[HN Gopher] Let's remove Quaternions from every 3D Engine (2018)
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Let's remove Quaternions from every 3D Engine (2018)
Author : lelf
Score : 489 points
Date : 2020-01-31 09:23 UTC (13 hours ago)
(HTM) web link (marctenbosch.com)
(TXT) w3m dump (marctenbosch.com)
| duke360 wrote:
| 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 wrote:
| 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.
| K0SM0S wrote:
| > _"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!
| radarsat1 wrote:
| 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.
| bwidlar wrote:
| Geometric Algebra in 2D - Fundamentals and Another Look at
| Complex Numbers:
|
| https://www.youtube.com/watch?v=PNlgMPzj-7Q&list=PLpzmRsG7u_...
| pauljurczak wrote:
| 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.
| gugagore wrote:
| 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.
| o_p wrote:
| 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
| BlueTemplar wrote:
| Rotations in 2D - absolutely !
|
| Rotations in 3D are a whole different beast...
| ChrisLomont wrote:
| 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.
| Epskampie wrote:
| Previous discussion:
| https://news.ycombinator.com/item?id=18365433
| crubier wrote:
| 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.
| diegoperini wrote:
| 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).
| VikingCoder wrote:
| 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?
| joppy wrote:
| It's puzzling - even in most of the mathematical literature,
| x_0 is used for the homogeneous coordinate.
| ColinWright wrote:
| 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?
| xg15 wrote:
| 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.
| ChrisLomont wrote:
| >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.
| lonelappde wrote:
| 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.
| aratakareigen wrote:
| SO(3) is definitely three-dimensional, and rotations don't
| involve magnitudes.
| papln wrote:
| 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_represen
| tat...
|
| 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
| ptero wrote:
| > 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.
| mlatu wrote:
| > 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
| ptero wrote:
| 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).
| Gene_Parmesan wrote:
| 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.
| kybernetikos wrote:
| > 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?
| xg15 wrote:
| 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.
| omaranto wrote:
| 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?
| wbhart wrote:
| 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.
| omaranto wrote:
| 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!).
| aratakareigen wrote:
| 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.
| omaranto wrote:
| 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.
| onedognight wrote:
| 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.
| papln wrote:
| But 3D is the D that matters for 3D games. video games are
| concrete mathematics, not abstract mathematics.
| BlueTemplar wrote:
| 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
| uulu wrote:
| I got me intrigued. Thanks! Could you point to a source that
| explains geometric algebra from this angle? Book, article,
| web source, anything?
| BlueTemplar wrote:
| Here you go : http://www.av8n.com/physics/clifford-
| intro.htm
| solinent wrote:
| 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 :).
| msla wrote:
| > 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.
| solinent wrote:
| 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).
| msla wrote:
| > 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.
| solinent wrote:
| 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.
| LanceH wrote:
| 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.
| gugagore wrote:
| 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://e
| n.m.wikipedia.org/wiki/Rotation_formalisms_in_three...
| ianai wrote:
| That's a notational change that incurs unneeded, increased
| cognitive load. That's admittedly a mile high view.
| mcnamaratw wrote:
| 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.
| friendlybus wrote:
| 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?
| Certhas wrote:
| 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.
| ColinWright wrote:
| > _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.
| yiyus wrote:
| > 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).
| BlueTemplar wrote:
| From the complex abacus the romans left us, it would seem
| that they used both base 10 and base 12 (for fractions -
| 360deg 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.)
| Certhas wrote:
| 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.... :)
| bonzini wrote:
| > 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?
| wbhart wrote:
| 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.
| bonzini wrote:
| 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).
| defen wrote:
| 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)
| HelloNurse wrote:
| 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"?).
| Certhas wrote:
| 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.
| papln wrote:
| In my college in Antarctica we were taught
| x ^ y = y x^T - x y exp(e x ^ y) v ~ v + e
| x ^ y * v = e y (x,v) - v + e x (y,v)
|
| because we use the left-hand rule down under.
| HelloNurse wrote:
| 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.
| bonzini wrote:
| 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.
| zupatol wrote:
| 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.
| stared wrote:
| 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!)
| cuspycode wrote:
| 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'.
| BlueTemplar wrote:
| API stands for Application Programming Interface. What _is_
| exactly an "API"? Why not just "interface" ? (One should
| generally avoid acronyms...)
| paulddraper wrote:
| Yes, it should be interface.
|
| No need to distinguish here between API/CLI/GUI.
| Koshkin wrote:
| 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...
| 6gvONxR4sf7o wrote:
| 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.
| moultano wrote:
| 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.
| BlueTemplar wrote:
| Uh, getting rid of pseudo-vectors ? Maxwell's Equations?
|
| http://www.av8n.com/physics/maxwell-ga.htm
| OliverJones wrote:
| 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.
| spacedome wrote:
| 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".
| adamnemecek wrote:
| 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 wrote:
| Never heard of this before. Thanks for the share.
| pflats wrote:
| 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.
| logfromblammo wrote:
| 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.
| Koshkin wrote:
| She probably hoped for a different answer.
| tudelo wrote:
| No, I bet this was exactly the answer they were looking for.
| After all, it was love at first sight.
| xg15 wrote:
| > _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)
| davidgl wrote:
| 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.
| paulddraper wrote:
| You can actually go as far as you want to with the Cayley-
| Dickson construction [1] of algebras.
|
| 1. Complex numbers have associativity and communitivity of
| multiplication. (That is, (ab)c=a(bc) and ab=ba).
|
| 2. Quaternions have associativity but not communitivity.
|
| 3. Octonions have neither.
|
| 4. Sedenions [2], trigintaduonions, and not associative,
| commutative, nor even alternative [3]. (Alternative is
| associative specifically when the middle value is equal to
| one of the other's; i.e. a(ab)=(aa)b.)
|
| [1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_co
| nstru...
|
| [2] https://en.wikipedia.org/wiki/Sedenion
|
| [3] https://en.wikipedia.org/wiki/Alternative_algebra
| pflats wrote:
| Thanks for writing this! I referenced it in a reply
| upthread for why I like the equation.
|
| https://news.ycombinator.com/item?id=22204995
| jjoonathan wrote:
| > 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.
| SkyBelow wrote:
| >Quaternions are an attempt to do that in 3D.
|
| I guess the question is, why does it then stop. Why not a
| 4D alternative. Or if you look at it going by scalars
| needed in a single value, it goes from 1 to 2 to 4. Why not
| 8 or 16 (or some other growth)? Why does it stop there?
|
| Also, is there as easy of a problem to understand
| introducing the 3D technique (be it quarternions or be it
| Gemoetric Algebra) that works as well as using sqrt(-1) for
| imaginary numbers?
| nitrogen wrote:
| I still haven't wrapped my head around quaternions, but
| 3Blue1Brown on Youtube has a good series of videos
| justifying and explaining the complex numbers and
| quaternions in terms not of sqrt(-1) but of
| transformations of space.
| nilkn wrote:
| It can be generalized, but doing so requires some
| subtlety. The naive approach (the Cayley-Dickson
| construction) can be repeated ad infinitum, but it
| doesn't continue to yield useful results for representing
| geometric interactions like rotations in high dimensions.
|
| Thankfully, this is a solved problem. The correct
| generalized structure for doing geometry is called a
| Clifford algebra. For n-space and any nonnegative
| integers p,q satisfying p+q=n, there is a corresponding
| real Clifford algebra Cl(R,p,q). Cl(R,0,1) turns out to
| be isomorphic to C (the complex numbers), and Cl(R,0,2)
| is a four-dimensional algebra that turns out to be
| isomorphic to Q (the quaternions).
|
| This is actually not that surprising, because the
| signature (p,q) more or less means the algebra is built
| by adjoining p generators that square to +1 and q
| generators that square to -1 in the base field. This is
| formalized by taking a quotient of the tensor algebra of
| the field. You might wonder though why we have (p,q) =
| (0,2) for the quaternions. That's because if the two
| generators that square to -1 are i and j, then we can
| build the third as k = ij, so we get it for free.
|
| A real Clifford algebra is known as a geometric algebra,
| and these give rise to objects called rotors. Rotations
| in an arbitrarily high-dimensional space can then be
| written as conjugation by a rotor.
| lanstin wrote:
| You can't get the associative property in higher
| dimensions. There is something called octonions (which is
| not associative but has some similar tho weaker
| properties) https://en.wikipedia.org/wiki/Octonion There
| is a sequence of such structures, but after the Octonions
| you get non-zero numbers that multiply to zero: https://e
| n.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...
| jjoonathan wrote:
| > I guess the question is, why does it then stop. Why not
| a 4D alternative.
|
| It doesn't stop. That's what motivated geometric algebra,
| which works in any dimension. Quaternions are a sub-
| algebra of geometric algebra. They represent 3D
| rotations, which makes them interesting in their own
| right.
|
| Asterisk: I believe there's a sign convention issue in
| mapping between quaternions and the even subalgebra of
| the 3D geometric algebra, so they aren't identical, just
| isomorphic.
|
| > Also, is there as easy of a problem to understand
| introducing the 3D technique (be it quarternions or be it
| Gemoetric Algebra) that works as well as using sqrt(-1)
| for imaginary numbers?
|
| That's an extraordinarily high bar. I don't believe
| anything reaches it. Part of the problem is that complex
| numbers are one of the most successful concepts in all of
| mathematics. The other part of the problem is that most
| of the useful facets of geometric algebra escaped the
| field of abstract mathematics under their own name before
| the unifying structure was discovered. The dot and cross
| product, quaternions, differential forms and the general
| Stokes' theorem are all examples. The remaining value
| proposition of geometric algebra lies mostly in getting
| rid of minor annoyances that come from this half-baked
| nature of traditional vector calculus tools:
|
| * Cross products break in more than 3 dimensions and they
| break if you reflect them (see: pseudovectors). Bivectors
| have no such issues. They represent rotations in any
| dimension, reflected or not.
|
| * Vector algebra with dot and cross products involves
| memorizing lots of new identities and applying creativity
| to work around the absence of division, while geometric
| algebra just has division and the same bunch of algebra
| tricks you already know. The geometric product isn't
| commutative, so it isn't perfect in this sense, but
| learning to deal with non-commutative algebra is a much
| more fundamentally useful thing than learning a bunch of
| 3D-specific identities.
|
| * Dot and Cross with one argument fixed "destroy
| information" mapping from their input to their output. If
| you put them into an equation, the equation does not
| fully constrain the free vector, so you are often going
| to need more than one equation to represent any single
| geometric concept. Not so with geometric algebra. Many
| concepts map to a single equation. Including Maxwell's
| Equation (I use the singular intentionally)!
| uryga wrote:
| > geometric algebra just has division
|
| do you have a good reference for this? i've looked into
| GA bit but don't remember seeing anything like this. e.g.
| what would dividing a bivector by a vector mean?
| gugagore wrote:
| I think the answer to that question is that it doesn't
| "stop", but I'll try to offer a reason that isn't
| "octonions exist", but instead goes in a different
| direction.
|
| Geometric Algebra can capture the structure of both
| complex numbers and quaternions, and also the structure
| of the dot products, cross products, and the different
| kinds of vectors that arise from those operations.
|
| To be clear, matrix multiplication can also capture the
| structure of complex numbers [1] and quaternions [2].
| There might also be a concise reference to matrix
| representations of some geometric algebras, but I didn't
| find one. So matrices are kind of one way to not "stop at
| 3D", but the structure is almost too uniform (which on
| one hand makes it too general, and on the other hand
| makes it not general enough), I'd say). Sure, with a
| matrix you can represent rotations in 4D, but you still
| need to operate on vectors only. Geometric algebra, if it
| does have a matrix representation, gives names to special
| kinds of matrices and special kinds of vectors.
|
| [1] https://en.wikipedia.org/wiki/Complex_number#Matrix_r
| epresen... and [2] https://en.wikipedia.org/wiki/Quaterni
| on#Matrix_representati...
| JackFr wrote:
| 8 https://en.wikipedia.org/wiki/Octonion
|
| 16 https://en.wikipedia.org/wiki/Sedenion
|
| 32 https://arxiv.org/pdf/0907.2047.pdf <-
| Trigintaduonions
| pflats wrote:
| 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
| jfengel wrote:
| 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.
| GlenTheMachine wrote:
| My PhD advisor was a stickler for citing original sources.
| Really, _really_ original sources. He made me cite some
| papers written by Lagrange in the 17th century in French,
| when neither he nor I nor nearly anyone else who would ever
| read my dissertation could speak French.
|
| I got to the point where I needed to cite an original
| source for the quaternion equations, so I cited the bridge.
|
| He got the message.
| pflats wrote:
| Thanks for writing this. It was indeed a large part of why
| I like it. I added more detail in a reply to the parent
| post.
|
| https://news.ycombinator.com/item?id=22204995
| kens wrote:
| For a summary of William Rowan Hamilton's life (including
| the bridge story), see this amazingly clever video based on
| the song from _Hamilton_ :
| https://www.youtube.com/watch?v=SZXHoWwBcDc
| Koshkin wrote:
| See, for example,
| https://math.stackexchange.com/questions/911807/what-is-
| the-...
| nilkn wrote:
| 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.
| JackFr wrote:
| When I took abstract algebra as an undergrad, we did a
| brief bit on the quaternions. Bursting with curiosity I
| asked the professor if 8 and 16 dimensional structures
| existed. "Of course! But just as you lose commutivity with
| Q, when you go to the octonions, you lose associativity,
| and the sedonions lack "alternativity" (had to look that up
| -- I didn't remember) and they're basically algebraic
| novelties with out any application."
| bonzini wrote:
| Right, but while the Cayley-Dickson construction mostly
| provides novelties (though I remember reading something
| about octonions and string theory[1]), Clifford algebras
| are derived differently; they are isomorphic to complex
| numbers and quaternions for two and three base vectors
| respectively, but they produce something else after
| quaternion. This "something different" can be used to
| represent, you guessed it, reflections and rotations in a
| 4D space. Because they are not obtained from the Cayley-
| Dickson construction they are not division algebras,
| however.
|
| [1] https://www.quantamagazine.org/the-octonion-math-
| that-could-...
| s_dev wrote:
| 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 wrote:
| > I would like to own an OpenGL kettle
|
| Do you mean the Utah Teapot?
| s_dev wrote:
| Yes -- I don't even know why I had the word kettle in my
| head.
| war1025 wrote:
| I knew what you meant and didn't even consider that it
| was the wrong name, for what it's worth.
|
| Kettle and teapot are synonyms as far as I'm concerned.
| bmn__ wrote:
| These are different equipment.
|
| A kettle is used for heating water. In earlier times, it
| was made out of metal and put onto a heat source (fire,
| stovetop). Nowadays it is almost entirely displaced by
| the electric kettle, which is commonly made out of
| plastic and contains a metal heating plate or spiral on
| the inside.
|
| A teapot is a ceramic pitcher where you put the boiling
| water and tea leaves to brew the tea.
| scott_s wrote:
| Earlier-time kettles may be more common than you think.
| war1025 wrote:
| Either way, a water kettle is super useful, often
| surprisingly so.
|
| Definitely a kitchen gadget I'd recommend to anyone.
| DonHopkins wrote:
| They're both called black.
|
| https://en.wiktionary.org/wiki/pot,_meet_kettle
|
| https://en.wiktionary.org/wiki/pot_calling_the_kettle_bla
| ck#...
| jfengel wrote:
| Kettles go on the stove (or have a built in heater), and
| are used for boiling water.
|
| You pour the boiling water into a teapot, usually made of
| ceramic, which holds the tea leaves.
|
| Not that it's important, but now ya know.
| kleer001 wrote:
| 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 wrote:
| I explicitly made a detour when I was in Dublin to take a
| picture of it the plaque on that bridge. Worth it.
| jchallis wrote:
| 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
| chadcmulligan wrote:
| 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 :-)
| Isamu wrote:
| Any idea why the name Hamilton appears to be deliberately
| defaced on the inscription?
| bregma wrote:
| 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.
| DonHopkins wrote:
| Then you could dress up for Halloween as Broome Bridge!
| tzs wrote:
| It's nice to see something other than e^(i)=-1.
|
| If I had to get a math tattoo, I think I'd go for lim n-[?]
| Q_n^(1/n) = e^(^2/(12 log 2)).
|
| That comes from a theorem proved by Khinchin and Levy. 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 Levy determined the value of that
| limit, which is now called either Levy's constant or the
| Khinchin-Levy 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
| pastrami_panda wrote:
| I love quaternions, but I have to ask: you'd rather have that
| formula than Euler's identity tattooed?
| FisDugthop wrote:
| 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.
| uryga wrote:
| > I don't know how I'd pick just one!
|
| picking one up to unique isomorphism should be good enough
| ;)
| klodolph wrote:
| 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.
| pflats wrote:
| Thanks for writing this! It's spot on. I referenced it in a
| post upthread[1].
|
| [1] https://news.ycombinator.com/item?id=22204995
| jayshua wrote:
| I'd probably go with "e^pi*i = -1". Kinda cliched, but I really
| love that equation.
| Mathnerd314 wrote:
| Then you would have worry about the tau-ists:
| https://tauday.com/tau-manifesto
| jayshua wrote:
| lol. I'm actually one of them. "e^tau*i=0" is my preferred
| form. I don't tend to bring it up because we're a little
| crazy and I don't want to draw attention to myself.
| evozer wrote:
| shouldn't it be e^tau*i = 1 if tau is 2pi?
| jayshua wrote:
| Oops, typo. You're right. You could also write "e^tau*i =
| 1 + 0" to relate the "5 most important numbers in math"
| but that form always seemed a bit forced to me.
| nybble41 wrote:
| If you write "-1 * e^(tau * i) + 1 = 0" you can
| reasonably claim to relate _six_ important numbers: -1,
| e, tau, i, 1, and 0. IMHO that looks a bit less forced
| than the version with "1 + 0", though of course it's not
| the simplest form. (I mean, that "+ 0" could have been
| inserted almost anywhere...)
| [deleted]
| Koshkin wrote:
| Or, better yet, e^pi*i + 1 = 0. (My personal preference is E
| = mc^2.)
| klodolph wrote:
| Why not E^2 = m_0^2 c^4 + p^2 c^2?
| pizza wrote:
| Someone once showed me this _" big brain"_ version of the
| equation: floor[pi] - ceil[e] = 0 :)
| jtolmar wrote:
| I'd probably pick the normal-normal conjugate prior, in
| precision form.
| [deleted]
| epicgiga wrote:
| Let's remove them from _one_ 3d engine first, and see how it
| goes.
| scythe wrote:
| 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.
| martin-adams wrote:
| I have no experience in this area. Would making a change like
| this inside a 3D Engine have any impact on performance?
| CrazyStat wrote:
| 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.
| bordercases wrote:
| 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.
| earenndil wrote:
| It's unclear that you really need to convert to matrices
| before sending off to the GPU. [1] finds that quaternions
| are faster; I find that questionable (does it scale?), but
| it shows that the two are comparable.
|
| 1: https://tech.metail.com/performance-quaternions-gpu/
| pjbk wrote:
| 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....
| DreamScatter wrote:
| 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 wrote:
| Holy cow, dude, I'm going to use this one to make people feel
| dumb. Craziest cryptononsense I've ever read!
| ComplexSystems wrote:
| What does the differential operator do and how does it relate
| to geometric algebra?
| Zenst wrote:
| 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.
| wbl wrote:
| Yes, the even Clifford algebra is the quaternions. So what?
| randName wrote:
| I wonder if this has anything to do with the 4D game that he is
| working on. Do quaternions work in 4D space?
| mcphage wrote:
| 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?
| matt-noonan wrote:
| 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 q _v_ q^-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.
| mcphage wrote:
| 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.
| gugagore wrote:
| 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.
| mcphage wrote:
| 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 :-)
| heyitsguay wrote:
| 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.
| mcphage wrote:
| Well, linear functions have the nice ability to be composed
| together, resulting in another linear function (which can
| then be composed, etc).
| heyitsguay wrote:
| Computable functions have that same property, and that's
| the space that programs work in. It doesn't matter if the
| composition elements are linear, only that they can be
| computed on your hardware within some time budget.
| cambalache wrote:
| This article suffers from the syndrome -I learned what a hammer
| is, everything now looks like a nail- .
| friendlybus wrote:
| In what way? The blanket application?
| DonHopkins wrote:
| I learned what math is, everything now looks like a number.
| rytill wrote:
| I learned what a relation is, everything looks like it can be
| compared to other things.
|
| I learned what a function is, everything looks like a mapping
| between two sets.
|
| Wait...
| DonHopkins wrote:
| It all boils down the fight between King Azaz -vs- The
| Mathemagician.
|
| https://thephantomtollbooth.fandom.com/wiki/King_Azaz
|
| >King Azaz, ruler of Dictionopolis: King Azaz the
| Unabridged is the ruler of words. His law is that words are
| much more important than numbers. He is the brother of the
| Mathemagician who is the ruler of mathematics. They came in
| agreement that numbers and words are equally important in
| the end. King Azaz is a lot like his brother, only the
| ruler of words. However, unlike his brother, Azaz is more
| like a stereotypical ruler. He takes the title of king, he
| lives in a palace, and he encourages his subjects to attend
| banquets.
|
| https://thephantomtollbooth.fandom.com/wiki/The_Mathemagici
| a...
|
| >The Mathemagician, ruler of Digitopolis: The Mathemagician
| is the brother of King Azaz and ruler of Digitopolis. His
| law in Digitopolis is that numbers are far more important
| than words, while King Azaz's law is that words are more
| important than numbers. The Mathemagician is pretty much
| polite but gets really mad whenever somebody mentions that
| numbers aren't that important or valuable, since numbers
| mean so much to him.
| [deleted]
| graphpapa wrote:
| This seems to be an excellent introduction to geometric algebra,
| a topic I have had difficulty picking up so far.
| yiyus wrote:
| 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.
| K0SM0S wrote:
| 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...
| dahart wrote:
| > 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.
| ur-whale wrote:
| 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 wrote:
| 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.
| earenndil wrote:
| 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/
| dahart wrote:
| 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.
| nautilus12 wrote:
| 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.
| jessermeyer wrote:
| 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.
| noobermin wrote:
| 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.
| joppy wrote:
| Rotors in geometric algebra form a "double cover" of
| rotations, for example using unit quaternions there are two
| different quaternions (q and -q) which will map to the same
| rotation matrix. This means that mapping from rotation
| matrices to quaternions is a bit fraught - you need to pick
| a "side", and if that side happens to be opposite to what
| another part of the code (perhaps the part only using
| quaternions) chose, you'll get some whacky stuff happening.
| The same idea applies for geometric algebra.
| [deleted]
| dahart wrote:
| 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.
| jessermeyer wrote:
| Without knowing more context it's a bit hard to respond,
| but from what you said, it sounds like they wrote an
| abstraction layer above matrices that observed GA
| semantics, which required a lot of run-time conversion.
| This is a flawed implementation, especially for a game
| engine. It's like writing a v-table for matrices to support
| every general case when you only need 3 or 4 specific
| instances anywhere in your code.
|
| Time spent -- You're probably right. For new technology,
| it's probably worth the research to see if code clarity is
| worthwhile...
| sdenton4 wrote:
| Not necessarily on top of matrices... I can easily
| imagine a part of an engine working with GA and then
| having to translate back or forth to matrices whenever it
| needs to interact with another (outside) part of the
| system.
| jessermeyer wrote:
| Translations like this are probably not a great idea in a
| game engine but I'm willing to be shown an example where
| it's the Thing You Want To Do.
| sdenton4 wrote:
| Oh, yeah, it's a terrible thing to do, absolutely. All
| I'm saying is that you can have a perfectly consistent
| /component/ using GA, but if it's living in a larger
| ecosystem of matrix-based code, you're going to have to
| do these translations somewhere. (eg, we rewrote the
| camera compoent in GA, but the dudes doing grass modeling
| haven't converted.) You get an all-or-nothing problem:
| it's efficient if the whole codebase is using the same
| abstraction, but converting around between a bunch of
| abstractions will come with a cost.
| jessermeyer wrote:
| Yeah, I don't work on codebases of that scale so I can't
| really comment knowingly of what cost it would take to
| move the whole parcel over the GA. Probably highly
| dependant on code / social org.
| quatquatquat wrote:
| I'm not sure you are fully understanding the situation.
| What jessermeyer is saying is that it's possible to use GA
| as a language for describing geometric operatiins while
| continuing to implement them under the hood using old-
| fashioned vector algebra.
|
| The major proponents of GA don't suggest doing this. I'm
| not an expert so I can't rule out the possibility that
| somebody, somewhere has done it successfully. Computer
| implementation of GA is still a research topic:
| http://geometry.mrao.cam.ac.uk/2016/11/ga-2016-lecture-7/
|
| There is a very strong flavour in the computational GA
| literature of directly implementing the GA operations--not
| translating to vector algebra. jessermyers is expressing a
| minority opinion when he says "don't do that".
|
| Finally (and this might be a bit rude) I'm dubious about
| your assessment of your coworkers fluency in GA. Maybe they
| seemed totally facile with the bits they knew. In general,
| though, non-speakers can't assess fluency (in any language,
| right?). Similarly, what are you actually saying when you
| say that you "believe" GA has insights to offer? What is
| that confidence/assessment based on?
|
| Applying GA is an active research field--it's not something
| people attain practical mastery in, just yet.
| jessermeyer wrote:
| For the record, I'm not saying everyone should not
| implement GA this way. The major proponents of GA are
| typically academic, who are interested in its abstract
| properties of computation. In this setting, it's probably
| a reasonable approach. But this is a completely different
| problem than engineering a game-engine which strongly
| encodes constraints into its solution. It's more like
| plumbing than math.
|
| In fact, the datapoint here is an argument against
| implementing GA in its general form to perform
| computations.
|
| There is a world of difference between understanding the
| properties of a computation and wanting to turn a wrench.
| bordercases wrote:
| It sounds like one shouldn't have expected much of these
| coworkers, or rather should have expected their error,
| given that the academics studying GA haven't found
| efficient compile-time translations for GA into a
| performant solution with matricies?
| sdenton4 wrote:
| Yeah, beating implementations of matrices is a tall order
| at this point in history...
|
| Similar to how electric engines are superior to ICE in
| many ways, but ICE has a century of optimization work
| already done to catch up to.
|
| From my gloriously irrelevant position in this arm chair,
| I wonder whether there's enough extra oomph that comes
| from GA to justify the conversion cost and catch up to
| the existing optimized linear algebra environment.
| dahart wrote:
| > I'm dubious about your assessment of your coworkers
| fluency in GA.
|
| I wouldn't call that rude so much as you making incorrect
| assumptions and jumping to conclusions on top of my story
| that is incomplete on details. It certainly would be
| better left out of your comment, as there's nothing to be
| gained from cross-examining my ability to assess whether
| they knew more GA than I did. They did, in fact, know
| more GA than I did. And while I'm framing myself as a GA
| noob, I've dabbled enough to know a bit about what I
| don't know.
|
| > it's possible to use GA as a language for describing
| geometric operatiins while continuing to implement them
| under the hood using old-fashioned vector algebra.
|
| You're missing my point. They did implement GA using
| linear algebra. They built classes for bivectors and
| rotors that use dot products and cross products under the
| hood.
|
| Once there are classes for GA objects, and they get used
| in the code, then everyone else has to use them and know
| how to use them. You can't use GA without knowing the
| algebra of GA types.
|
| The implementation of the GA objects is not the question
| here at all.
| oppositelock wrote:
| 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 wrote:
| 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.
| jacobolus wrote:
| Quaternions (a.k.a. rotors) are by far the easiest
| representation to interpolate and compose.
|
| Matrices and Euler angles are both horrific to interpolate.
| oppositelock wrote:
| You can't solve the problem of avoiding gimbal lock in
| arbitrary rotations, so you rig the game not to ever have
| arbitrary rotations. You cheat, basically. Gimbal lock is a
| huge problem in software where free form rotations are
| allowed, such as 3D modeling packages, CAD, but in games,
| since we control the world, we can set it up not to have
| these problems.
|
| Quaternion interpolation works well, but it introduces
| twist, which is sometimes not what you expect. When you
| start composing many quaterions, you get some wild
| rotations, going the long way, or doing an additional 360
| twist, and whatnot. Mind you, I'm digging 20 years back in
| my brain here, I don't remember many specifics anymore.
| banachtarski wrote:
| There are a number of problems with your assessment in
| modern day engine design I think.
|
| First, because of IK, we _cannot_ control orientations
| exactly. Newer techniques like motion matching /IK can
| generate new orientations on-the-fly, depending on what a
| character is doing and the character's environment.
| Gimbal lock matters for camera movement as well. Looking
| up with Euler angles is a great way to induce a seizure
| if not done correctly.
|
| Second, the "wild rotations" you mention has a very
| simple solution employed by every engine I've worked
| with. Basically, you just constrain the real part to be
| positive which fixes your interpolation on one half of
| the Lie-manifold which ensures the arc taken is as short
| as possible.
| ErotemeObelus wrote:
| > instead of rotating from orientation A to B, they'll go
| through a C in a very different place.
|
| This is because H double-covers SO_3, not just simply covers.
| salty_biscuits wrote:
| 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).
| notduncansmith wrote:
| Thank you for the anecdote! Sorry to leave such a small comment
| but I believe you meant "foisted it on everyone", not
| "hoisted".
| dahart wrote:
| 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 wrote:
| 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 wrote:
| > 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.
| moralestapia wrote:
| 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 wrote:
| You mean there's a way to establish yourself as an arcane
| code wizard _and_ make yourself irreplaceable? Sign me up!
| moralestapia wrote:
| It's more like:
|
| "Why would we use a framework that nobody understands but
| you, to accomplish the same stuff a few SQL statements can
| do?"
|
| The former option never wins.
| crimsonalucard wrote:
| 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.
| agumonkey wrote:
| 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 wrote:
| 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.
| BlueTemplar wrote:
| 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...
| SkyBelow wrote:
| 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.
| madrox wrote:
| 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.
| [deleted]
| Chirono wrote:
| 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 wrote:
| > 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.
| pfranz wrote:
| I haven't worked on large gaming projects, but I've worked
| on large CG feature films. I was a little horrified about
| how aggressively they adopted some technologies. The theory
| was that Pixar proved a lot of their technologies in their
| short films (Disney a bit of that, too, in previous years).
| The studio I was at didn't do that and both Disney and
| Pixar seem to have moved away from that at least a decade
| ago. Very very rarely, they'd repurpose old projects or
| chose a sequence or specific department for releasing
| something--that often was a mess (resources spent at the
| "boundaries" instead of shoring up the new tech).
|
| I imagine AAA games are similar. You have 4 years and ship
| one monolithic game. Maybe you could prove out this idea in
| a small area like artist's tools or an engine fork (similar
| to the approaches above). Trying to incrementally adopt
| something would take 2 or 4 games (10+ years) and I just
| don't see technology roadmaps like that. Especially if it's
| not a huge win.
| BlueTemplar wrote:
| IIRC matrices are still required in Geometric Algebra ? Maybe
| by "matrix math" you mean "Vector Algebra? "
| gugagore wrote:
| 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".
| BlueTemplar wrote:
| OK, after refreshing my memory on GA, it seems that one of
| the nice things about it is that you don't have to think
| about the "row" vs "column" vectors present in "Vector
| Algebra". Which are usually represented in the form of
| matrices. But you don't "need" them in GA, since you can
| directly work with vectors (= arrays) ! (Which makes it
| even more curious as to why matrices would be more
| efficient for 3D work ??)
| gugagore wrote:
| I think I understand roughly what you mean when you
| distinguish row and column vectors. For example, let [x,
| y] represent a row vector and [x, y]^T represent a column
| vector. If you have a function f that maps [x, y]^T -> z,
| (you might write it z=f(x,y)), then the gradient(f) is a
| function [x, y]^T -> [x, y]. That is to say, the gradient
| is a _row_ vector. It 's a different kind of vector than
| the input to f. And it transforms different (c.f.
| https://math.stackexchange.com/a/3200912/)
|
| So using that kind of thinking, the cross product of two
| column vectors is a row vector:
| https://www.youtube.com/watch?v=BaM7OCEm3G0
|
| As you say, Geometric Algebra doesn't talk about row
| vectors and column vectors. For example, in 3D GA,you can
| choose a representation in R^8. That's 1 scalar, 1
| pseudo-scalar, 3 column-y components, and 3 row-y
| components.
| aspaceman wrote:
| From reading a lot of comments it seems like
| misinformation. One thing worth keeping in mind is that
| game developers are highly risk averse. For good reason.
| nbulka wrote:
| "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 wrote:
| So what else does this operator do? It's not commutative. Does
| (x * y) * z still equal x * (y * z)?
| yiyus wrote:
| The geometric and exterior products used in GA are non-
| commutative too.
| timerol wrote:
| 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.
| alpineidyll3 wrote:
| 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.
| eigenspace wrote:
| 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'.
| nmca wrote:
| Did you read the article? Irrc you can lerp rotors and the
| results are smooth and interpretable...
| moultano wrote:
| You should read the article before commenting.
| moron4hire wrote:
| 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.
| nine_k wrote:
| Not knowing what's behind the hood in floating point numbers
| and their limited precision leads to all kinds of surprises.
| moron4hire wrote:
| Rarely
| gvjddbnvdrbv wrote:
| Often. Often missed though.
| moron4hire wrote:
| Almost every component of a game is built on the premise
| of leaky abstractions that go unnoticed. Texture
| compression, lighting techniques, physics, audio filters,
| network latency filters.
| Fredej wrote:
| There are other uses of 3D than games. 3D Engine != Game
| engine.
| friendlybus wrote:
| Try building a space game at real-world scale on 32 bit
| floats. Star Citizen tore apart Cryengine to make it have
| 64 bit positioning for this very reason.
| kaibee wrote:
| Kerbal Space Program manages it.
|
| (by moving the world around the origin)
| friendlybus wrote:
| Yeah in singleplayer with lower fidelity that is
| passable.
| rikroots wrote:
| 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.
| Koshkin wrote:
| 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.)
| xg15 wrote:
| 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.
| moron4hire wrote:
| 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.
| baq wrote:
| the math is the same so technically nvidia gpus already
| implement rotors.
| krajzeg wrote:
| Shader languages do not include quaternions as primitives,
| so if you do have quaternions in your GPU shaders, it's
| your own code (or a library), not NVidia's. What is usually
| done in shaders is encoding all transformations (rotation,
| scale, translation) in 4x4 matrices, which are a primitive
| in all shader languages I know of.
|
| From my personal experience, it can pay off immensely to
| understand the internal structure of the representation
| you're working with. I can look at a 4x4 matrix and
| immediately identify some stuff (does it include a
| translation component? does it scale and is this scale
| uniform? is it rotated and around which axis?).
|
| Meanwhile, I can't do this with a quaternion. I know what
| they do and can understand how, but I have no intuition for
| what the four numbers mean.
| gugagore wrote:
| They didn't say "understand everything", but that's the
| interpretation you're replying to, and it gives me some
| thoughts.
|
| Different people will be okay with different levels of
| understanding. You put "deep" in quotes. You should also
| put "heuristic" and "easier", and many more, because all of
| these terms are up for analysis now.
|
| A very heuristic operation of technology is "power
| cycling"--"have you tried turning it off and on". Another
| version is "factory resetting". This is very useful, but
| without a slightly deeper understanding of what is _does_ ,
| or how computers _work_ , it's easy to waste time doing it.
| Like if I get a cloudflare message saying some website is
| unavailable, I'm not going to log out and back in. I'm not
| going to turn my computer off. I'm not going to do
| anything. But that requires a slightly deep understanding.
|
| I wouldn't try to argue what's easier and what's harder for
| people so generally. There are lots of different kinds of
| people. That's why both the quote and the grandparent are
| explaining their personal motivations and describing
| themselves. I mean, you kind of acknowledge this after the
| fact when you talk about "arbitrary lines", but it still
| sounds like you're dissing someone's "arbitrary"
| personality. I mean, yeah, even if it were arbitrary drawn
| at GPU---that's the sense in which we are individual
| people...
| bvda wrote:
| > 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.
| moron4hire wrote:
| 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.
| gugagore wrote:
| > Needing to "understand" root construction before "being
| able to use" is just procrastination.
|
| since I feel personally attacked by that statement (I say
| in jest), it's more charitably viewed as a risky
| investment. Understanding now might help you use it more
| efficiently later.
|
| 3D Rotations are traditionally very tricky to represent in
| a computer! It's only in "modern" times that basically
| everyone has settled down on quaternions as the best
| parameterization. I think the comparison to anything about
| real numbers misses the point. It's not about rotors
| themselves, or real numbers themselves, it's about how they
| model something we care about. Money isn't a real number,
| but we model a balance in an account using one. And an
| accountant definitely needs to understand operations such
| as "debiting", "crediting", "accruing interest". So they
| need to understand addition, negative numbers _, and
| multiplication. But that 's just because of the choice of
| the model.
|
| _funny enough, it seems like lots of accounting existed a
| while before negative numbers were obvious, so there are
| all sorts of to-me funny ways of representing negative
| numbers, or subtraction (I don't have any clear evidence to
| back this up). Everyone was doing accounting just fine
| before, but really it just feels more obviously elegant to
| use a negative number to represent a deficit.
| BlueTemplar wrote:
| Why then engineers _do_ get taught that?
|
| Also :
|
| You "can" be a "good" engineer while thinking the Earth is
| flat. (Unless you're working on space-related projects of
| course.)
|
| You "can" be a "good" scientist without knowing anything
| about epistemology, Popper, Kuhn...
|
| (But can you, really, be a good one ?)
|
| (Also IMHO most of today's economists are just charlatans
| akin to the astrologers of old, misusing math because math
| gets your more respect, and it's probably related...)
| friendlybus wrote:
| That's fine if you're building something that's already
| been made. Games don't do that, when you push the
| boundaries you need to know how your math works. You won't
| get more performance than the competition without
| understanding the tools you are using.
| moron4hire wrote:
| And here we see the other common conceit in the game
| industry, that graphics make a game.
| friendlybus wrote:
| I didn't say that? There's more ways to push a game than
| with graphics? That's not the words in my mouth.
|
| Star citizen needed 64 bit positioning for real world
| planet scales in game, that had to be hacked into
| cryengine...
| moron4hire wrote:
| It didn't _need_ it. That was just the solution the devs
| created. They could have done it with 32-bit floats, or
| hell, 32-bit integers, if they wanted, given the right
| representation. But _they_ chose 64-bit floats to solve
| the problem, the problem did not choose 64-bit floats.
| ArtWomb wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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/
| adam-a wrote:
| 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.
| GlenTheMachine wrote:
| 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.
| rationalfaith wrote:
| Dude Le-performance is always important. Helper functions can
| be added for readability but perf perf perf perf.
| cygx wrote:
| _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.
| overgard wrote:
| 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.
| DonHopkins wrote:
| 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!
| danmg wrote:
| 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 wrote:
| What research field is this? Why hasn't one of the researchers
| made the conceptual leap you propose?
| danmg wrote:
| It has to do with metallurgy. Some people do use them, but
| most work is done with euler angles.
| 6gvONxR4sf7o wrote:
| 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.
| thrower123 wrote:
| 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 wrote:
| He's not actually proposing any change. He just gives a
| different construction of the usual quaternions.
| playing_colours wrote:
| What is a good book you can advise for someone with maths
| background to dig into GA?
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