Understanding Quaternions

https://www.3dgep.com/understanding-quaternions/

by njn • 8 years ago

208 98 8 years ago

3dgep.com

adamnemecek• 8 years ago

Dual quaternion are even whackier. They are the best formalism for reasoning about 3D space developing over time.

Here’s a cool example http://www.chinedufn.com/dual-quaternion-shader-explained/

umanwizard5• 8 years ago

Possibly off-topic: why does practically any description of quaternions include the anecdote about somebody carving something into some bridge in Ireland?

I’ve learned about plenty of mathematical concepts while having no idea who discovered them or under what circumstances. Why are quaternions the exception?

brandonmenc• 8 years ago

somebody

umanwizard5• 8 years ago

What exactly is your point? I said "somebody" for effect, not because I don't know who Hamilton was.

twtw6• 8 years ago

I don't know, but here are my ideas:

- quaternions maybe have a more interesting backstory than other constructs. Not everything was carved into stone.

- quaternions never really became part of mainstream mathematical education. This makes them more niche and strange, and worth telling a story about. It's not as interesting to tell a story about something commonplace.

Entalpi• 8 years ago

People should read up on Galois theory and the story behind it. It involves a french youth, the french revolution, groundbreaking maths, and tangentially Poisson, Fourier and Cauchy, Gauss, Jacobi.

His life should be a movie. :-)

GlenTheMachine• 8 years ago

My doctoral dissertation used quaternions to describe spacecraft attitudes. My advisor was a stickler for not only appropriately citing references, but citing the most original (in the sense of oldest) references. This was something of a problem because the original references to some of the mathematical techniques I used were several hundred years old and not in English, and under the theory that a big part of the purpose of a citation is to help the reader understand the backgroud material this seemed a bit excessive. Nobody is going to learn French in order to read about Lagrange multipliers from Lagrange's original papers.

But for quaternions, it was easy: I actually cited the Brougham Bridge inscription. One cannot, of course, check the bridge out of the engineering library to check the citation, but clearly this was the original “publication” of quaternion multiplication.

My advisor finally got the point.

ModernMech• 8 years ago

Nobody is going to learn French in order to read about Lagrange multipliers from Lagrange's original papers.

You'd think, but I had a professor in physics who learned German just so he could read Boltzmann's original works.

iantrt• 8 years ago

If someone learned French to read about Lagrange multipliers from Lagrange's original papers, then they would be sorely disappointed: it was actually Euler who first used the Lagrange method! (And by some curious coincidence, the L/lambda could just as easily be a homage to Leonhard as to Lagrange...)

nj65537• 8 years ago

I think the story is told as a sort of cultural marker of what a Big F-ing Deal this discovery was/is. Mathematics, collectively, struggled for a long time to find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers. (The cross and dot products are unsatisfying, because they don't have division.) The shock that this can be done in four, -- not three -- dimensions is still sort of reverberating, and that's what I think this story marks. It's a short stand-in for the longer story I've just summarized, and it evokes (or is meant to evoke) the mind-shattering thrill of discovery.

I don't necessarily think the story accomplishes this -- your question is but one piece of evidence that it doesn't -- but I think for those who spend a good amount of time with these kinds of algebra questions, it comes to take on that role, and that's why I think it's repeated.

(Teaser -- if you want to know more about these kinds of questions, Google for "real division algebras". There are not very many, and they way they are organized is not, I think, something one would expect.)

jf-2• 8 years ago

To find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers

What does that mean? My understanding was that Hamilton was searching for a way to make the manipulation of points in space easier, such as rotation, and noticed that the imaginary part of the complex numbers could be manipulated in the way he wanted. He then created a rather artificial tool in the form of the quaternions that allowed this.

blt3• 8 years ago

I think it's more like a meme. Countless other mathematical insights in the same time period have been at least as important and counterintuitive.

xelxebar3• 8 years ago

Also, in Hamilton's time they didn't have the view of mathematics as axiom systems that could be played with. Math was a way of finding Truth, so the idea of making up multiplication rules at your convenience probably seemed like an extremely non-obvious move.

The discovery of complex numbers and quaternions probably played a big part in getting people to question what math is, leading to Hilbert's program to study Foundations etc. Hamilton's story is a nice, rare single instance we can point to, symbolizing this discovery.

jf-2• 8 years ago

People think it’s a cute story and there’s a plaque on the bridge. It doesn’t need to be anything more than that.

samontar• 8 years ago

It wasn’t when we learned it in high school, but it’s just that it’s a story that’s easy to tell. Just like the Königsberg reference for Eulerian graphs; Kekule and the Benzene ring; Lorenz, the weather, the butterfly, and deterministic chaos.

umanwizard5• 8 years ago

Thanks - out of all the replies to my comment, I think this is one of the most plausible hypotheses. I indeed think the Konigsberg and benzene stories are as commonly trotted out as the Hamilton one.

garmaine• 8 years ago

Surely you read about Galois fatal dual? I think it's just that these are instances where there is an interesting story attached.

KboPAacDA3• 8 years ago

And why do descriptions of quaternions include a rigorous proof? I trust quaternions work, so skip the proof and show me how to use them in a practical, efficient way.

umanwizard5• 8 years ago

In mathematics often proofs lead to understanding why something works (on the other hand, many proofs aren’t written in a way that make the main ideas clear or easy to tease out from details)

joppy6• 8 years ago

The fact that the quaternions “work” (are a finite-dimensional real vector space with an associative real-bilinear multiplication, and a division operation) is quite special. In fact there are only three objects satisfying this criteria: the real numbers themselves, the complex numbers, and the quaternions.

billfruit2• 8 years ago

Very curious indeed why it is so, esp since there is normally zero historical background explained about vectors and matrices.

beautifulfreak• 8 years ago

Maybe you haven't heard of Oliver Heaviside. https://en.wikipedia.org/wiki/Oliver_Heaviside

Retra• 8 years ago

There are millions of abstractions in mathematics, and the truth is that the human brain has a much better time remembering stories than it does math. You teach the stories so people have some context to associate complex ideas with.

edflsafoiewq7• 8 years ago

Anyone know an easy way to show that multiplication by a unit quaternion is a unitary operator?

joppy6• 8 years ago

Unitary in what sense? Clearly multiplication by a unit quaternion preserves the quaternion norm.

edflsafoiewq7• 8 years ago

In the sense of preserving the dot product, x.y = (ux).(uy).

xelxebar3• 8 years ago

A little more exotic, but if you look here

https://en.m.wikipedia.org/wiki/Plate_trick

you can read about some simple rotating systems that, in some sense, end up reversed after a 360 degree rotation amd require another 360 to turn fully around. The article above has a nifty gif of this.

Formally, this kind of thing is studied using what are called spinors and gets used a lot in quantum mechanics and friends when talking about quantum spin. Behind the scenes these are described using things called spinors.

As it turns out, quaternions (and octonions) are capable of describing such things while simple rotations of any kind are insufficient. There are deep connections between spinors of different dimensions and the quaternions/octonions. For those interested, the term here is "Bott periodicity".

chombier3• 8 years ago

You just have to show that the quaternion norm is multiplicative, which follows easily from its definition |q|^2 = q * conj(q).

aaaaaaaaaab2• 8 years ago

Protip: learn Geometric Algebra.

dbcurtis• 8 years ago

Can you recommend any good references?

aaaaaaaaaab2• 8 years ago

Here’s an appetizer from Eric Lengyel in the context of game development (or 3D graphics in general): https://m.youtube.com/watch?v=WZApQkDBr5o

And here’s the in-depth stuff:

Geometric Algebra for Computer Science: http://www.geometricalgebra.net

pjbk• 8 years ago

http://bleyer.org/dw/doku.php?id=geometric_algebra

nraynaud4• 8 years ago

I had a question about quaternions: does anyone use them for anything else than multiplying a rotation by a scalar?

In particular, it feels a bit like a waste of coding space to always use unit ones.

colechristensen2• 8 years ago

Some control systems for air and space craft are designed with quaternions - here the problem being solved isn't a programming problem but a math problem involving the dynamics of the craft and/or orbital dynamics.

nraynaud4• 8 years ago

As far as I know they are also only doing SLERP for attitude control.

blt3• 8 years ago

There does not exist a R^3 parameterization of the 3d rotations without some problem. https://en.m.wikipedia.org/wiki/Charts_on_SO(3)#Parametrizat...

edflsafoiewq7• 8 years ago

But quaternions are pretty good. They're a double cover but angles are an "infinity cover" of SO(2) and everyone understands angles. For animation purposes, being a single-cover is arguably a problem. An example in 2D is to suppose you have a rotation sequence

    0° -> 90° -> 180° -> 270° -> X

and you want to return to the original state. Well, you can do that either by continuing to rotate in the same direction, corresponding to X=360° or by going back along the path you came corresponding to X=0°. These alternatives are unrepresentable in a single cover. (For angles you also have X=-360°, 720°, etc. corresponding to making any number of revolutions in either direction before coming to rest at the desired target, which if you think is weird makes quaternions an even better chart on SO(3) than angles are on SO(2)).

blurbleblurble5• 8 years ago

The remarkable thing about quaternions is that you can describe rotation around any arbitrary axis! Try doing that with real rotation matrices and you will bump into a major problem:

https://en.wikipedia.org/wiki/Gimbal_lock

Because matrix multiplication is not commutative, you can't easily compose a matrix representing rotation around an arbitrary axis from "component rotation matrices".

With quaternions/clifford algebras, you can say "here's the vector I want to rotate about and here's this is how much I want to rotate", and it just magically works.

billfruit2• 8 years ago

I find that many articles explaining how to describe the orientation of a rigid body rotated along an axis by an angle in a confusing manner.

Normally in 3d space you will have to construct quaternions for rotations along yaw, pitch and roll, and then take their product to get the quaternion of the orientation of the rigid body.

That is, when using quaternions to describe orientations,we are actually describing the rotations done to bring the body from its default orientation to its present orientation.

civility• 8 years ago

The remarkable thing about quaternions is that you can describe rotation around any arbitrary axis! Try doing that with real rotation matrices and you will bump into a major problem

https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matri...

nraynaud4• 8 years ago

how is that an example of using quaternion for something else than rotations?

xaedes• 8 years ago

Not quite...

Gimbal lock is no problem of rotation matrices but of euler angles. Just look up the wikipedia page you posted: "The problem of gimbal lock appears when one uses Euler angles in applied mathematics"

You can represent any (any!) rotation with rotation matrices.

Quaternions are neither commutative. Fortunately you don't need commutativity to build up arbitrary rotations.

1wd• 8 years ago

You can also use the for making pretty pictures of Quaternion Julia fractals. It uses non-unit quaternions, but typically intersects a plane with the 4D solid:

http://paulbourke.net/fractals/quatjulia/

Apparently there's also an application in multiantenna radio transmission: http://www.ece.ualberta.ca/~hcdc/Library/StCommClass/Hugh00....

foobarbecue2• 8 years ago

In case you missed it, 3B1B has a brilliant video introducing quaternions: https://youtu.be/d4EgbgTm0Bg

raffael-vogler• 8 years ago

His videos are really incredibly intuitive and well-made. He deserves an award for his channel.

foobarbecue2• 8 years ago

Yep, and all the source code is developed in the open as well! https://github.com/3b1b/manim

jasonincanada• 8 years ago

Apparently part 2 is interactive and his patreon supporters already have access to it

choonway5• 8 years ago

Please use Lie Groups/Algebra instead.

formalsystem• 8 years ago

Do you mind on elaborating on that? Any nice books or tutorials you've seen on the subject?

choonway5• 8 years ago

Skipping the formalism you can get directly into the practical aspects in this book.

Modern Robotics by Lynch and Park Chapters 3 and 4

pre-preprint of book / more info available here http://hades.mech.northwestern.edu/index.php/Modern_Robotics

edflsafoiewq7• 8 years ago

Instead? Non-zero quaternions are a Lie group under multiplication...

choonway5• 8 years ago

Yes, you are referring to SO(3) but SE(3) includes translations.

bnolsen• 8 years ago

left handed thingies. go to geometric algebra for the real meal deal.

paulgrant9992• 8 years ago

I have no idea why they insist on using such horrible graphs to explain a simple idea.

Go to wiki/quaternions. scroll down to about 2/3rds of the way ;)

kuwze• 8 years ago

I remember being introduced to quaternions recently by this post[0] which recommended this book[1].

[0] https://www.haroldserrano.com/blog/best-books-to-develop-a-g...

[1] https://www.amazon.com/Quaternions-Computer-Graphics-John-Vi...

amai• 8 years ago

Sorry to spoil the party, but this is the old 19th century way of teaching quaternions (and also complex numbers). It is much easier to start with some https://en.wikipedia.org/wiki/Group_theory and then you understand that quaternions are simply matrices of a specific form: https://en.wikipedia.org/wiki/Quaternion#Matrix_representati... . Quaternion multiplication is simply matrix multiplication of these matrices. And that's it. No mysteries, this is just simple linear algebra (you do't even need complex numbers, the real representation is enough and makes the connection to 4d rotations manifest).

gmadsen• 8 years ago

it shouldn't be an either or with geometry and algebra. both are valid and it helps to know both. representing H as a matrix is certainly valid and makes computations easier, but I dont really think that is building intuition

earthicus2• 8 years ago

The modern approach is not lacking all geometry, and the 19th century presentation is not lacking all algebra. Any respectable treatment will include both, the question is which makes the relationship clearer? The abstract algebraic approach talks of 'the rotation group' which maps on to geometric concepts as cleanly and directly as possible. Then we talk of different parameterizations or representations of this group, with the unit quaternions being the 'double cover' of the rotation group (which gives rise to the primary difficulty in understanding them - that a rotation by 360 degrees reverses orientation, and 720 degrees returns us to where we started).

I think the modern approach is much clearer - the geometric ideas appear more directly, and the algebra is far less messy.

twtw6• 8 years ago

Do you have any books you could recommend that present this approach?

earthicus2• 8 years ago

Stillwell[1] discusses quaternions in chapters 1 & 2, although the meat and potatoes of the book is more generally about lie groups and lie algebras. It's pitched at a lower level than most books on this topic by restricting attention to the classical matrix groups.

[1]https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math...

kwoff• 8 years ago

If you want to go the opposite direction, Hamilton's "Elements of Quaternions" is long out of copyright: https://archive.org/details/elementsquaterni00hamirich/page/... (I doubt most people get past the impressively long table of contents. :)

joppy6• 8 years ago

A lot of open problems in mathematics boil down to trying to analyse "matrices of a specific form" - I think while this viewpoint is important to recognise (essentially: quaternions are a finite-dimensional real associative algebra), it might not give as much intuition as other interpretations. It's kind of like how in computer science, many many hard problems can be restated as problems about some graph constructed from the input, but now you have two problems...

splittingTimes• 8 years ago

Previous discussion

https://news.ycombinator.com/item?id=7364442

Those who like to have a print version:

https://github.com/frankMilde/interesting-reads/blob/master/...

wink• 8 years ago

I found these videos very helpful: https://www.youtube.com/watch?v=SCbpxiCN0U0

Understanding Quaternions

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