Let's remove quaternions from every 3D engine (2018)

https://marctenbosch.com/quaternions/

by lelf • 6 years ago

627 322 6 years ago

marctenbosch.com

Epskampie• 6 years ago

Previous discussion: https://news.ycombinator.com/item?id=18365433

ColinWright2• 6 years ago

I've only skimmed the article, but it seems to be saying this:

Instead of using this thing you don't understand and investing the time to understand it, why don't you use this other thing you don't understand, and invest the time to learn that instead.

Indeed, later in the article the author says:

We can notice that 3D Rotors look a lot like Quaternions ... In fact the code/math is basically the same! The main difference is that i, j, and k get replaced by y∧z, x∧z and x∧y, but they work mostly the same way.

Have I got that right?

ianai• 6 years ago

That’s a notational change that incurs unneeded, increased cognitive load. That’s admittedly a mile high view.

mcnamaratw2• 6 years ago

Sure, people who are happy customers of quaternions and don't feel driven to understand what's under the hood, why change?

The pitch in the OP is (I think) intended for people who feel pressure to understand why it works. Geometric algebra can be built up yourself by picturing actual rotations, one dimension at a time. It's physical from the start. Quaternions are a finished math system that you have to reverse-engineer to understand. Some people like one approach or the other better.

Some geometric algebra people are kind of true believers. That's probably not helping the notation get traction.

friendlybus6• 6 years ago

It's possible to plot the rotation of planets around the earth using very complex systems. Why didn't we stay with that?

Why not change? Comfort?

Certhas3• 6 years ago

That's not correct. Mathematically as they describe the same thing they have to be translatable into each other. But one is a natural mathematical definition, the other exploits a completely unintuitive incidental mathematical isomorphism.

If you would read a few sentences on from where you quote, this becomes evident.

    ij = k

is arbitrary, needs to be memorized and just happens to do what you want for reasons that require a lot of working out.

    (xy)(yz)=x(yy)z=xz

On the other hand requires zero memorization.

ColinWright2• 6 years ago

ij = k is arbitrary, ...

It's probably my mathematical background speaking, but I find this far from arbitrary. Quaternions don't spring from nothing. It seems to me that (xy)(yz)=x(yy)z=xz needs just as much justification and memorisation ... why should y^2=1?

I guess my point is this. If people put in as much effort to understand quaternions as has been expended in this article, then quaternions would probably be just as easy to grasp and the system being described.

zupatol• 6 years ago

I haven't read that particular article, but from other sources I've read quaternions appear to be mysterious/magical even to people who know them, while the rotors have a very concrete geometrical interpretation in geometric algebra.

Geometric algebra also addresses much more than just rotating stuff, it's just that when you want to do rotations it leads naturally to something very similar to quaternions.

xg154• 6 years ago

Your opinionated summary implies that both concepts are equally hard to understand, but the whole point of the author is that rotors are potentially a lot easier to understand and reason about than quaternions.

He also gives a concrete reason: For 3D objects, rotors operate fully in 3D, whereas quaternions are in 4D. This means you can visualize rotors and imagine "how they work" whereas with quaternions, you have to more or less blindly trust the formulas.

ptero2• 6 years ago

rotors operate fully in 3D, whereas quaternions are in 4D. This means you can visualize rotors ... whereas with quaternions, you have to ... trust the formulas.

This is a bit of a stretch -- by the same logic any operation on more than three numbers means I have to trust the formulas. We have many methods to understand, visualize, develop intuition, etc. in such cases, such as "fix a, look at what happens when you vary b, c and d". When I am working to understand dynamics of an object with three variables in MATLAB or similar, I seldom (probably never) plot it in 3D (which gets projected on the surface of a flat monitor anyway). Instead I usually play with numerous 2D and even 1D plots. My 2c.

omaranto3• 6 years ago

I don't understand the phrase "rotors operate fully in 3D". What does operating in 3D mean?

If it means rotors can be used to give a simple formula for rotations in 3D space, then it also applies to quaternions.

If it means the space of rotors is three-dimensional, then it's false: rotors have 4 coordinates, a + b x^y + c y^z + d x^z, just like quaternions.

Does it mean something else? Something that applies only to rotors and not to quaternions?

lonelappde• 6 years ago

That's not true. Rotations in 3D space are 4D objects: a 3D orientation and a 1D magnitude.

Rotors use a.b with aXb, and quaternions use magnitude and 3D direction vector.

ChrisLomont2• 6 years ago

rotors are potentially a lot easier to understand and reason about than quaternions

Quats are quite easy to reason about: a unit quat holds a rotation about an axis: rotation about angle theta an axis given by direction (x,y,z) is quat q = sin(t/2) + cos(t/2)(xi+yj+zk).

Composing quats q1 * q2 is the rotation you get by applying q2 then q1 to an item.

So think of the scalar as holding info about the angle rotated, and the vector part as the direction of the axis.

I don't see how with rotors or quats you can easily visualize composition or more complex actions (and I've used both extensively). "Blindly" trusting the formulas can be replaced by working through a proof enough times until you feel how they work, just like linear algebra, algebra, quadratic formula, etc.

onedognight• 6 years ago

Yes, you have that right. The underlying operations are the same. What you are missing is that the ideas from geometric algebra work in any dimension! Quaternions are a “trick” that only works in 3D.

You can write down Maxwell’s equations using the geometric algebra easily, and it will make them better! They will be obviously coordinate independent.

Pretty much anything with a cross product will be better written using geometric algebra.

solinent5• 6 years ago

The exterior/wedge product is actually very closely related to the cross product, and works to generalize the cross product to n dimensions. You can read Spivak or Munkres, "Calculus on Manifolds" and "Analysis on Manifolds" respectively.

One of their main uses is to prove the generalized Stokes theorem in n dimensions.

3D rotors are isomorphic to quaternions, so go ahead and rename your variables :).

uulu• 6 years ago

I got me intrigued. Thanks! Could you point to a source that explains geometric algebra from this angle? Book, article, web source, anything?

BlueTemplar22• 6 years ago

Maxwell's equations : much more consistent without the god damned pseudo-vectors that are vectors except when they aren't ! http://www.av8n.com/physics/maxwell-ga.htm

papln3• 6 years ago

But 3D is the D that matters for 3D games. video games are concrete mathematics, not abstract mathematics.

LanceH• 6 years ago

The best 3d location/vector method I've used allowed rotations using normal x,y,z coordinates, but you could do rotate an object in the xyz of the engine itself, the object, or the object that object was currently attached to. This made things like shooting a lob from a forward facing gun easy, since you could raise the shot from the gun rather than the world and no quaternions were involved. I don't know if quaternions were used underneath.

gugagore8• 6 years ago

I don't understand what you mean, but I think you're describing exponential coordinates, or said another way, choosing an angular velocity and getting a rotation by following that angular velocity for a unit of time. https://en.m.wikipedia.org/wiki/Rotation_formalisms_in_three...

moron4hire7• 6 years ago

You don't need to understand the root construction of Quaternions to use them in game code anymore than you need to understand the root construction of the Reals to be able to do arithmetic in game code. Just treat them as opaque values that have certain operations you can perform to get certain results and be done with it.

xg154• 6 years ago

The author is arguing against this exact mentality - and I believe, this was one of the main motivations this article was written:

Personally, I have always found it important to actually understand the things I am using. I remember learning about Cross Products and Quaternions and being confused about why they worked this way, but nobody talked about it. Later on I learned about Geometric Algebra and suddenly I could see that the questions I had were legitimate, and everything became so much clearer.

I tend to agree with the author. I find it a lot harder to work with concepts I don't understand: I'm forced to "fly blind" and just plug in formulas and hope everything works. At the latest when you have to debug something, this can go horribly wrong and leave you without a lot of options.

moron4hire7• 6 years ago

So you know exactly how everything in your computer works, because otherwise it'd be too hard to use? I wager the exact opposite is the case: it's much easier to use a computer through heuristics of operation rather than any sense of "deep" understanding.

Besides, your GPU shader code implements fast quaternions, and you aren't going to get NVidia to replace them with rotors. So the game is lost.

Oh, unless you don't mean "understand everything" and are going to draw your arbitrary line at the GPU.

bvda• 6 years ago

Personally, I have always found it important to actually understand the things I am using. This article is aimed at readers who do find understanding the constructions they use important.

moron4hire7• 6 years ago

That is the point of my comparison to the Reals. No accountant or mechanical engineer or architect need ever read, let alone understand, any proofs that the Reals exist or the fundamental construction of arithmetic. I know several highly successful economists who don't know the construction of statistics, anymore than knowing that the proofs exist and could be found somewhere, if it ever really mattered (it doesn't). As a developer, you are not a theoretical mathematician. You use math.

Game development is not fundamentally more hard than any other field of applied engineering or mathematics. It isn't theoretical mathematics. You just use the math.

Needing to "understand" root construction before "being able to use" is just procrastination.

nine_k• 6 years ago

Not knowing what's behind the hood in floating point numbers and their limited precision leads to all kinds of surprises.

moron4hire7• 6 years ago

Rarely

rikroots• 6 years ago

I added quaternions to my Javascript canvas library. They scared the heck out of me - I coded up the functions from scratch in JS, partly to try and 'understand' the concepts behind quaternions, but mainly because I was stupid and over-confident. I am not a math/physics genius!

I doubt using the concept of rotors in place of quaternions would've made my experience any better. It's not the concepts that really frighten me - I can read about them and nod my head, pretending that the information is somehow sinking into my brain; the things that scared me were the equations which, the article implies, are pretty much the same.

In the end, the things (seem to) work as intended in my library, and I can now sleep at night knowing I'll never have to revisit quaternions (or rotors) again in my life.

Koshkin6• 6 years ago

That's not how it works in programming in general. Many - if not most - abstractions turn out to be extremely leaky. (For example, you do want to know the difference between the "root construction of Reals" as it is done in mathematics and what is taken to be their representation in the computer.)

stared• 6 years ago

Quaternions are 3D rotations (plus, possibly, rescaling). The author notes that they have the same API (I like this term used in mathematical context).

So, the actual point is that the word quaternions (and "these strange i, j, k") is confusing. Rightfully (at least for anyone without a background in maths or physics).

"Let's remove Quaternions from every 3D Engine" -> "Let's remove the word 'quaternion' from every 3D Engine"

(Nice explanations and visualization, anyway!)

BlueTemplar22• 6 years ago

API stands for Application Programming Interface. What is exactly an "API"? Why not just "interface" ? (One should generally avoid acronyms...)

paulddraper2• 6 years ago

Yes, it should be interface.

No need to distinguish here between API/CLI/GUI.

cuspycode• 6 years ago

I agree, it's mostly a matter of words. But just for the record, quaternions in general are 4D isoclinic rotations (with optional scaling). Left-multiplication by a unit quaternion corresponds to a left-isoclinic rotation, and right-multiplication corresponds to a right-isoclinic one. By combining them you can produce any 4D rotation. A subset of this is the set of 3D rotations that rotate ix+jy+kz into ix'+jy'+kz'.

adamnemecek• 6 years ago

Check out http://bivector.net, a new community on geometric algebra.

Check out the demo https://observablehq.com/@enkimute/animated-orbits

Join the discord https://discord.gg/vGY6pPk

yantrams• 6 years ago

Never heard of this before. Thanks for the share.

graphpapa2• 6 years ago

This seems to be an excellent introduction to geometric algebra, a topic I have had difficulty picking up so far.

K0SM0S2• 6 years ago

I suggest you look at Mathoma's videos¹ on the topic on YouTube. His explaining is so simple (even too simple, some would say) that GA becomes a no-brainer. The magic really happens with this teacher (for me). A lot like Khan, for comparison.

[1] https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ...

yiyus3• 6 years ago

As someone who has to teach GA, may I ask what you tried and what you found difficult? I see problems with the non-standard notation, for example. I also think that it is much easier to understand 3D-GA starting with 2D or even 1D, and of course using a non-formal approach (as we do with vectors). It is also hard to find nice examples between the very obvious and very difficult stuff.

graphpapa2• 6 years ago

I think it is mainly the extra time spent on the foundational definitions - ie. multiple paragraphs and multiple interactive widgets just to clarify what is meant by a bi-vector.

I don’t claim to have delved very deep in the topic before, but of course it is standard to start with definitions - and I found these definitions particularly clear & disambiguated.

duke360• 6 years ago

quaternions (<3) are extremely useful and not so hard to learn as other have pointed out. it's just a matter of lazyness and intesrest in solving a problem

wccrawford• 6 years ago

Learning to use Quaternions is easy. Learning how they work is much, much harder.

Likewise, IMO, learning to use Rotors is easy, but learning how they work is much harder... It's just that it's easier than learning how Quaternions work.

And, also IMO, you don't need to know how either of them work to use them in gamedev. It's fine to use a library that understands them and just continue making the important parts of your game.

K0SM0S2• 6 years ago

“It's fine to use a library that understands them and just continue making the important parts of your game.”

Oh that's indeed absolutely what a game dev should do.

A 3D engine dev however, might do well to eat the math leading to the understanding of quaternions, and by extension Clifford algebras (the underlying/original theoretical structure leading to geometric algebra). You get to understand how particular variations in n-dimensions of this structural framework are isomorphic to all numbers like R, C, H and much more. (hyperbolic! dual!)

It really paints a whole arch-picture, a meta-framework to unify all possibly kinds of numbers in one's mind (including the geometry of these numbers and ring operations, with a 1:1 equivalence between geom and algebra).

Note that this is why, I think, some strong proponents of GA (which I find myself agreeing with in that regard) would have it enshrined in K-12 education in lieu of linear algebra — because the intuition of GA is really great / second-to-none, and intuition is all that most math students in high school will ever retain afterwards (they won't do math again, ever, not really). The argument being that people who need more (from linear algebra for calculations notably) can learn that complicated and non-intuitive stuff later (university), building on top of a good base intuition nurtured in GA / Clifford.

So, the 3D engine maker, people in robotics, anyone working with spatial representations of any kind (even abstract, like research with multilinear models) would do themselves a fantastic favor for a lifetime to learn these topics. It's a no-brainer, really, from the other side.

It's delicious math!

pauljurczak2• 6 years ago

In order to make geometric algebra really nice, you have to go 5D, see versor.mat.ucsb.edu. This imposes a significant computational overhead. Nice things are expensive.

gugagore8• 6 years ago

I think that's a different point. To get rotations in 3D you don't need to go to 5D conformal GA.

To be able to unify many geometric objects, like lines and spheres and point pairs and represent the duality e.g. the "meet" of two lines constructs a point (possibly a point at infinity) and the "join" of two points constructs a line (not sure what happens if the two points are identical), .... then you need 5D which is really like 2^5 = 32D in my mind.

But if all you're trying to do is stop being confused by two different things that both look like vectors, but transform differently under spatial transformations (i.e. any vector that is the result of the cross-product is really a different type of vector than the argument vectors, or e.g. normal vectors) then 3D GA is fine. Though really it's more like 2^3 = 8D (1 scalar, 3 regular basis vectors, 3 "axial" basis vectors, and one pseudo scalar).

An efficient implementation would likely need to use a type system avoid representing 8 dimensions directly. Like the cross product of two vectors will produce an element where only the "axial" components are non-zero.

pauljurczak2• 6 years ago

You are right, 5D conformal GA goes way beyond nice rotations in 3D space, but why stop there? We can have two worlds: computationally expensive, high level, powerful abstractions and computationally efficient, messy abstractions. In some cases of rotation, even quaternions are too much.

pflats5• 6 years ago

I'm a pure math dude at heart, even if I don't get to do it much any more.

Two years ago, my wife asked me, "If you had to get a math equation tattooed on your body, what would it be?" I answered, "i^2 = j^2 = k^2 = ijk = -1".

I felt a brief flush of anger when I saw this headline.

This is an extraordinarily good article that should be read by pretty much anyone doing graphics programming.

s_dev2• 6 years ago

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

You're like this Irish bridge that has the notation inscribed on it as well.

I would like to own an OpenGL kettle with the expression on it.

andybak• 6 years ago

I would like to own an OpenGL kettle

Do you mean the Utah Teapot?

bregma• 6 years ago

When I visited Dublin that was the one spot I absolutely had to visit. For some folks it was the Temple Bar, for others the James Joyce trail. For me, it was the plaque on the Broombridge and the Trinity College Library.

jchallis• 6 years ago

Friesland (formerly part of Melitta) sells the kettle for about 37 euro. If you have a way to add the messaging, you are well on your way.

https://frieslandversand.de/teekanne-1-4l-weiss-utah-teapot

Isamu• 6 years ago

Any idea why the name Hamilton appears to be deliberately defaced on the inscription?

kleer001• 6 years ago

Hahaha! That reminds me of the time I couldn't recall the name for "leaf blower" and called them an "air rake".

If anyone's curious the story is here: https://en.wikipedia.org/wiki/Utah_teapot

CountHackulus• 6 years ago

I explicitly made a detour when I was in Dublin to take a picture of it the plaque on that bridge. Worth it.

chadcmulligan2• 6 years ago

If you ever happen to be near a Siggraph the render man guys hand out little walking Utah teapots - a tradition going back many years apparently. Worth the price of admission :-)

DonHopkins3• 6 years ago

Then you could dress up for Halloween as Broome Bridge!

Koshkin6• 6 years ago

She probably hoped for a different answer.

tudelo• 6 years ago

No, I bet this was exactly the answer they were looking for. After all, it was love at first sight.

xg154• 6 years ago

I answered, "i^2 = j^2 = k^2 = ijk = -1".

Could you explain why? For someone without a math background, it seems indeed like a pretty arbitrary thing to define.

(I can understand the idea behind complex numbers and how the multiplication rules followed from the desire to define the square root of a negative number - however, so far, I don't get the motivation of introducing even more "special" elements)

Koshkin6• 6 years ago

See, for example, https://math.stackexchange.com/questions/911807/what-is-the-...

nilkn2• 6 years ago

By the Frobenius theorem, there are only three possible structures for a real finite-dimensional associative division algebra. Those structures correspond to the real numbers, the complex numbers, and what are called the quaternions. So essentially the above definition is not arbitrary because it's the only other possible way (besides R and C) to get that sort of algebraic system. Of course, this is not obvious at all. C famously is algebraically closed as a field, which makes it a ripe playground for much of topology, algebraic geometry, and analysis. There are some nonobvious generalizations of algebraic closure for the quaternions. (Naively, the quaternions are not algebraically closed in the classic sense because, evidently, ix + xi - j has no root.)

As for why one might want to consider such a noncommutative division algebra in the first place, the answer I suppose is just that it manages to pop up in a variety of areas in mathematics. We've already seen the connection with rotations in 3-space (the topic of this post). Here's another. The 3-sphere (that is, a sphere in 4-dimensional space whose surface is itself 3-dimensional) can be realized as the multiplicative group of unit quaternions spanned by {1,i,j,k}. Consider the circle H = {cos(theta) + i * sin(theta)} for real values of theta; H is a subset of the 3-sphere. If r is any unit quaternion, then the coset rH is another circle. But given a subgroup H of any group G, the left cosets of H in G form a partition of G. Therefore, these circles just described form a partition of all of the 3-sphere (the Hopf fibration).

Speaking of rotations, the involvement of quaternions should not be surprising. Indeed, complex numbers are intimately involved in rotations in 2-space (multiplication by a unit complex number e^(i*theta) corresponds to rotation about the origin by theta). Quaternions can similarly express rotations in 3-space, but one cannot just left- or right-multiply but must instead use conjugation. In general, one can generalize this using the techniques of geometric algebra.

jjoonathan3• 6 years ago

the multiplication rules followed from the desire to define the square root of a negative number

That's a bit reductionist. You don't just get the square root of a negative number, you get the Fundamental Theorem of Algebra (an Nth degree polynomial has N roots), which is a mathematical power tool if ever there was one.

Complex numbers dramatically simplify a bunch of proofs in linear algebra, give us tons of nifty integration techniques in complex analysis (the techniques are relevant for real numbers, they just use C), provide a representation of 2D rotations that can be manipulated using the rules of algebra (this is the most relevant to the thread), and give physicists, electrical engineers, and signal processing people an abstraction to represent oscillations (energy sloshing between two buckets = two elements of a complex number, which you can then do algebra with). They're a workhorse.

Quaternions are an attempt to do that in 3D. The dot and cross product of vector calculus are other pieces of those efforts. Unfortunately, vector calculus escaped the "math lab" before it was complete and got written into other fields and engineering books, so even though the underlying concepts were eventually sorted out (it's called Geometric Algebra), everybody just uses the half-baked abstractions (quaternions, dot product, cross product) which are Good Enough. It's a perfect example of "worse is better" affecting something other than software engineering.

jfengel2• 6 years ago

As other replies have said, the math is kind of important. The idea of a tattoo harkens to the story of the discovery of quaternions: Rowan Hamilton was out for a walk in Dublin, trying to figure out how to generalize complex numbers. He was walking under a bridge when he came up with that equation, and carved the equation on the bridge.

His carving, if it ever existed, is gone. But there is a plaque on the bridge commemorating the event. It reads:

Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i² = j² = k² = ijk = −1 & cut it on a stone of this bridge.

davidgl• 6 years ago

I'm not a mathematician, but I think it's about extending the idea of a scalar and a single rotation (Complex numbers) into a scalar + 3 rotations (Quaternions). The idea can be extended further to a scalar with 7 rotations - https://en.wikipedia.org/wiki/Octonion, but no further, for reasons I don't understand.

pflats5• 6 years ago

I got busy, so I wanna say thanks to everyone who tagged in for me. So many great answers. I'll link a few that speak the most to my own feelings.

Why do I like it? I am, as klodolph notes [1], a dyed-in-the-wool algebraist. It's where I find the most beauty and joy in mathematics.

[1] https://news.ycombinator.com/item?id=22202606

This invention/discovery is a fundamental development in abstract algebra, not a terminal one. Quaternions are just a jumping-off point, and I've always found the Caley-Dickenson construction that pauldraper explains[2] absolutely beautiful.

[2] https://news.ycombinator.com/item?id=22202513

Why would I want it specifically as a tattoo? jfengel points out the special history of that specific equation[3]: it was (allegedly) carved into a bridge in Dublin when Hamilton stumbled onto it, but the carving is gone. Kinda fitting to give it new permanence.

[3] https://news.ycombinator.com/item?id=22202513

So, putting it all together: it's a fundamental development in abstract algebra, which is my jam. It's could have been permanently inscribed in a bridge, but that's been lost to time, so giving it new permanency seems fitting.

Also, it's practical. My first thought was actually the Cayley table for the Klein four-group[4], but that would be a lot harder to get tattooed in a nice visible way. How I went from there to Hamilton's quaternion equation is left as an exercise to the reader. (If you're new to Cayley tables, they're just fancy times tables. Replace "e" with 1.)

[4] https://en.wikipedia.org/wiki/Klein_four-group#Presentations

jayshua3• 6 years ago

I’d probably go with “e^pi*i = -1”. Kinda cliched, but I really love that equation.

Mathnerd314• 6 years ago

Then you would have worry about the tau-ists: https://tauday.com/tau-manifesto

Koshkin6• 6 years ago

Or, better yet, e^pi*i + 1 = 0. (My personal preference is E = mc^2.)

pizza2• 6 years ago

Someone once showed me this "big brain" version of the equation: ceil[e] - floor[pi] = 0 :)

pastrami_panda• 6 years ago

I love quaternions, but I have to ask: you'd rather have that formula than Euler's identity tattooed?

klodolph2• 6 years ago

Today, mathematicians in the most general sense divide into algebraists and analysts. Tattooing Euler’s identity identifies you as a member of the analyst tribe, you live and breathe limits, sequences, and measures. A tattoo of Hamilton’s i^2 = j^2 = k^2 = ijk = -1 would identify you as a member of the algebraist tribe, who lives and breathes commutators, cohomologies, and quotients.

FisDugthop• 6 years ago

As another algebraist (category theory and computational complexity), this makes a lot of sense. Euler's identity is capricious and Euclidean to me, and far from the most beautiful equation, although it is still remarkably elegant. I don't have any tattoos, but I might consider some categorical diagram; I don't know how I'd pick just one! Perhaps there is some cool way to draw the Snake Lemma with a realistic-looking snake.

tzs2• 6 years ago

It's nice to see something other than e^(ᴨi)=-1.

If I had to get a math tattoo, I think I'd go for lim n→∞ Q_n^(1/n) = e^(ᴨ^2/(12 log 2)).

That comes from a theorem proved by Khinchin and Lévy. Khinchin proved that for almost all real numbers if you take the sequence of convergents of their continued fraction expansion, {P_1/Q_1, P_2/Q_2, ...}, then the sequence {Q_1, Q_2^(1/2), Q_3^(1/3), ...} approaches a limit, which is the same limit for almost all real numbers. Then Lévy determined the value of that limit, which is now called either Lévy's constant or the Khinchin–Lévy constant.

If not that, then this (in standard math notation rather than the verbose notation I'm using here):

Line 1: Let H_n = sum i=1 to n 1/n

Line 2: Hypothesis: sum d|n d < H_n + e^H_n log(H_n) for all n > 1

That's neat because that hypothesis is true if and only if the Riemann hypothesis [1] is true [2].

The Riemann hypothesis is a conjecture about complex numbers, and is widely considered to be the most important unsolved problem in pure mathematics. That it turns out to be equivalent to a such a simple conjecture involving just integers and a couple real functions from pre-calculus is a surprise.

[1] https://en.wikipedia.org/wiki/Riemann_hypothesis

[2] https://arxiv.org/abs/math/0008177

BlueTemplar22• 6 years ago

I'm curious about that "almost all" ?

logfromblammo• 6 years ago

Mine would be the Fano plane mnemonic for octonion multiplication, using two curves of constant width instead of the triangle and the circle. That's got the quaternions covered with the inside curve.

It can go next to the skeletal formula for benzaldehyde on my imaginary nerd canvas.

BlueTemplar22• 6 years ago

Brilliant !

jtolmar• 6 years ago

I'd probably pick the normal-normal conjugate prior, in precision form.

martin-adams• 6 years ago

I have no experience in this area. Would making a change like this inside a 3D Engine have any impact on performance?

CrazyStat• 6 years ago

No, they're mathematically exactly the same. Just a different way of thinking about the same mathematics that the author argues (probably correctly) makes more intuitive sense.

bordercases2• 6 years ago

The development environment would have to be set up such that the part of the language that uses GA primitives could be efficiently transpiled into their corresponding matrix abstractions to prevent loss of performance on the GPU at runtime.

mcphage4• 6 years ago

I'm confused—the author keeps referring to rotations. The very first sentence is "To represent 3D rotations graphics programmers use Quaternions", even. But I don't think you don't need quaternions to represent 3D rotations. You need quaternions to represent translations, because translations aren't linear functions in 3D. (To be a linear function, v * 0 = 0, but that isn't true for translations). My understanding was that the extra dimension was to be able to represent translations. Am I mistaken?

heyitsguay2• 6 years ago

You don't need to restrict yourself to mathematically-nice algebras of operations or whatever if you're doing 3d graphics though. If you want to represent a translation of x by vector v in 3D, just do x+v.

mcphage4• 6 years ago

Well, linear functions have the nice ability to be composed together, resulting in another linear function (which can then be composed, etc).

matt-noonan• 6 years ago

Yes, you are mistaken. You need 3 dimensions to represent a 3D rotation (2 for picking an axis, plus one for picking an angle). To represent translations and rotations together, you need 6 dimensions.

This also shows that the talk of needing to visualize 4d to understand quaternions is disingenuous. The formula for using a quaternion to rotate a vector is qvq^-1, from which it is immediate that changing the length of a quaternion does not change the rotation it represents. So you can just deal with unit-length quaternions, which form a 3D space.

mcphage4• 6 years ago

Hmm, I seem to be mixing up using 4d linear operations (ie, a 4x4 matrix) to represent translation, rotation, scaling, etc, with their combinations, with representing a rotation itself. Thanks for your help—I'll dig more into this.

gugagore8• 6 years ago

I wonder if you're getting confused with what is called "a dual quaternion" which has 8 basis elements, and can represent both a translation and a rotation.

mcphage4• 6 years ago

Doing some reading, I seem to be mixing this up with https://gamedev.stackexchange.com/questions/72044/why-do-we-.... The later bit I understand, but I think it shows I don't really get what this article is all about. I'll have to read it further. Thank you :-)

diegoperini• 6 years ago

Putting content aside (it's great actually), I love the format used for the article. Each header is also a link to a time point in the accompanying video. There are interactive canvases that are also used in the video. The only missing part is a teacher AI which I can ask questions to (just kidding).

alpineidyll3• 6 years ago

The author is missing a huge very real advantage of quaternions, that they are smooth functions of thier parameters and don't suffer from gymbal lock singularities. All 3d representions do, as can be shown with the math this guy is too lazybrained to do.

That really matters for defining smooth paths, quaternions simply have the right topology and rotations don't.

nmca• 6 years ago

Did you read the article? Irrc you can lerp rotors and the results are smooth and interpretable...

moultano2• 6 years ago

You should read the article before commenting.

eigenspace• 6 years ago

The author is using geometric algebra to create mathematical objects with all the same properties as quaternions, but to explain and understand them in a more comprehensive, sensible way. These things have all the same rotational properties as quaternions.

In particular, he’s explaining that quaternions are not vectors, they’re actually oriented planes and showing how their multiplication rules arise in a way that’s not ‘out of thin air’.

ArtWomb• 6 years ago

When I was learning graphics, the reasoning around quaternions was to avoid "gimbal lock". It was just taken as orthodoxy without question.

I think going forward next gen 3D engines will have to account for the improvements in GPU hardware realized by advances in machine learning. Mixed precision matrix multiply at massively parallel scale. As well as the demands of next-gen games. Things like real time ray tracing of deformable meshes ;)

phlakaton• 6 years ago

Same critique as I have for the article, though: just because the subject is poorly taught doesn't necessarily mean that the subject needs to be replaced. The key here is that gimbal lock isn't a feature of quaternions, it's a feature of Euler angles: https://math.stackexchange.com/questions/8980/euler-angles-a...

alejohausner• 6 years ago

Quaternions can be normalized, thus eliminating numerical errors than can accumulate if you multiply a rotation matrix many times. These numerical errors make the matrix non-unitary, and introduce weird stretches and skews into the transformation.

iorrus• 6 years ago

The reason many people have trouble understanding quaternions is that they reason about them incorrectly.

Quaternions are actually a separate 'thing' when compared to vectors (which is why their multiplication seems off and the square is a negative number).

Quaternions should be considered a versor (a rotation around great circles), that is a change in direction which is different from a vector.

See this work: https://archive.org/details/cu31924001506769/

DonHopkins3• 6 years ago

From the title, I was expecting to read an article by a crazy old retro mathematician shaking his fist and yelling at clouds that we should all go back to using Euler angles. But this was a much better article than the title suggested!

dahart5• 6 years ago

As a side note, Geometric Algebra contains more than just Rotors, and is a very useful tool to have in one's toolbox.

I worked in a game engine where a pair of mathy programmers fell in love with Geometric Algebra, and use this same argument that quarternions ought to be replaced to overhaul all of the math code in the engine using Geometric Algebra. The character rigging system removed matrices and used GA instead.

This caused several large problems in the code base:

For one, it slowed the code down a little because the GPU interface is all matrices, so there were conversions to matrices all over the place, rigging in particular.

And only two guys in the studio knew Geometric Algebra, and they didn’t invest time in teaching it or helping people understand it, they just hoisted it on everyone. All the rest of the programmers knew matrix math but not Geometric Algebra, so they would end up avoiding touching any of the GA code, i.e., any code that dealt in transformations. The two guys ended up with a lot of support of their own creation, but they were short with their answers, in part because they got so many questions, so the problem never went away.

The third problem is this whole rewrite was unnecessary. Fixing gimbal lock with quaternions is a tiny corner of the game engine, whereas using GA throughout is a massive rewrite. Matrices work really well for 98% of the code, and it’s not really a huge problem to have one or two routines that convert to quaternions and back while they do a rotation. It is a problem when any transform at all involves bivectors and rotors and you have no idea what the hell those are nor do you have time in your schedule to take a math class at work.

Personally, I’m intrigued by GA and have wanted to learn it for a while, but having used it in production, I’m mildly against replacing quaternions with GA, and very wildly against replacing matrices with GA.

jessermeyer5• 6 years ago

These problems are not inherent of GA, but of (flawed) implementation details and deeper social problems. GA code boils down to essentially the same computations as 'raw' Linear Algebra, and should be implemented as such. Your math centered programmers fell in love with the idea without ever figuring out how to really use it.

dahart5• 6 years ago

I completely agree it wasn’t inherent to GA, but I wouldn’t summarize it as they didn’t figure out how to use it. They were fluent, the problem is the rest of us weren’t.

The problem with what they did, and what this article is doing, is suggesting we “fix” something that takes 2 or 3 math classes to understand by replacing it with something that takes an entire semester to understand.

I honestly believe that GA has insights to offer, but I don’t believe it will save any time to upgrade the use of quaternions in any 3d engine.

noobermin• 6 years ago

Right, as a physicist, this may be super naive but matrices are just representations of the underlying algebraic structure so I'm not sure why it isn't possible to merely do both once you've decided on a way to convert between the two.

notduncansmith• 6 years ago

Thank you for the anecdote! Sorry to leave such a small comment but I believe you meant “foisted it on everyone”, not “hoisted”.

dahart5• 6 years ago

Foisted is indeed a better choice there, whatever I thought I meant. Thanks! I thought about editing, but I’ll leave it so as not to disturb your comment.

mrspuratic• 6 years ago

I'm sure everyone's been in a situation where so much sh*t was dropped on them that a hoist must have been involved ;-)

smcameron• 6 years ago

Fixing gimbal lock with quaternions

Huh? I thought quaternions don't have any problem with gimbal lock. That's a problem with Euler angles and matrices.

Edit: I think you meant using quaternions to fix gimbal lock problems with matrices, which makes sense.

ur-whale• 6 years ago

The wikipedia page for GA, and in particular the introduction, is an absolutely perfect example of the horrible quality of math pages on wikipedia.

GeorgeTirebiter• 6 years ago

Thanks for saying this. On other math topics on wikipedia, I've read, re-read, and re-read them and still felt like I didn't understand. Maybe it wasn't all my fault.

WAHa_06x362• 6 years ago

Math on Wikipedia is just absolutely atrocious.

harmoat• 6 years ago

Remember, wikipedia is an encyclopedia, not a textbook

SkyBelow2• 6 years ago

I feel like this requires the same concern when replacing any system. Even if the new tool is better in theory, does it have enough people using it/experts working on it that it is better in practice and does the costs of replacing it justify the improvement.

Just look at something like a 50 year old IBM mainframe vs. the newest Sql Server with .Net Core running on the server of your choice. Is the latter the better choice for a new application? Yes. Yet many very successful businesses don't replace their mainframes because the benefits do not justify the costs.

BlueTemplar22• 6 years ago

The main issue here is that they tried to port what sounds like a mature code base. Imagine changing programming languages! They should have restarted from scratch, and with a team fluent in Geometric Algebra...

BlueTemplar22• 6 years ago

IIRC matrices are still required in Geometric Algebra ? Maybe by "matrix math" you mean "Vector Algebra? "

gugagore8• 6 years ago

I'm not the grandparent. I'm not sure what you mean by "required", but, no, I don't think so.

You can represent a lot of algebraic objects and operations using matrices and linear algebra operations. The most accessible example of that is probably complex numbers: https://en.wikipedia.org/wiki/Complex_number#Matrix_represen...

and I wouldn't say that "matrices are still required in complex numbers" any more than I'd say "matrices are still required in geometric algebra".

agumonkey• 6 years ago

Although the fact that GPUs are not ready for GA .. forcing a model that cannot fit onto hardware is quite a massive naive mistake.

janoc• 6 years ago

That's not about "not being ready", the GPU really doesn't care whether the numbers it is multiplying are coefficients of a matrix or a bivector.

The problem is that all libraries, drivers, etc. use matrices, quaternions and vectors. So you would have to constantly convert back and forth, which is both error-prone and a non-trivial performance problem.

Of course, you could build your own libraries for everything, but that's just crazy. Who has time for doing that?

Not to mention that writing a mathematical library like that is a highly non-trivial task. Not just the algebra part but also numerical stability and accuracy are a big deal. That requires a very skilled person, naive implementations will rapidly blow up in your face.

And doing all this what for, exactly? So that the GA explanation of rotations doesn't require 4 dimensions while the math complexity of rotor algebra ends up being the same as with quaternions? So there isn't really a performance benefit neither.

oppositelock2• 6 years ago

I'm a former game engine lead from many years ago, from around the time that quaternions were becoming popular, since we never used them before the early 2000's, really, as 3D was young and rotation matrices sufficed.

Quaternions came into favor to solve the problem of gimbal lock in composed Euler rotation matrices. This happens when you create a rotation which rotates one axis into another, and end up with a matrix that loses one axis (it loses an eigenvector), and you become "trapped" in the new rotated frame and can't ever rotate out of it. Quaternions don't suffer from this problem, but they're also tricky to work with and reason about, and their rotations can become funny when composed - instead of rotating from orientation A to B, they'll go through a C in a very different place. You need to create heuristics to keep rotations looking sane with quaternions. In the games that I've worked on, we took other precautions to avoid gimbal lock in rotations and stuck to Euler matrices. For example, in a flight simulator, you always computed the final rotation matrix for the plane location directly from its heading, pitch and roll, and so, you'd never suffer gimbal lock.

Why stick to Euler matrices and not some better Geometric Algebra or Quaternions? Because it's really easy to interpolate, and think about plain old rotations about a vector, and you can teach anyone to avoid gimbal lock. It's easier to avoid that problem than to train a bunch of junior engineers on higher level math.

pfedak• 6 years ago

Could you elaborate more on interpolation and weirdness of composing quaternions? I thought a big selling point for quaternions was the ease and naturalness of just using slerp, whereas with euler angles a similar approach gives bad results. When you mention rotations about a vector, do you mean you decompose the desired rotation matrix into axis/angle? Otherwise I'm very impressed by your ability to visualize compositions of arbitrary rotations.

salty_biscuits2• 6 years ago

Gimbal lock is a problem for euler angle representation of rotations, not rotation matrices themselves. You can fix it for euler angles the same way they used to for old mechanical gimbal gyroscopes. By having a fourth rotation of the frame that points the singularity out of the way. This is a typical solution in navigation for dealing with the poles of the earth (a wander angle frame).

structural• 6 years ago

Quaternions are also quite a bit more efficient (both in flops especially when composing rotations frequently) as well as storage space (4 values in a quaternion vs. 9 in a 3D rotation matrix).

It's interesting to compare disciplines, since I've never worked on a game engine but have done a lot of simulation of physical systems: there, quaternions have been standard for at least fifteen years, and I've seen geometric algebra frequently for about the past five.

Interestingly, where we saw most gains from thinking about geometric algebra wasn't in simple rotations, but when starting to look at things like computing kinematic chains composed of dozens of joints: Now we're at 4x4 matrices to store each transformation, which is the traditional way and works well. From this, some seriously deranged mathematicians introduced the concept of an eight-element dual quaternion (essentially, pair of quaternions specially constructed) that can represent this. Again, super efficient, but even more intractable for newcomers than quaternions. At this point, starting to express both concepts in terms of geometric algebra has been able to keep most of the performance improvements as well as only having to teach one concept.

When I left that position a year ago, we were starting to see recent graduates that already had pre-exposure to geometric algebra concepts before even starting to work on the codebase.

Wistar• 6 years ago

Grass Valley Group used quaternions in their Kaleidoscope real-time digital effects system which was launched in late 1987. It was a fabulous machine.

nautilus12• 6 years ago

Isn't the data structure that supports quaternions just one dimension up from matrices? Why not continue using matrices and retain that last dimension seperately somehow? Then it doesn't require having to retool the GPU/matrix side of it. IE you will have multiple matrices where you would have had one otherwise.

madrox• 6 years ago

As an engineering manager who's had to push adoption of new technologies, I feel this argument in my bones.

That said, taking your argument to the extreme, we all probably should've stuck with PHP, since everyone understood it and, well, was it really worth the switching cost? Naturally that's dumb, so there's a certain amount of "how do we get there?" with any new technology that's probably a good idea.

That's probably not going to get answered by the principal engineer who says This Tech Is The Future. It's going to get answered by people like you who think about switching costs and the impact new tech has on people.

Chirono• 6 years ago

I would hope any good principal engineer spends a decent amount of time considering switching costs and impacts of new technology. Sounds like this isn't your experience though?

jaw• 6 years ago

taking your argument to the extreme, we all probably should've stuck with PHP, since everyone understood it and, well, was it really worth the switching cost?

There are multiple paths for honing a new technology, without dumping it on a large group of developers while it's in its experimental stages:

- Use it for a series of side-projects

- Use it in a startup or startup-like team in which all the developers are bought into it, and happy to work through the obstacles

- Use it at an organization that's both able and willing to devote a large amount of resources to making it work (e.g. devoting a full-time team to its development & support & related training)

Once the kinks have been sufficiently worked out in one of those contexts, and a solid ecosystem with good documentation exists, then there's a much better chance that the benefits will be worth the switching costs for the average project.

moralestapia4• 6 years ago

This is more or less the thing with "Category Theory".

Very fancy and neatly structured code but in the end you accomplish pretty much the same; and now nobody on the team understands a bit about what you're doing.

asdfman123• 6 years ago

You mean there's a way to establish yourself as an arcane code wizard and make yourself irreplaceable? Sign me up!

crimsonalucard8• 6 years ago

That just means the concept is ahead of its time. People scoff at things they don't understand before that thing takes over the world.

earenndil3• 6 years ago

GPUs don't have dedicated matrix hardware anymore, making it questionable whether you really need to convert your GA structures to matrices first. [1] finds that quaternions are faster than matrices on modern hardware; I find that questionable, but it at least shows that they are comparable.

1: https://tech.metail.com/performance-quaternions-gpu/

dahart5• 6 years ago

The APIs (D3D, etc.) do use matrices, and that's all that matters from the point of view of an engine developer. Composing quats together is certainly faster than composing matrices, but quats are limited (no translation / skew / perspectivce / etc.), so there is no direct apples-to-apples comparison between quats and mats.

nbulka• 6 years ago

"We just accept their odd multiplication tables."

What is odd about the quaternions' multiplication tables? Is it the fact that the commutative property of multiplication is violated? For the sake of argument, let me assert that it is.

What if at the highest level of abstraction x * y had no obligation to equal y * x?

rytill• 6 years ago

So what else does this operator do? It’s not commutative. Does (x * y) * z still equal x * (y * z)?

yiyus3• 6 years ago

The geometric and exterior products used in GA are non-commutative too.

timerol• 6 years ago

It is definitely not that, since the article shows that a^b = -b^a. I suppose that it's the choice that ij = k as opposed to -k?

I'm not sure, but the article's implication that nobody else cares why things work to be distracting and insulting.

Personally, I have always found it important to actually understand the things I am using.

danmg2• 6 years ago

Those who don't understand quaternions are condemned to reinvent it, poorly.

I work in a research field that uses rotations heavily, and trying to use things like Euler angles (with 4 or 5 competing representations) and axis angles has created nothing but confusion among people. If people used quaternions from day one, it probably would have saved, cumulatively, the time of several dozen phds.

jessaustin• 6 years ago

What research field is this? Why hasn't one of the researchers made the conceptual leap you propose?

danmg2• 6 years ago

It has to do with metallurgy. Some people do use them, but most work is done with euler angles.

6gvONxR4sf7o2• 6 years ago

I think you misunderstand the article. It's not saying to replace them with euler angles or axis angles or anything analogous. GA is among other things a more intuitive mental model of quaternions.

OliverJones• 6 years ago

As we kiss these quaternions goodbye, let's remember how they came to be, and their role in building the discipline: Ivan Sutherland's early hardware graphics pipelines.

wbl• 6 years ago

Yes, the even Clifford algebra is the quaternions. So what?

Koshkin6• 6 years ago

They say Geometric Algebra is "the new language of physics" and whatever else; my problem with this is that, unlike, say the calculus of differential forms (or tensors), it does not make sense in more general manifolds which are at the heart of the modern theoretical physics. Geometric Algebra, therefore, is more like something that, sure, could be taught in high school in an attempt, for example, to streamline elementary vector algebra and thus eliminate some nasty questions (should they arise); yet, there is some mental load to it that may make it not worth the effort...

moultano2• 6 years ago

Can you elaborate on that or provide a link? The OP made it seem like it was strictly more general, and I was getting excited to dive in through that route.

Koshkin6• 6 years ago

Well, nothing wrong with getting excited about and learning this stuff. Point was, in more complicated cases like (non-flat) [co]tangent bundles GA, as an intuitive framework, loses any advantage, perceived or real, over differential forms.

6gvONxR4sf7o2• 6 years ago

I found that learning geometric algebra in a nice flat space made differential forms much more intuitive for me. I don't think they're so at odds, and teaching one then the other may benefit both.

BlueTemplar22• 6 years ago

Uh, getting rid of pseudo-vectors ? Maxwell's Equations?

http://www.av8n.com/physics/maxwell-ga.htm

zrkrlc• 6 years ago

Huh, it absolutely does. Alan Macdonald wrote a book about geometric calculus and made a companion video series for it. Here’s the section defining manifolds: https://www.youtube.com/watch?v=jfwdlW7Yr_I

That said, tensors are in fact more general than geometric algebra’s multivectors (in that the latter form a quotient algebra of the former). But not all tensors correspond to manifolds as we know them in physics and I argue that GA keeps track of the geometric properties we want better.

thrower123• 6 years ago

Nice thought, perhaps, but I doubt anything would come of it. Quaternions came into vogue because the problems of Euler angles were very apparent in practice, but we still think, at best, in yaw/pitch/roll, and for a lot of people, roll is a little on the sketchy side.

Now we have twenty, twenty-five years of code and resources that make use of quaternions. In some ways, game development is incredibly hide-bound and conservative, and for the most part eschews rigid correctness for performance, convenience, and a loosey-goosey good 'nuff feel.

edflsafoiewq• 6 years ago

He's not actually proposing any change. He just gives a different construction of the usual quaternions.

VikingCoder• 6 years ago

While we're at it, can we please move the homogeneous coordinate to be the first value in memory? That way, a vector or a point, if they're represented sparsely in the data structure, just work.

[1], a point at the origin, is the same as [1 0] - a point at the origin in 1 dimension, or as [1 0 0], a point at the origin in 2 dimensions, or [1 0 0 0], a point at the origin in 3 dimensions.

Similarly, [0] is a zero vector. [0 0 0 0] is a zero vector in 3 dimensions.

Having to know [0 0 1] is a point at the origin in 2 dimensions (with a homogeneous coordinate), while [0 0 1] is a z vector in 3 dimensions (without the homogeneous coordinate), is just silly.

Why did we do this to ourselves?

joppy3• 6 years ago

It’s puzzling - even in most of the mathematical literature, x_0 is used for the homogeneous coordinate.

orbots• 6 years ago

Memory layout. You can do a hacky "cast" from a homegeneous 4-vector to a ideal 3-vector for free. Same pointer.

spacedome• 6 years ago

Amusingly the discovery of Quaternions by Hamilton was also followed by arguments within the mathematical community as to whether mathematics should be rewritten in the language of quaternions, as Hamilton proposed to do and subsequently spent the rest of his life pursuing. Initially many mathematicians found them confusing, Hamilton's book is unusually difficult to read and there were no others. The work of Grassmann was somewhat neglected at the time, and eventually the Gibbs/Heaviside notion of vectors (essentially what we use today) emerged as a competitor to the quaternions. It seems it was a particularly bitter mathematical divide, up there with Newton vs Leibniz. Here is a quote by Tait, one of the "quaternionists":

"Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassman"

See "History of Vector Analysis" by Crowe or "Hamilton, Rodrigues, and the Quaternion Scandal" by Altmann. Nice to see the author cites these!

The Geometric Algebra comes from Clifford Algebras, which where an attempt to combine Hamilton's Quaternions and Grassmann's forms, and in fact contains both as sub-algebras. In the case of 3D rotations calling them Rotors or Quaternions seems mostly like a different way of thinking about the same thing.

I think this would be more kindly put as "reimagining" quaternions and not "removing" them. The additional geometric intuition from GA does seem useful, and even as someone who has used quaternions extensively (though in a very different context), I would also choose to work with Geometric Algebra as a framework for geometry over Quaternions.

The visualizations are quite good here, it is a good way to understand bi-vectors, you can wiggle them about a bit in 3D instead of just staring at parallelograms on a page. The only criticism I have is that they say quaternions and the cross product come "out of nowhere", but then the way they present the geometric product is equally "out of nowhere".

randName• 6 years ago

I wonder if this has anything to do with the 4D game that he is working on. Do quaternions work in 4D space?

gliese13373• 6 years ago

Yes, it does; and no, they don't.

cygx2• 6 years ago

But they do? A 4d rotation can be decomposed into a left-isoclinic and right-isoclinic rotation, which in turn can be represented by left- and right-multiplication with unit quaternions.

adam-a• 6 years ago

This is a very cool article which has left me wanting more. It seems to stop just short of the end though, there is no section to actually explain the component values in a rotor, or an interactive diagram deconstructing the component parts of a rotor. Or did I miss it?

I'd also be really interested to see some more applications. I wonder what rotor interpolation looks like for example? I know I've had problems with quaternion interpolation in the past.

o_p• 6 years ago

Maybe the blog poster should try to learn how imaginary numbers work instead of trying to stick with reals, as an EE im probably biased but rotations -> complex numbers, as they make it much easier to work with it

BlueTemplar22• 6 years ago

Rotations in 2D - absolutely !

Rotations in 3D are a whole different beast...

epicgiga• 6 years ago

Let's remove them from one 3d engine first, and see how it goes.

DreamScatter• 6 years ago

Check out my geometric algebra software

https://grassmann.crucialflow.com

The Grassmann.jl package provides tools for doing computations based on multi-linear algebra, differential geometry, and spin groups using the extended tensor algebra known as Leibniz-Grassmann-Clifford-Hestenes geometric algebra. Combinatorial products included are ∧, ∨, ⋅, *, ⋆, ', ~, d, ∂ (which are the exterior, regressive, inner, and geometric products; along with the Hodge star, adjoint, reversal, differential and boundary operators). The kernelized operations are built up from composite sparse tensor products and Hodge duality, with high dimensional support for up to 62 indices using staged caching and precompilation. Code generation enables concise yet highly extensible definitions. The DirectSum.jl multivector parametric type polymorphism is based on tangent bundle vector spaces and conformal projective geometry to make the dispatch highly extensible for many applications. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.

skratlo• 6 years ago

Holy cow, dude, I'm going to use this one to make people feel dumb. Craziest cryptononsense I've ever read!

dang• 6 years ago

Please don't be a jerk in comments here. It's against the spirit of the site (and also its rules).

https://news.ycombinator.com/newsguidelines.html

ComplexSystems• 6 years ago

What does the differential operator do and how does it relate to geometric algebra?

ajkjk2• 6 years ago

There's a lot of connections between projective geometry and geometric algebra (well-- at least exterior algebra. Not sure about 'geometric', because I don't know what the geometric product means). If you implement _oriented_ projective geometry in homogenous coordinates (so a point (x,y) is a vector (x,y,1)), then the meet and join operators are implemented as ∧ and ∨. You can make a little dictionary:

Join = ∧, Meet = ∨. Vector = point, Bivector = line, Trivector = area, etc. The figure spanned by points (a,b,c) = a ∧ b ∧ c. The boundary of the figure = ∧^(k-1) of the metric (a,b,c), equal to ∂(a,b,c) = a ∧ b + b ∧ c + c ∧ a.

I have a very amateur blog that I never publicize about this stuff and I had a long post about this, but I've taken it down for now to rework it, or I'd link it here. Suffice to say there's a lot of connections and I feel like there are even more here that haven't been discovered yet.

The book "Oriented Projective Geometry" by Stolfi has a lot of this, although it doesn't explicitly talk about geometric algebra or the wedge product -- but it uses all the same symbols. I'm on the lookout for a better reference that bridges the gap.

[I have so far not figured out what the exterior derivative means in projective geometry, besides being dual to ∂; I believe that if derivative operators are just dual to basis vectors, then d is literally just dual to ∂. Not sure. I also have no idea what the geometric product means, and tend to be skeptical of it for that meaning.]

ajkjk2• 6 years ago

Do you have a reference for all the operations you've implemented here? I've found random papers that make reference to ∨, d, and ∂ wrt geometric / exterior algebra, but I haven't really found a comprehensive summary of them, so I've been trying to figure it out myself.

Zenst• 6 years ago

Bit like arguing that another keyboard layout would be better and it would if your starting from scratch, but people tend to go with what they know and that propergates as accepted and good enough, a standard. Hence, whilst there are better layouts for keyboard, we still have QWERTY.

Much the same with argument here, sure GA would be better, and maybe that may well come about, but alas quaternions are somewhat known by the many over the few and a bit of a QWERTY situation plays out.

playing_colours• 6 years ago

What is a good book you can advise for someone with maths background to dig into GA?

bwidlar• 6 years ago

Geometric Algebra in 2D - Fundamentals and Another Look at Complex Numbers:

https://www.youtube.com/watch?v=PNlgMPzj-7Q&list=PLpzmRsG7u_...

radarsat1• 6 years ago

Hm. I don't claim to fully understand quaternions, but I simply think of them as an obfuscated version of axis-angle representation, which seems to serve me well in reasoning about operations on them.

pjbk• 6 years ago

Reading some comments here...

* GA provides some nice savings when it comes to rotor/3D calculations, provided that the underlying data structures and architecture supports them. Check out the publications by Dietmar Hildenbrand and his team for several examples:

- https://www.researchgate.net/profile/Dietmar_Hildenbrand

* Quaternions start to exhibit several limitations when dealing with complex objects, even in 3D. It provides just the necessary structure to store quadrature information to avoid the "gimball lock" issue, for example. However the fact that it collapses the scalar with the pseudoscalar presents several problems again in calculations with dual quaternions and higher dimensions (projections, for once), and it's not much different from the hurdle of having to discriminate axial and polar vectors in Vector Calculus. The main issue is that its handedness doesn't scale well and has problems capturing the geometric nature and physics of the world in several dimensions (ie symplectic geometry).

* IMHO the real usefulness of GA is that, as it name implies, it's an algebra. That is what Clifford, Ball, Lie and Klein realized while extending the work of Grassmann on exterior algebras, screw theory, vector spaces and differential forms. Matrices, quaternions and some forms have awkward behaviors when they are treated symbolically as algebraic objects, being "leaky" on information or having singularities just because they are not the best representation. GA fixes that allowing you to formulate problems symbolically, and then you can almost blindly solve the equations with high confidence that the result will be sound. You can then convert the objects back to your favorite representation. For good examples, check out Terje Vold papers on:

- Rigid body dynamics: https://www.researchgate.net/publication/241273951_An_Introd..., and

- Electrodynamics: https://www.researchgate.net/publication/245345681_An_introd...)

[Note: Watch out for some typos]. For comparison, take a look at Featherstone's 6D Spatial Vector representation, which is similar to screws and I think shows the best you could do with Vector Calculus objects:

- http://bleyer.org/files/A%20Beginner's%20Guide%20to%206-D%20....

scythe• 6 years ago

His criticism of the cross product seems more poignant than his criticism of quaternions in computer programming. One might ask -- why not teach bivectors in introductory math instead of cross products?

Historically it seems like 3D geometry and particularly cross products in the context of electromagnetism fomented the primary demand in mathematics education for students to be taught vectors. Unfortunately the mathematics curriculum (in Western countries) has not really been updated in decades to better prepare students for the jobs of today; we still enroll all students in a sequence that culminates in differential equations and particularly in second-order linear differential equations, which just so happen to be crucial to control problems in electrical and mechanical engineering. While many people's jobs involve some kind of mathematics somehow, only a few jobs involve the mathematics of electrical engineering, and I suspect that is part of the reason why so many students are bored in math class.

BlueTemplar22• 6 years ago

Puzzles me too.

And even further :

IIRC, you don't need cross-product for electrical engineering, and IIRC it (thankfully) isn't (generally?) taught in high school.

Furthermore, later, in college, it would seem that electromagnetics get easier when taught through bivectors rather than cross products (and you avoid the traps with pseudo-vectors and, later, gimbal lock with Euler angles ?).

Maxwell's multiple equations seem to condensate to a single one under GA !

(What if we teach GA it will open the possibility of teaching EM in high school ?)

GlenTheMachine2• 6 years ago

A point that I don't think has been mentioned here: quaternions as a representation of attitude are really just a funky parameterization of an Euler axis/angle representation. Quaternions happen to be parameterized in a way that lets you avoid the evaluation of trigonometric functions when expressing their dynamics. This used to be important, e.g. in aircraft control systems, because computing power was extremely limited and you didn't want to have to evaluate sines and cosines in real time.

The biggest disadvantage of using quaternions over Euler axis/angle representations is that quaternions are basically impossible for humans to visualize, whereas Euler axis/angle representations are easier than transformation matrices, Euler angles, or any other representation.

So why not just use Euler axis/angle representations instead? Nobody cares any more about evaluating cosines at 1 kilohertz, and there would be none of this complicated geometric algebra stuff that nobody understands.

cygx2• 6 years ago

So why not just use Euler axis/angle representations instead?

Composition of two rotations in axis/angle representation basically proceeds via the quaternionic formula, ie in terms of half-angles. So if you need to do that a lot, it makes sense to go fully quaternionic to avoid having to work with both full and half angles.

crubier3• 6 years ago

I am impressed at the number of people reacting with « Euler angles are fine » or « quaternions are fine, we always did this way ». Having worked extensively with them it is obvious that Euler angles are broken beyound repair (singularities, 12 competing variations...) and quaternion are too (they only work in 3D). This article is extremely interesting and absolutely true. If geometric algebra had been discovered earlier we would never have needed a lot of these unnatural constructs.

QuesnayJr4• 6 years ago

Because we're going to start doing computer graphics in 4D?

crubier3• 6 years ago

We already do! You seem to ignore the widespread use of projective geometry in all 3D engines. For me it shows that you don’t know enough about the topic to have a useful opinion on this.

We do 4D stuff in 3D engines because it leads to massively nicer and simpler math, like replacing extremely large trigonometric calculations with a few additions and multiplications. It is actually the right way to do things. And bonus fact: it actually makes logical/intuitive sense when you actually try to understand the math.

ChrisLomont2• 6 years ago

Having written engines using both GA and matrices+quats, as well as writing articles on GA (one intro one in an old Games Gems book), I'd say GA is a terrible idea for 3D engines. GA is inherently slower to manipulate, as items require more storage and more memory touches. They add almost zero benefit.

If the idea is use a higher, more pure math structure, then one can go to even more abstract math formalisms, such as coord-free calculus and bigger algebras, but these, like GA, add more computational overhead to solve problems that don't exist.

Hestenes et. al., the main popularizes of GA in the math/programming intersection, have papers on writing raytracers in both, and they too clearly demonstrate loss of performance using GA.

overgard• 6 years ago

Worth noting that the author is the guy behind this extremely interesting project to create a 4D puzzle game: https://miegakure.com/

He's given a lot of talks on subjects like this. I imagine that what he's doing mathematically is a lot more intricate than most games need though.

ahaferburg• 6 years ago

Oh, thanks for pointing that out!

He posted an update on the game recently: https://marctenbosch.com/news/2020/01/miegakure-update-end-o...

thisrod• 6 years ago

I like this as an introduction to exterior algebra. It would be nice if the article mentioned the Hodge star, and explained why you can identify any form with a vector in 3D, even when it might be clearer not to.

QuesnayJr4• 6 years ago

This is the exact kind of clickbait title that HN normally changes to something less inflammatory. Every time I see the title I feel my blood pressure going up.

WAHa_06x362• 6 years ago

Isn't the only difference between rotors, as defined in this article, and quaternions the sign of the second 3d component?

And this only happens because the planes are defined as "xy", "xz" and "yz", rather than the more consistent "xy", "yz" and "zx"?

If you just changed the definition of the planes at the start of the derivation, it seems you would end up deriving the exact same operations as you would use with quaternions? I'm not sure if there is a good argument for not doing that.

skywal_l• 6 years ago

Is the outer product the same thing as the exterior product? Wikipedia gives different definitions. That's why I was a little confused by the article initially. Something to keep in mind.

chombier• 6 years ago

It is somewhat strange that all these articles/blogs claiming that Geometric Algebra is inherently superior to Quaternions spend so much time on how Quaternions are isomorphic to the even-subalgebra of GA3 (hence for all intents and purposes they're the same) and so little time on the odd-subalgebra (exterior algebra IIRC), which as far as I can tell is the main (only?) advantage of GA over Quaternions.

On the other hand, almost nobody mentions the nice geometric perspective that unit quaternions offer that are somehow "lost in translation" in GA: as a compact Lie group, unit quaternions come endowed with a bi-invariant Riemannian metric which means you can do interpolation, clustering, blending, statistics with them in a metric-consistent manner. And since the metric is compatible with the group structure, the geodesics are cheap to compute.

amai• 6 years ago

Let's also remove bivectors and instead use skew-symmetric matrices ( https://en.wikipedia.org/wiki/Skew-symmetric_matrix ) . The modern matrix notation is so much more powerful and flexible than all those 19th century inventions like complex numbers, quaternions, bivectors, dual numbers, ... that we should really stop teaching all those old confusing stuff.

Let's remove quaternions from every 3D engine (2018)

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