Hacker News Reader

Comment by ColinWright | Hacker News Reader

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?

View on HN

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.

> 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.

> rotors operate fully in 3D

By this, the author means: you can visualize a rotor FULLY in 3D, no extra dimension neccessary.

I suspect you never have tried to think about how quaternions work mechanically, I suggest this neat video:

https://www.youtube.com/watch?v=d4EgbgTm0Bg

Thank you for the link, very cool video!

I do get the theory part -- I know how quaternions and rotors work (and have a PhD in "pure" math). My objection (and a pretty firm one) is with the "if a phenomenon has more than three variables we cannot visualize it" logic. While lower dimensions make things a little easier to understand and develop an intuition for, a convenient abstraction or a model is much more important. Most engineers work with things described by multiple variables all the time and "reduce dimension to three" is seldom the main goal.

If the claim is that rotor is a better model, more convenient for software engineering, we can examine (and debate) that. But we should not recommend switching just because it has one less variable. For someone with an algebraic rather than geometric view, the ability to multiply quaternions on a piece of paper may be a strong benefit. Double-checking, say the result (1+i)*(j-k) by hand takes 10 seconds and a piece of paper; try mentally computing the rotor composition -- you would probably fall back on algebra (I certainly would).

No math degree here, but personally I think the author's main point was not purely that they could be visualized without resorting to 4D. Instead I think it was that the concept of rotors can be explained from first principles, whereas he feels that currently, programmers look at quaternions as black boxes for which they have no intuition. Here's the relevant quote from the video:

> But instead of defining Quaternions out of nowhere and trying to explain how they work retroactively, it is possible to explain Rotors almost entirely from scratch. This obviously takes more time, but I find it is very much worth it because it makes them much easier to understand!

I can't speak to his argument really as, when it comes to 3D game dev, I'm purely a hobbyist. I do believe though that in the end, programmers want to program. They want to be handed a library with an API that makes sense. The ones who care deeply about the whys and hows of the math will always take the time to learn it; others just want to know how to write the code. For those people, I'm not sure the rotor equations I saw are any more intuitive at first blush than the quaternion equations.

In short, I feel like by the time you're at the point where you're explaining the details of a topic like geometric algebra, you've likely already lost the people who just want to code, even if the description you provide is more intuitive.

Having said that, I still found the video fascinating.

I think different people have different approaches. I can only say that, for me, I usually need some kind of visual understanding of what an object represents - which may be 2D, 3D, a graph structure or whatever else, depending on the task at hand.

In this case, a 3D visualisation is the right tool because the problem is all about objects in 3D space. So rotors let you mentally work with the same objects you're thinking about anyway instead of switching to some other concept or representation.

I don't see that "it's technically 2D anyway because it's projected to the surface of my monitor" is a valid argument. My visual system knows pretty well how to deal with pseudo-3D objects, thank you very much.

> 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.

This is a very interesting point. Have you tried VR? Do you think this preference for 2d and 1d is just because 3d display technology is still inconvenient or is it something deeper than that?

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?

It means the things in the basis, x^y, y^z and x^z have an easy to understand interpretation in ordinary 3D space, namely the parallelogram between x and y (in the case of x^y), for example.

It's also easy (after reading the article) to understand the operations that can be performed on these things.

It's not as obvious what i, j and k in the quaternions correspond to, or why they have the multiplication table that they do. It's an algebraic construction, not a geometric one and hence more difficult to visualise.

You're probably right, that's probably what was meant. I guess I'm so used to thinking geometrically about quaternions, I don't see much advantage to geometric algebra for 3D rotations (but geometric algebra does have other advantages, like working in any number of dimensions!).

The space of rotors and rotation quaternions are both three dimensional because the coordinates are restricted to unit magnitude.

Quaternions and rotors are exactly the same in practice, but the intuitions are very different. The intuition behind rotors involves planes and lines in 3D, whereas the intuition behind quaternions typically involves a unit hypersphere; there's a 3B1B video on quaternions where you can learn more about them.

You're right that the space of rotations is 3-dimensional. When I said 4, I meant without restricting to the unit hypersphere. This causes no problem for rotations because replacing q by a multiple tq leaves the formula x --> qxq^(-1) unchanged --although of course q^(-1) invloves dividing by the square of the norm so it's nice if the norm is 1.

Note that even if you do restrict q to have unit norm, q and -q still denote the same rotation! (In other words, the space of rotations isn't really S^3, but projective space RP^3.)

I'll be sure to check out the 3B1B video, I keep seeing them recommended and I think they might a good resource to recommend to my students.

>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.

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.

SO(3) is definitely three-dimensional, and rotations don't involve magnitudes.

I miswrote. Rotation calculations are represented (in quaternions and in geometric algebra calculations) as a 3D subspace of R^4, with the "extra" dimension accounted for by constraining quaternions to the unit hypersphere.

Rotations do involve magnitudes when interpreted as an axis (2 dimensional manifold embedded in R^3) and a magnitide (0 to 2pi)

https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat...

SO(3) is a coordinate-free abstraction that isn't relevant to the topic of GA vs Quaternions for calculating concrete values in computer graphics

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.

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 :).

> The exterior/wedge product is actually very closely related to the cross product, and works to generalize the cross product to n dimensions.

So, to be as opinionated as the g'parent comment:

The cross product is a hack which only works in certain dimensionalities, whereas the wedge product is the underlying idea, which works in all circumstances.

To be less opinionated:

The cross product is inconvenient because what it gives you aren't the "usual" vectors, they're axial vectors, which behave differently under mirror reflection than all of your other vectors do.

This is incorrect, if you read Spivak, you can define and (n-1)-ary product which generalizes the cross product. You give it (n-1) vectors and it gives you a vector orthogonal to all of them.

Whether you call it the cross product or not is just semantics, but it does exist in terms of the exterior product.

It's difficult to find online, but it's constructed directly in Spivak. A quote:

"It is uncommon in mathematics to have a "product" that depends on more than two factors. In the case of two vectors v,w in R^3, we obtain a more conventional looking product, v X w in R^3. For this reason it is sometimes maintained that the cross product. can be defined only on R^3" - Calculus on Manifolds, pg.84

Most graduate students read this (or did five years ago).

> This is incorrect, if you read Spivak, you can define and (n-1)-ary product which generalizes the cross product. You give it (n-1) vectors and it gives you a vector orthogonal to all of them.

I never said the wedge was the only way to generalize the cross product, I just said the cross product itself wasn't general.

It's just semantics: what you call the generalized cross product, I call the cross product.

We used to think as negative numbers being a generalization of the integers, so that's some food for thought. I'm sure once quantum mechanics dominates solving eigenvalue problems will be a high-school level problem, so we'll end up having complex numbers losing their complexity. We don't call them "real" numbers outside of math circles anymore.

To say that the cross product itself is a hack is a bit of a stretch though, it can easily be generalized and I think it's quite natural.

Good points about the cross product, but :

> We used to think as negative numbers being a generalization of the integers

"used to" ?

> We don't call them "real" numbers outside of math circles anymore.

Sure we do ?

These words all change meaning over time, mathematical definitions change, old words are used to describe new objects and new words are used to describe old objects. I'm using the word integer with it's archaic meaning here for rhetorical effect, but I'm probably being too obtuse ;)

To clarify, we used to think of integers as just the natural numbers. Integer was a colloquial word meaning "whole, entire", so there was presumably a discussion about how negative numbers were whole or entire, though I vaguely remember this historical story. My point is just that at some point negative numbers were seen as a advanced extension of the natural numbers. Even zero was seen as an unnatural extension, which is why there is a confusion to this date as to whether "natural" numbers include zero.

ref: http://mathforum.org/library/drmath/view/57212.html

Not the best reference, but I've read it elsewhere, so maybe it's a myth, but someone can trace it to the source if they want.

> We don't call them "real" numbers outside of math circles anymore.

I mean laypeople don't know what the real numbers are in distinction to the rationals, for example.

Oh, missed that part about integers.

Then again, with these conversations, the N < Z < Q < R < C... classification automatically pops in your mind if you did mathematics in last classes of high school (that's going to be a LOT of laypeople ! Of course, a lot of them, those that rarely encounter them, might then forget about this classification.)

And integers are still called "natural integers", and negative numbers are NOT called "natural" ?

"five years ago" ?

I have no idea what the kids are reading these days, I've since ceased my aspirations towards pure mathematics.

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

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

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

Here you go : http://www.av8n.com/physics/clifford-intro.htm

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.

> 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.

I disagree, but then I used bivectors extensively in the geometric part of my mathematical research (back when) so I would. ;)

y^2 = 1 is because it's a unit vector. The fact that bivectors are the correct way to view rotations is evident if you look at higher dimensions. Bivectors work just the same way as described here when you want to work in R^n, you just add basis vectors beyond x, y, z. But the algebra they describe no longer forms an associative division algebra over the reals. There is no way to get to the algebra of the (generators of) rotations in higher dimensions from the quaternions.

This is why i would say the quaternions (as built up from the complex numbers) are a non-sequitur and incidental to the structure of rotations.

Then again, I would always argue that one should use complex numbers to think of rotations in R^2, so.... :)

> I would always argue that one should use complex numbers to think of rotations in R^2, so.... :)

That's not so strange, it's basically e^i*theta isn't it?

> 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.

You can do anything you do with our numbers with roman numerals. You could argue that our base-10 system is as arbitrary as roman numerals, or that knowing how to do arithmetic in one system will allow you to do arithmetic in the other one. But that does not mean that learning arithmetic with roman numerals has the same difficulty as with base-10 numbers, and the only reason we find it more difficult is because we do not put enough effort.

I have introduced a few grad students to quaternions and GA. We eventually use quaternions most of the time, but they do not understand quaternions until they see GA (the same as someone may need some base-10 theory before completely understanding roman numerals arithmetic).

From the complex abacus the romans left us, it would seem that they used both base 10 and base 12 (for fractions - 360° in a circle is an extension of that). Base 12 seems to be slightly better than 10, but switching over would be WAY harder to "finishing" algebra... (and we wouldn't have the quite important these days 10^3~2^10 approximation.)

But they wouldn't extend to 4D.

You can treat complex numbers as 2D rotors (a 2D rotor is cos alpha + xy sin alpha, so xy can be replaced with i) and quaternions as 3D rotors. But if you build Cayley numbers from quaternions they don't have the same geometric meaning as 4D rotors.

y^2 = 1 because it is the sum of the dot product (1) and the outer product (0). The latter is because the parallelogram between y and y has zero area. The former is just the length of the vector y.

Some memorisation is required, but it is in terms of things that are easily visualised.

But why would the area be expressed as a length? This always struck me as a coincidence, and having read on bivectors it just makes sense that it would be (because a vector represents a distance on a line, a bivector represents an area on plane, a trivector represents a volume on a 3D space and so on).

Area is not expressed as a length. In general you can't simplify an expression that is a sum of an area and a length (this is one of the features of geometry that GA handles for you algebraically). e.g. "one meter plus one meter" can be simplified to "two meters", but "one square meter plus one meter" cannot be simplified. However if one of the terms is zero you can remove it, so "zero square meters plus one meter" is "one meter".

So x² = xx = x^x + x·x = 0 (area) + 1 (length)

xy = x^y + x·y = x^y (unit area) + 0 (length)

Right, with bivectors as in the article it all works.

What I was saying is that summing areas and length is exactly what happens with vector product. The k vector (or the k imaginary unit in quaternions) is the third unit vector, so it has length 1. But when I compute i⨯j=k, I suddenly interpret it as the area of the parallelogram formed by a and j, and at the same time k is the direction perpendicular to both i and j so its coefficient must be a length.

Likewise for triple product which computes a volume but it expresses it as a number (i.e. a length). We study all of these things in vector calculus and don't pay attention to these inconsistencies, but they are there.

Ah, apologies, I see what you're saying now :)

It's even more confusing because the triple product is actually a signed volume (pseudoscalar). I admit that I also never noticed these problems until learning GA.

ij=k means that the product of two of the three unit vectors must equal the third, which is a fundamental, non-arbitrary property, and without loss of generality the signs can be chosen so that ij=k (rather than ij=-k). Adding ii=-1,jj=-1,kk=-1 it's easy to deduce ijk=-1, j=ki, and i=jk, and ij=-ji, ik=-ki, jk=-kj.

Having to remember, out of the six possible permutations, that one of the three correct formulas is the one with the symbols in alphabetical order is much better than remembering the "right hand rule" (or was it "left hand rule"?).

But why would you remember a right hand or left hand rule? bivectors don't require you to.

x ^ y = x y - y x^T is the generator of the rotation that rotates a vector in the direction y into the direction of x. There is absolutely nothing to remember, and you can work this out immediately from just multiplying out the matrices:

exp(epsilon x ^ y) v ~ v + epsilon x ^ y * v = v + epsilon x (y,v) - epsilon y (x,v)

We add a little bit in the x direction and remove a bit in the y direction.

The whole view of rotations happening round an axis is just a coincidence of 3d space, it doesn't make sense in higher dimensions. On the other hand rotations always do (compose into ones that) happen on planes.

In my college in Antarctica we were taught

  x ^ y = y x^T - x y 
  
  exp(ε x ^ y) v ~ v + ε x ^ y * v
  = ε y (x,v) - v + ε x (y,v)

because we use the left-hand rule down under.

That's my point, you need to remember "ijk" in order (i.e. i=jk or ij=k) in a purely algebraic definition instead of a meaningless and complicated "handedness" rule involving rotations.

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.

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?

Why not change? It's a good question. Learning about GA is an investment, getting it coded up and tested is an investment, and the obscurity of GA is going to make the code almost encrypted to people who aren't fairly mathy. Although quaternions are definitely crazier than GA, the sad fact is they're much more popular.

There's some return on all that investment. It might pay off or it might not. I think it's a design decision.

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.

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.

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...

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

Reply on news.ycombinator.com