suffer from gimbal lock because it's trivial to interpolate between 3d two vectors using linear interpolation.

Anonymous

Anonymous Tue 30 Nov 2021 23:33:22 No.13908859 Report

Quoted By:

>>13907238
lol I remember trying to understand this shit one day, spending an hour totally confused watching stupid visualisation of 3d figures moving in gyros and just went "oh shit it's that thing that happens when I play FPS games"

Anonymous

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Anonymous Wed 01 Dec 2021 05:35:26 No.13909852 Report

Quoted By: >>13911519

>>13908720
>which can be encoded in a single 3d vector whose direction is the axis and magnitude the rotation angle

Anonymous

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Anonymous Wed 01 Dec 2021 18:25:08 No.13911519 Report

Quoted By: >>13911582

>>13909852
things get fucky when you recognize that a rotation by 180' along one direction/axis is the same as a rotation by -180' along the opposite direction/axis. so all rotations can be represented by spherical volume with radius pi (180'). this means rotation space is connected up in a really strange way that leads to things like the belt trick

Anonymous

Anonymous Wed 01 Dec 2021 18:43:23 No.13911582 Report

Quoted By: >>13911805 >>13911970

>>13911519
Yeah i have never seen a formal explanation of how the belt trick is related to the trickiness of SU(2), SO(3) and all these things. Almost as if its just hinted mystery and the expounder just does the trick and drops some terms from topology.

Anonymous

Anonymous Wed 01 Dec 2021 18:57:13 No.13911628 Report

Quoted By:

>>13907215
You should know quaternions only if you plan on doing robotics or astral math, both cases are cringe, just learn how to use them it's not that hard but operations are tedious, so they are left to calculators

Anonymous

Anonymous Wed 01 Dec 2021 19:01:45 No.13911647 Report

Quoted By: >>13911907

>>13907215
https://youtu.be/AC-ifrf3wig

Anonymous

View Same Google iqdb SauceNAO belt_trick.png, 25KiB, 1412x482

Anonymous Wed 01 Dec 2021 19:45:20 No.13911805 Report

Quoted By: >>13911843

>>13911582
i'm still putting the pieces together myself. this is what i gather:

consider the spherical volume used to represent rotation space. draw a continuous straight line path that connects one side of the sphere to the opposite side. because opposing points on the sphere represent the same rotation this path is closed (forms a loop), however there is no way to continuously shrink the path to a point. this means the space of rotations is not simply connected, in a way similar to how on a torus you can't shrink all closed paths to points.

now consider a continuous closed path like in the first case, but in this case it traces out the path twice before looping (i.e. north pole -> origin -> south pole -> north pole -> origin -> south pole). being careful to keep opposing points identified as the same point (i used the same color on opposite sides in my figure to indicate this), you now can freely move one set of opposing points on top of the other set, and then shrink the path to zero.

i need to think more about how this is precisely connected to the belt trick.

Anonymous

Anonymous Wed 01 Dec 2021 19:49:57 No.13911817 Report

Quoted By:

>>13907215
>(I don't understand how gimbal lock is a problem)
https://youtu.be/zc8b2Jo7mno?t=2m30s

Anonymous

Anonymous Wed 01 Dec 2021 19:55:20 No.13911837 Report

Quoted By:

>>13907215
The complex numbers can be constructed as ordered pairs of real numbers with a special multiplication rule that makes them different from 2-vectors.

The quaternions can be constructed in as ordered pairs of complex numbers with a special multiplication rule that makes them different from rotation matrices.

Anonymous

Anonymous Wed 01 Dec 2021 19:57:40 No.13911843 Report

Quoted By:

>>13911805
I read this in my book on lie groups (Gilmore). SO3 is represented by a sphere of radius Pi. Its not simply connected ok, double loop ok. Homotopy group of paths with 2 elements, ok. Still cant understand how that's linked to the belt trick. Keep thinking.

Anonymous

Anonymous Wed 01 Dec 2021 19:57:50 No.13911844 Report

Quoted By:

>>13907215
complex numbers, but squared.

Anonymous

Anonymous Wed 01 Dec 2021 20:27:05 No.13911907 Report

Quoted By:

>>13911647
Literally any comment? Does this represent gimbal lock or the lack thereof?

Anonymous

Anonymous Wed 01 Dec 2021 20:45:27 No.13911970 Report

Quoted By: >>13912158

>>13911582
>Yeah i have never seen a formal explanation of how the belt trick is related to the trickiness of SU(2), SO(3) and all these things. Almost as if its just hinted mystery and the expounder just does the trick and drops some terms from topology.
The formal explanation is that a belt when you rotate it has its start be the beginning of a loop through the space of all possible rotations and the end go back to the trivial rotation(imagine the belt cut up into slices with each slices rotation being a point in SO3), with the movements you make without rotating the ends be homotopies of loops. Then the belt trick shows that the 720 degree rotation is homotopic to the trivial rotation and 360 degree isn't - showing that pi_1(SO3) has an element or order 2.

Anonymous

Anonymous Wed 01 Dec 2021 21:35:24 No.13912158 Report

Quoted By: >>13912257

>>13911970
>The formal explanation is that a belt when you rotate it has its start be the beginning of a loop through the space of all possible rotations
Yeah im gonna need you to prove this because its not self evident at all. You are just talking about the geometry of these groups but then drop "and then this element of abstract math its the belt rotation and then this rotation is like this loop". Nobody believes this.
The actual mathematics of lie groups is much easier to understand than this belt thing.

Anonymous

Anonymous Wed 01 Dec 2021 22:17:57 No.13912257 Report

Quoted By: >>13912271 >>13912280 >>13912282

>>13912158
>The actual mathematics of lie groups is much easier to understand than this belt thing.
The mathematics of the belt thing is about the description of how if you have a smooth embedding of an oriented square into Euclidean 3 dimensional space you will have in an obvious way a ordered orthonormal basis in E^3 associated to each point on the ribbon, giving a smooth map from the ribbon into SO(3), which if you restrict to a path on the ribbon will give a smooth map from the unit interval into SO(3). Because in the belt trick the actual elements of SO(3) the start and end of such a path are mapped to one element and your only allowed isotopies fix the ends of the ribbon in place - such isotopies of the ribbon will end up mapping to homotopies of loops in SO(3). However, the loop you get in the belt trick is exactly the generator of pi_1(SO(3)), meaning that it isnt homotopic to the identity. This concludes the proof that you cant untwist a belt with a 360degree twist while holding its ends in place.

Anonymous

Anonymous Wed 01 Dec 2021 22:20:43 No.13912271 Report

Quoted By:

>>13912257
P.S Something something an embedding of an oriented ribbon gives a section of a trivial SO(3) principal bundle over it, something something.

Anonymous

Anonymous Wed 01 Dec 2021 22:22:59 No.13912278 Report

Quoted By:

>>13907215
>How the fuck do they work?
The original reason Hamilton came up with them is that he wanted something like the complex numbers $\mathbb C$ but instead of just one $i$ with two or more of them. All this other stuff like rotations came after.
> (I don't understand how gimbal lock is a problem)
Again all this stuff came long after Hamilton invented the quaternions, but to see gimbal lock as a problem you just have to imagine a literal gimbal. In the basic configuration (all parts 90 degrees from each other) it can move around pretty easily; but at the other extreme (all parts lined up) it has to move a bit to free itself up before it can move around smoothly again.

Anonymous

Anonymous Wed 01 Dec 2021 22:23:33 No.13912280 Report

Quoted By: >>13912285

>>13912257
>This concludes the proof that you cant untwist a belt with a 360degree twist while holding its ends in place.
P.P.S holding the ends parallel*
The actual rigorous mathematics of the belt trick are basically trivial and irrelevant differential topology. The argument presented here isnt 100% rigorous from that view, but thats how it goes.

Anonymous

Anonymous Wed 01 Dec 2021 22:24:06 No.13912282 Report

Quoted By: >>13912298

>>13912257
>The mathematics of the belt thing is about the description of how if you have a smooth embedding of an oriented square into Euclidean 3 dimensional space you will have in an obvious way a ordered orthonormal basis in E^3 associated to each point on the ribbon, giving a smooth map from the ribbon into SO(3)...
I challenge you to actually do and present all of this, in a rigorous way. These ordered basis, these mappings. Translate the belt trick into mathematics and show the equivalence.

Anonymous

Anonymous Wed 01 Dec 2021 22:25:10 No.13912285 Report

Quoted By: >>13912298

>>13912280
>trivial and left as an exercise to the reader
You cant be serious

Anonymous

Anonymous Wed 01 Dec 2021 22:30:37 No.13912298 Report

Quoted By: >>13912334

>>13912282
>>13912285
Do it yourself, I gave you the idea of why the belt trick relates to the fact that pi_1(SO(3))=Z/2Z. The rest would just be me attaching a lot of differential topology jargon to a visually obvious point, which I am not doing, as Id rather just sleep.

Anonymous

Anonymous Wed 01 Dec 2021 22:33:15 No.13912306 Report

Quoted By:

>>13907215
The 3b1b video is shit.

Anyway, it's too late for me to read up on the thread after there's 20 posts. The OP doesn't really have a concrete question..

Anonymous

Anonymous Wed 01 Dec 2021 22:39:37 No.13912334 Report

Quoted By:

>>13912298
I know the mathematics of the Lie group SU(2) and all its quotient groups and algebras. That still has nothing to do with the belt trick except "spooky rotation". You have failed at establishing a connection and it must be embarrassing you since you allege its easy and trivial and a teaching aid for laymen and newbies and not something for professionals. By telling me to do it myself you only show you know nothing. You dont explain anything by claiming that "a mapping exist". You are a pseud and should be ashamed.

Anonymous

Anonymous Thu 02 Dec 2021 04:47:16 No.13913371 Report

Quoted By: >>13913389

can you help me? I am using quaternions to animate a character model. I also have some data that represent 3-axis of a limb rotation that form an orthonormal basis. how do I turn this set of basis vectors into a quaternion without gimbal lock? I know how to make a transformation matrix and find the axis of rotation, but will it lock?

Anonymous

Anonymous Thu 02 Dec 2021 04:54:50 No.13913389 Report

Quoted By: >>13913497

>>13913371
this shows you how to go between quaternion and axis-angle format.

>will it lock
depends on how you parameterize your rotation. if you use quaternions/axis-angle, then no. if you use euler angles, then yes.

Anonymous

Anonymous Thu 02 Dec 2021 05:53:39 No.13913497 Report

Quoted By: >>13913742

>>13913389
>depends on how you parameterize your rotation
I'm not parameterizing it. thats the problem. If I were parameterizing it I'd just make the quaternion no problem. the motion data comes from motion capture. all I have are three vectors that describe the local coordinate system of the limb.

Anonymous

Anonymous Thu 02 Dec 2021 07:55:04 No.13913742 Report

Quoted By:

>>13913497
Quaternions describe rotations with a redundancy, for every rotation there's two quaternions that work as equivalent descriptions. I think that if Q describes it then -Q also.

Anonymous

Anonymous Thu 02 Dec 2021 18:28:51 No.13915566 Report

Quoted By:

>>13907215
It's a point that represents an infinite series of lines in a limited space or volume. x^2+y^2+z^2+i^2 is an example of a quaternion.

Anonymous

Anonymous Thu 02 Dec 2021 18:54:04 No.13915644 Report

Quoted By:

>>13907215

Complex numbers are 2 dimensional "numbers". Quaternions are 4 dimensional "numbers".
The problem I'm having with the word "number" ist, that real numbers form an algebraic object called field and when trying to do that on the set of pairs, or 4 tuples, the task is to define a proper multiplication, basically. And quaternions are your 4d tuple with a proper multiplication. Very oversimplified, tho

Anonymous

Anonymous Thu 02 Dec 2021 18:56:31 No.13915650 Report

Quoted By:

it's just like a pretty autistic algebra

Anonymous

Anonymous Thu 02 Dec 2021 19:03:30 No.13915670 Report

Quoted By:

unrelated but what are grassmanian numbers and why are they important? Only intuitive explanation plz because i too have wikipedia

Anonymous

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Anonymous Thu 02 Dec 2021 19:04:02 No.13915671 Report

Quoted By:

>>13907215
>>13908674
I don't know why you guys are making this so complicated. Quaternions are just the natural embedding of four-vector arithmetic.

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