A Swift Introduction to Projective Geometric Algebra
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- čas přidán 31. 05. 2024
- This video is an introduction to Projective Geometric Algebra, which is a flavor of geometric algebra that allows for manipulating objects like points, lines, and planes, including operations like meets, joins, projections, and rigid transformations. This is done by treating lines/planes as their own linear space and doing geometric algebra on it. PGA provides a simple and powerful framework to do dimension-independent geometry.
In case you were curious, this is finally the big project I've been teasing several times on my channel. It's also my #SoME3 submission.
Here's my video on the linear space of lines: • Two-Dimensional Lines ...
In case you don't really know anything about geometric algebra, here are three videos of mine that I would suggest:
• A Swift Introduction t... (My original introduction)
• Addendum to A Swift In... (The addendum to the introduction)
• An Overview of the Ope... (The general definition of all of the operations used in this video)
Here are some links to other introductions of PGA that might be useful/interesting:
• GAME2020 0. Steven De ... (The first resource that I saw that takes the "Rigid Transformations" approach to PGA)
• PGA Tutorial SIBGRAPI2021 (A playlist that covers PGA in a similar manner to the previous video but goes in more detail, and it culminates in describing rigid body dynamics)
arxiv.org/abs/1101.4542 (One of the first sources to describe PGA, and it uses projective geometry)
I would link to sources for the applications shown at the end, but you can find sources for most of them at bivector.net, so you can just go there instead.
The notes got too big to fit in the description, so I moved them to a pinned comment.
Discord: / discord
Patreon: / sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
LoganMP
p11
Richard Penner
Rosario
trb
Sections:
00:00 Introduction
02:15 The Linear Space of Lines
07:04 Basic Definition of 2D PGA
09:43 2D PGA Bivectors/2D Meets
13:47 2D PGA Points
14:57 More 2D Meets
16:15 2D Joins
17:50 2D Inner Product
19:35 2D Projections
21:51 2D Reflections
24:16 2D Rigid Transformations
26:27 2D Rigid Transformations on Points
28:03 2D Bivector Exponentials
31:16 2D Rigid Transformations Without PGA
32:54 2D Summary
33:27 3D Introduction
36:13 The Linear Space of Planes
37:32 Basic Definition of 3D PGA
38:07 3D PGA Bivectors and Trivectors
41:08 3D Meets
42:30 3D Joins
43:59 3D Inner Product
47:03 3D Projections
49:00 3D Rigid Transformations
49:46 3D Summary
50:25 nD PGA
50:54 Demonstration
52:10 Applications
Here are the notes that got too long to fit in the description:
Note 1 (2:04): Credit for this "note" idea goes to the CZcamsr Bismuth.
Note 2 (4:59): You might notice that there could actually be two different angles that two lines can make. So which angle is the one we use? It actually depends! You see, I have neglected to mention that each line actually has a hidden orientation that can have an impact on calculations. I don't mention it here because it doesn't impact most of the calculations in this video, but this is one of the few exceptions. The dependence of the inner product on the orientation of lines won't matter much to us because the main thing I want the inner product for is the definition of parallel and perpendicular (which I'll get to in a moment), which doesn't depend on the orientation of the lines.
Note 3 (12:58): If you think that this looks like homogeneous coordinates, that's right, because this is precisely homogeneous coordinates! In fact, this is the main point of connection between PGA and projective geometry. But in PGA we prefer to make lines/planes our vectors rather than points because the geometric and inner products don't work as we want them to if we make points our vectors.
Note 4 (15:12): You might be wondering about why I picked (b, -a) rather than (-b, a), and the reason is that this is another case where the orientation of a line comes in. However, given that the two points at infinity are the negative of each other, they are scalar multiples of each other, so in some sense they represent the same point. Thus, it actually doesn't matter which of (b, -a) and (-b, a) we choose, so I just picked one.
Note 5 (20:22): You might notice that I swapped an inner product while fusing the two projection equations. Your first instinct might be that this is fine because the inner product is symmetric, but then you might realize that this is not fine because the inner product in geometric algebra is not symmetric. So what gives? Well, between homogeneous multivectors, the inner product is always either symmetric or antisymmetric, and because scalar multiples don't affect what object our multivectors represent, the minus sign won't really matter to us. Thus, for our purposes, we can swap the inner product without really worrying about whether or not we should be introducing a minus sign.
Note 6 (24:54): You might be wondering why I write the rotor transformation law as R†aR rather than RaR†, which is what many others use. In the end it's just a convention, and I prefer R†aR because many things feel more straightforward to me when using this convention. I might make a video or a short in the future expounding on this a bit.
Note 7 (28:30): Not only is the magnitude of the lines important, but their orientation is as well! In fact, the relative orientation of the lines affects whether you rotate in a clockwise or counterclockwise direction. This is probably the only time in the video that the orientation of lines really matters. However, in my own work, I'll admit that I still don't really think of the orientation of lines/points when I'm exponentiating. I'll just look at the result, and if I see that something's off, I just throw in a minus sign then.
Note 8 (51:21): You might be wondering how the symbols map to GA operations. * is the geometric product, ^ is the outer product, and & is the regressive product (while it wasn't used here, | is the inner product). The GA in this demonstration was done using the Python library kingdon: github.com/tBuLi/kingdon
your 8th note answered my question. And Note 7 made me really chuckle hahaha. Thank you for all this.
I am unable though to find the library in a quick search, can you link to it?
Is there an intuitive reason why making hyperplanes our vectors works better than just making (homogeneous) points vectors?
@@HEHEHEIAMASUPAHSTARSAGA Because reflecting across 3 orthogonal planes is the same as reflecting across the point where they intersect. Thus it makes sense that composing three planes can result in a point. The other way around however does not apply. You can generate translations by reflecting across two points, but there's no way to generate rotations, and rotations are nice.
@@housamkak8005 Sudgy's notation given here seems to be standard, though I personally use a different convention.
Outside of geometric algebra, ∧ and ∨ are the standard symbols for AND and OR respectively. There are also contexts outside of GA that use meets and joins and respectively use the symbols... ∧ and ∨. It might also be worth mentioning the intersection and union, which are both closely related to the AND and meet, and OR and join respectively, and use the symbols ∩ and ∪, which look a heck of a lot like the prior symbols.
With this in mind, most programming languages represent AND as & and OR as |. Those are the symbols that I've come to prefer for the outer product and regressive product. The main thing it lacks is the visual similarity between ∧ and ^, though since ^ is usually used for XOR, it seemed as good a choice as any for the inner product. Oh, and also that it conflicts with the established approach, but I don't really care about that as much.
I can't believe this is all free!
It is not your maths or his maths, her maths or theirs!
It is *our maths*
Union indestructible free republics
Союз нерушимый республик свободных
United forever Great Rus'
Сплотила навеки Великая Русь
Long live the one created by the will of the peoples
Да здравствует созданный волей народов
United, mighty Soviet Union!
Единый, могучий Советский Союз!
Hail, our free Fatherland
Славься, Отечество наше свободное
Friendship of peoples is a reliable stronghold!
Дружбы народов надёжный оплот!
Party of Lenin - the power of the people
Партия Ленина - сила народная
It leads us to the triumph of communism!
Нас к торжеству коммунизма ведёт!
Through the storms the sun of freedom shone for us
Сквозь грозы сияло нам солнце свободы
And the great Lenin lit the way for us
И Ленин великий нам путь озарил
He raised the nations to a just cause
На правое дело он поднял народы
Inspired us to work and deeds!
На труд и на подвиги нас вдохновил!
Hail, our free Fatherland
Славься, Отечество наше свободное
Friendship of peoples is a reliable stronghold!
Дружбы народов надёжный оплот!
Party of Lenin - the power of the people
Партия Ленина - сила народная
It leads us to the triumph of communism!
Нас к торжеству коммунизма ведёт!
In the victory of the immortal ideas of communism
В победе бессмертных идей коммунизма
We see the future of our country
Мы видим грядущее нашей страны
And the red banner of the glorious Fatherland
И красному знамени славной Отчизны
We will always be selflessly faithful!
Мы будем всегда беззаветно верны!
Hail, our free Fatherland
Славься, Отечество наше свободное
Friendship of peoples is a reliable stronghold!
Дружбы народов надёжный оплот!
Party of Lenin - the power of the people
Партия Ленина - сила народная
It leads us to the triumph of communism!
Нас к торжеству коммунизма ведёт!
math is free, getting hints/learning sometimes isnt.
Yeah, now that I realize how expensive MATH-books are, this feels more and more like watching Robin Hood.😮😮
"Everything in this video can be derived from what is currently on screen."
I have never seen this used before, but this is definitely the best way to establish the simplicity of the concept!
Like the ancient Greeks
Not specific to this video, but I'd like to give a mention to the surprisingly consistent color scheme between your videos for the basis blades. They even have a near perfect match with the color scheme for the basis blades in a Swift Introduction to Spacetime Algebra, though with the e₀₁, e₀₂, and e₀₃ terms using the colors γ₁, γ₂, and γ₃ respectively, and the reverse. e₀ and γ₀, along with e₀₁₂₃ and γ₀₁₂₃, have completely different colors, but that's understandable since while they have similar relationships with the rest of the algebra, they also behave somewhat differently.
For those not yet convinced of PGA, I recommend doing some reflection, and eventually you will see. ;)
Is reflection a pun? Nice! I like thinking of reflections as (180 degree=pi radian) rotations. Thinking of translations as rotation around a point of infinity is new to me, as is the point of infinity itself. I find PGA fascinating too, I’m not a mathematician, just like learning things every now and then
@@daniellewilson8527 To be specific, a 180° rotation is a reflection across a "hyper-line" you could say. (Whatever a N-2 D subspace is when working in ND.)
The further away said "hyper-line" is, scaling the angle down proportionally to the distance makes a rotation around said "hyper-line" become more and more straight and more closely resemble a translation more so than a rotation. A "point at infinity" (point being a "hyper-line" in 2D) is basically just taking the limit as the distance from this line approaches infinity. An odd quirk that makes more sense in elliptic geometry or when visualizing the N+1 dimensional embedding is that when the point in question actually "reaches" infinity, it becomes indistinguishable from a point at infinity in the exact opposite direction.
the part where you showed the code totally blew me away. i'm just speechless
Very excited to see this! Your introduction to VGA and STA were awesome, and I have no doubt this will be similar!
Edit: and I should mention Zero to Geo is amazing too! Building up an entire field of mathematics from the ground up is no small feat!
I took a graphics class before and discussed briefly about projective geometry. The whole things was kinda magical with the introduction of homogeneous coordinate that just kinda solve everything. Never thought how extended algebraic system could encapsulate all the geometric operations so elegantly... Thank you so much for opening my plebeian's mind
I was pretty familiar with VGA and Projective Geometry, but this video (especially the shot of all the different traditional formulas) is probably the best non-calculus argument for why GA is really useful, rather than merely fun/interesting. Excellent work!
I keep coming back to this channel for more geometric algebra all the time. It's so clear.
51:38 and 51:53 are the absolute killer app for PGA. So dang excited
I love it!!! I think I will spend some hours with my analytic geometry textbook redoing all the excercices with these new tools!!
Thank you very much. This is awesome
You're the Geometric Algebra channel! Watching that video made me the most excited I've ever been about math!
@53:45 "it allows for a simple way to do n-dimensional rigid body dynamics"
Now you're just flexing! I'm really looking forward to 7D video games with a realistic physics engine (that would of course be programmed for 2D then changing a single character to 7 at the beginning of the file). Geometric Algebra truly is powerful
Thanks for this awesome video
This is your best video yet, fantastic work!
Dude, thank you so much for all your hard work on this! Your OG GA video is one of my favorite math explainers of all time, and STA was up there too. I’m wondering what’s next, conformal GA?
So amazing! Thank you for all your work. In by opinion, your videos are the best source for approaching GA. This video makes me so excited because there is almost no approachable information on the different flavors of GA on CZcams. I think the least amount of basic info available for any flavor is conformal geometric algebra. So I’m so excited to see if you will make a similar video for CGA. That being said, I get so pumped every time you upload a video and love what you are doing. Thank you so much!
This is incredible, thank you so much for doing these videos!
Just came across this series and it has been a wonderful introduction to PGA. It’s so clear and I’m amazed how beautiful and elegant PGA is.
Your videos are so great, thank you!
this is fire and lit my soul to learn as much as possible, thank you!
Man you are the father of this topic on youtube by now. I really would like to see rigid body dynamics videos using pga. Amazing work I swear. How can I code that as well? U used an Algebra object, where was that defined?
Look into the bivector CZcams channel, they have stuff on rigid body dynamics using PGA. The Algebra object was defined in the python package kingdon: github.com/tBuLi/kingdon
I love PGA, it's a standard method in Russian School program, that is allowed to be used on Government Attestation Test to complete your school diploma. I love the simplistic approach to it in this video. We use a different notation, and our most used thing is formula for plane, normal and vector sum, and projection to 2D plane is a backbone to Vector Stereometrics. Things like perpendicularity and parallelism was always considered a time consuming process to formally notate at exams by PGA formulas and we was strongly advised to use traditional methods at that. I'm supper happy to watch this different view at that topic.
Which schools teach PGA? It is definitely not a part of a standard high school curriculum, and even not a part of most university curriculums (unless, may be, you major in mathematics).
@@onebronx in Russia it's a part of high school program. There is a separation: language education schools - "gymnasium", history and literature - "licey", and physics and mathematics - "physmat", there are also religion and sport focused schools as well, to specifically learn PGA in Russia you need to go do "physmat" schools or in "competetive math" classes in "gymnasium" or "licey" schools. We have right now 3 Government standard and licensed books for high school students about PGA and basic Vector Geometry. The one that i read was: Alternative ways of solving problems by Isaak Kushmir, Solving geometric problems with vector method - G.A. Klekovkin, and the most complete: Vectors at Exams - S.A. Shestakov. But all High schoolers is allowed to use PGA and Vector Geometry methods on Government Attestation Test. And small nuance about GAT - you need notate your solving process even if it's a test, because test question is usually "yes" or "no", but you can have a high variation of answers depend on method of solving. Enumeration Method is also valid, and you also allowed to use degrees and radian in answers. You can google: "примеры решения профильной математики с ЕГЭ" to look for some answers sheets.
@@d1namisI looked into the last one (С.А Шестаков, Векторы на экзаменах. Векторный метод в стереометрии, 2005, available online), and I do not see any PGA there, just a standard vector algebra in R^3. The book tells nothing about multivectors, their different products (outer, geometric, "sandwiching"), projective space, linear space of lines, points/lines at infinity etc. Just a regular vector algebra (vector basis/coordinates, scalar product, dot product -- that's it), i.e. exactly what PGA tries to escape from.
@@onebronx There is projections and infinite lines in that book, as far as i remember, but yes, most of that book is made for idiomatic solving thing approach.
@@d1namis yes, there are projections in the book -- but those are just regular dot products, not projective spaces, big difference. There is no any Geometric Algebra in the book at all. I believe you just confused Geometric Algebra with Analytic Geometry (cartesian coordinates, lines, planes, conics etc).
BTW, I also found a PDF of the Kushnir's book (Исаак Кушнир, Альтернативные способы решения задач (Геометрия), 2006) -- it is all regular planimetry and stereometry, no traces of PGA at all, too. And, as a participant of high school math olympiads and later a graduate of a polytechnical institute (not math major, but pretty math-heavy engineering, including computer graphics, where I'd certanly got benefits from GA), I never heard about [P]GA, Clifford algebras etc. Most probably only math majors and, may be, theroretical physicists studied them, but not high school/gymnasium/lyceum students.
Another great video! Looking forward to a future one on Conformal GA!
1:45 I've also seen "Euclidean PGA" referred to as "Parabolic PGA" when contrasting "Elliptical PGA" and "Hyperbolic PGA". So in this way these are sometimes abbreviated PPGA, EPGA, and HPGA without "Euclidean" and "Elliptical" being ambiguous
Thanks for the clarification 👌
What a wonderful effort to bring down beautiful math to us...visuals..
You are a math angel man!!
figuring out quaternions almost feels like unlocking the cordless drill of the physical sciences
but this shit is literally magic
It's really interesting to watch math videos as I progress through my studies. I can actually understand something now.
Amazing video, thanks!!
Your videos are gold 🤩
This is really remarkable stuff. I am working on a computing environment, based on a "souped up Forth" system, and I want to incorporate this stuff into it, along with several other things I've stumbled across. It's just a dream right now, but... we'll see.
I was waiting for this one!
you are TOO good at this oh my god
Thank you from deep heart❤️
Nice video. I hope you will eventually cover ideals of Clifford algebras and spinors! They might be hard to visualize though.
woah, conformal geometric algebra might be just what I needed to become better at geometry
finally I will be able transport my algebra skills to geometry in math competitions
This is awesome.
Always looking forward to more geometric algebra lectures. Speaking of vectors, can we apply this topic to statistics?
What a treat!
I'm excited!
Your intro sounds like a too-good-to-be-true ad 😀A thousand times yes. Better not turn out to be the Springfield mono-rail.
YES!
THANK YOU!
this is the big one!
Let's gooooooooooooooooo!
Learning about projective algebraic geometry in class... come home and see this in my feed... maybe tomorrow :')
Very nice! I grasped the overview of PGA in some De Keninck's and Dorst's videos and books, but I find their style to be a bit impenetrable. Yours makes everything much clearer (eg. the profound reason why the *outer* product was actually appearing to *lower* the dimension of objects in PGA, which seemed backwards).
One question: how would you compute the regressive product in PGA though? You can't just use A \/ B = (Ai /\ Bi) i^(-1) because i is non-invertible due to the degenerate metric.
I know you mentioned a basis-dependent way of dualizing in your video on operators of GA, but is there a more principled way to do so in PGA?
Annoyingly, in degenerate metrics, the dual actually *has* to be basis-dependent. So to calculate the dual and the regressive product in PGA you have to do something similar to what I showed in that video.
Your videos are really good at exposition. To the point where one might fool oneself that one understood the content. Are there any exercises that you would recommend? Are you planning on composing a list of such exercises to verify one's knowledge?
Not yet, but I'm planning on doing this when I eventually reach this point in From Zero to Geo.
how does this all translate into embedding this into the basic geometric algebra? i'm having a vague intuition that you need to dualize everything then project onto a plane distance 1 from the origin, but i'm not sure how that interacts with the different products in the algebra
That's actually pretty much exactly one way to do it, although there are other ways.
Finally started to understand GA thank you so much. Please enable thanks button
You can find that button in the description: www.patreon.com/sudgylacmoe
Your content is amazing!! Eye opening to me and presumably to many!! Can you please do a video on where are we on explaining general relativity and quantum field theory using a flavour of geometric algebra? Maybe possible unification hope there? Would love to watch that from you. Thanks!!
Question, since a vector in PGA is equivalent to a scalar multiple of itself, doesn't that mean that the magnitude of a vector would then be non-unique? Or are you saying that they geometrically represent the same line, but not algebraically?
I'm saying the latter.
36:13
What can you expect, these are different kinds of objects!
**Anyway**
48:45
Get this man an ambulance that was brutal
I really need to recreate the PGA cube rotation code.
This is absolutely amazing
Hey, I knew I had seen that username somewhere! I remember hoping for a more elegant and general framework some years ago in vector calculus class, and this seems really promising. Just having one set of general operations that will work in any dimension, no weird specifics like the cross product, it's definitely a much more intuitive and satisfactory approach. Would be interesting to see the different dimensional rotations you would get in the conformal extension.
@@Saturos02 What I nice suprise to see you here as well! Yeah, GA is something that has intrigued me for years, ever since I had seen the video on rotors and the paper on n-dimendional rigid body dynamics by Marc ten Bosch.
I need to read into and experiment with it more, try and see if I can make some demoscene effects more cleanly with the PGA stuff.
@@TheFerdi265 Sounds like fun! I've only experimented a bit with fragment shaders, implemented a Möbius transform and some stuff. Didn't know demoscene was still actually a thing. I watch crystal dream 2 from time to time though, amazing time capsule :)
Great video. The penny hadn’t dropped until today that, because scaling doesn’t change anything in PGA, (A.B)B^-1 is essentially the same as (A.B)B. Neat! I’m not an experienced Discord user, I joined, but seem to be struggling to find the notes! Thanks.
Ah, found them in the comments.
One really cool observation is that the coefficient of e_0 is the (perpendicular) distance of the line/plane from the origin, multiplied by the magnitude of that line/plane;
or conversely the perpendicular distance of a line from the origin is the coefficient of e_0 divided by the magnitude of the line, and similarly for a plane.
In general, the magnitude of the join of two normalized objects is the perpendicular distance between them. Taking the join of a point (like, say the origin) with a mirror (line in 2D, plane in 3D) gives a scalar, and the magnitude of a scalar is just its absolute value. If the point is the origin, all of the non-e0 components of the mirror get multiplied by 0, and the e0 component gets multiplied by 1. So the join of a mirror with the origin is 1 times the e0 component, plus a bunch of 0s, resulting in a scalar whose absolute value is there distance from that line to the origin.
There are no coincidences in math; only special cases.
Oh snap, I just realized that the entire video went by without explaining how to compute a distance.
Bro I truly live your videos.❤
Can you tell us the name of application that you make the animation of your videos and the visual parts please?
21:26 There seems to have been a mistake where x and y got swapped. I plotted the point it in GeoGebra and it gave the wrong point, but on like a different line. I then tried switching the formula for x and y, and it then correctly projected the point onto the line.
So seemingly the left expression is the y-value and right is the x-value.
I assume this is the big video you've been teasing forever?
Yep!
Awesome video! Is it true then that in (3,1,1) GA (PGA + time vector) one can also do the whole Poincare group with rotors by adding Lorentz boosts? Is it the same as CGA?
This is an active area of research, and I'm looking forward to seeing what is found out. I at least know it's not the same as CGA because CGA doesn't have a basis vector that squares to zero.
@@sudgylacmoe But you can combine positive and negative vectors to make two independent null vectors (+)+(-) and (+)-(-), so choosing a single null turns {4,1,0} into {3,1,1} or {4,0,1}, and choosing both turns it into {3,0,2}, right?
@@NikolayMurzinWhile you can generate two null vectors from a positive and negative pair, those two null vectors aren't actually orthogonal to each other like they would be in Cl(3,0,2), nor would they be orthogonal to either of their terms like they would be in Cl(3,1,1) or Cl(4,0,1). They can still be interesting to examine on their own though and likely have some similarities.
13:35 Git yer MEAT PRODUCTS fresh OUTSIDE!! lmao im never gonna forget it now.
(legitimately a fantastic and super informative video, i actually learned a lot)
one concern i have with this inner product is that the same two lines yield a different inner product, depending on which coordinates they are represented by. In fact, you can always make the inner product equal to either 0 or 1 (however this would also change their magnitudes, so the definition of angles is unaffected)
Thanks for putting time into these videos. I have two questions: 1. You showed how to rotate geometry. Can these techniques implement shears? 2. Is conversion between PGA representation and 4x4 transformation matrices straight forward?
1. In this case, no. PGA only does rigid transformations by design.
2. Converting from a PGA rotor to a matrix is easy: See how the rotor affects the basis vectors, and that's the columns of your matrix. The other way is impossible in general because not all matrices are orthogonal.
A course on conformal GA would be awesome.
Are you going to create a video for CGA as well? Most of the concepts are similar so I presume it would be shorter than this video.
While I think this is a great idea, I currently don't know much CGA.
*Is there an operator symbol for the geometric product?* It’s probably a stupid question. (I mean like + is the operator symbol for addition.) Because × is taken, ⋅ is taken, * or ∗ are essentially taken as well, so is there really no operator symbol for the geometric product?
In handwritten math, I never use a symbol for it. When programming I use *.
@@sudgylacmoe Programming is probably the only context where you absolutely must have one.
But good to know that I'm not crazy/stupid for not finding one somewhere online.
Thanks for your response.
53:04 :0 that thing you told me when I asked about geometry problems using GA
This is so great, GA seems a bit clunky to me, but PGA is awesome
Literally the vector representing the Kernel of a linear form, giving the hyperplan, which in itself gives the "projective" property (invariance by scalar multiplication)
Also mid video I wondered if you could do circle and inversion, well now I'm happy 😊
Actually affine form
And for the circles, i'd say quadratic forms ?
A good thing to keep in mind is that the PGA mindset also carries over to all other GA’s. For example, also in ordinary 3D GA, you can think of vectors as planes *through the origin*, bivectors as lines *through the origin*, and the point as *the origin*. This plane based view makes it much easier IMHO to reason geometrically in any GA.
Circle inversion is just a special type of reflection, or perhaps reflections are just the special case of circle inversion when the circle in question has an infinite radius. In Conformal Geometric Algebra with the same interpretation as PGA (Most CGA sources treat grade-1 blades as points, though they're really 0-radius circles/spheres), grade-1 blades are circles/spheres, and can be reflected across just like a line/plane in PGA to get circle/sphere inversion.
What most graphics programmers really care about though is scaling, which, at least for uniform scaling, can be achieved with two concentric circle/sphere inversions. In some cases, it may be useful to ignore most of CGA and instead just borrow its uniform scaling into PGA, which works since planes are just special cases of spheres, and so PGA is fully embedded withing CGA (though joins can be a tad weird if you try doing them the CGA way), and uniform scaling away from or towards a point keeps planes, lines, and points as planes, lines, and points.
There is an alternate interpretation of CGA that lacks circles/spheres and has other objects in their place. It still has uniform scaling, though instead of the hyperbolic "boosts" that would result from two non-concentric circle/sphere inversions, it instead acts like a projection matrix, which is particularly useful when trying to use GA while taking advantage of the GPU.
@sudgylacmoe Hi, do you plan to release the python implementation that you demonstrated as open source code?
The GA library used is kingdon: github.com/tBuLi/kingdon/tree/master/kingdon. Other than manim, that, and what's shown on the screen, there isn't much more, and the few things left out aren't that difficult to make.
When I hear "meet" and "join," my brain goes to lattices. Is there some kind of (pre)order in this algebra for which these are meets and joins in that sense? (That is, glb and lub operators)
GA meets and joins are related to the equivalents from order theory, but they're not quite the same. In particular, the meet is the largest subspace contained in the set of inputs, which implies that the meet of an object with itself is itself, but in GA, the meet of two n-1 dimensional subspaces (grade 1 vectors) is always an n-2 dimensional subspace, and if the intersection is still n-1 dimensional, then it instead gives 0. There are some algorithms to factor k-vectors allowing you to find the proper meet and join essentially as the least-common-multiple and greatest-common-divisor, though most libraries don't implement this since it's a lot more complex, and you generally know the grades of the objects you're working with, letting you find alternate ways of computing the desired result. The ordering in this context would probably be the dimensionality of the subspace, which is the reverse ordering of the grades.
The symbols used in PGA for the meet and join are the same as in order theory though: ∧ and ∨ respectively, which are also shared with the boolean operators AND and OR. (Pay no attention to the demo code and its usage of & to implement the join ∨.)
@@angeldude101 interesting! Thanks for the thorough response!
It seems to me like this is just for drawing fancy computer graphics. Are there any deep theorems in projective geometric algebra that say counterintuitive things, or was the whole idea behind this video to show it's all just a handful of formulae that are the same in each dimension?
PGA is useful for more than computer graphics, like robotics and physics. But in general, the reason for doing geometric algebra is not that it enables us to do anything new, but that it enables us to do things that we already could do but in a much simpler way.
Also, theoretically, PGA is pretty pesky because of that null vector. PGA is focused much more on applications than on theory. When I'm working with GA theoretically I don't focus on any one particular flavor because each individual flavor isn't that interesting by itself in theoretical terms.
Question: what is regressive product? I seem not understand how you joined two points, what is the definition for regressive product
I made a video right before this one explaining all of the operations, including the regressive product: czcams.com/video/2AKt6adG_OI/video.html
@@sudgylacmoe Thank you!
Great as always! Would you consider making a video about geometric calculus? It would be very interesting.
I've been asked this question enough now that I just added it as question 16 of my FAQ: czcams.com/users/postUgwGXciowesPuiRUlBJ4AaABCQ
@@sudgylacmoe Thanks for letting me know. Also I thought I was late and you wouldn't see my comment, it's nice that you check the comments on your videos once a while. Also I would love to see a geometric calculus series regardless of your mastery on it, I think it would be fun regardless of the number of mistakes or anything and you could just put a disclaimer at the start saying it is not a professional video but of course if you don't feel you're good enough no pressure.
Can't wait for chapter 5 now. I really like GA and I wish I was better at it.
Ooooo, loved the video ! Really nice and thorough, really helped me wrap my mind around it.
One thing stands out though. I've been reading Doran and Lasenby's Geometric Algebra for Physicists, and in there they define the extra vector we add to get PGA as having magnitude +1 rather than 0. I'm having trouble wrapping my mind around what that changes, except for the fact that they don't use any kind of magnitude in the corresponding chapter. Do you have any pointers on this ? I'm not even sure how I could make the jump between both representations, or if it's even productive to do so
If the standard euclidean bases square to +1, then adding an additional basis vector that squares to +1 results in elliptic PGA, which is geometry on the surface of a hemisphere where opposing points on the equator are considered the same point. Make the basis vector's square -1 and you get hyperbolic PGA, letting you do geometry on the surface of a hyperboloid, whose "translations" look suspiciously like Lorentz boosts. If you're not on a hyperboloid or hyper-hemisphere and would rather stick to the familiar flat euclidean space, then the extra basis vector should square to 0. (Or you could add multiple additional basis vectors which have a linear combination that squares to 0, which is what's done in CGA.)
I think the magnitude is +1, but it squares to 0. If this seems odd, consider the 2x2 matrix of form [0, a, 0, 0]. It's not just the 0 matrix, but when you square it, you do get the 0 matrix. Note that this leads to an algebra where things are generally non-invertible. I wonder if there is then a connection to the sedenions. Many view this non-invertibility as a huge negative "feature," but when you think about it, the derivative as an operation is not invertible (thus we add a +C when integrating), but clearly the derivative is an important operation! For more info, check out dual numbers online. Also, Wildberger has some interesting related videos in his "Famous Math Problems" playlist.
@@angeldude101but I thought quaternion i j and k's square to -1 and this forms a unit algebra that is the double cover of the space of rotations in 3D (SO(3)), thus not a hyperbolic space at all.
@@AkamiChannel Quaternions do square to -1. They're also _not_ the basis of their algebra. They're the even subalgeba of Cl(3), which has 3 basis vectors that square to +1. Since the basis vectors all have the same square, the products of two of them all square to -1 and thus generate rotations. If one of the basis vectors squares to +1 and another squared to -1, then their product squares to +1 resulting in hyperbolic rotations. If either of them square to 0, then you get translations as seen in this video.
The rigid transformations look like group conjugations gng^-1 found in normal subgroups. Having projection be one operation also would simplify the Gram-Schmidt process. Projective geometry also works well with exterior algebra.
Indeed, the conjugation we use in GA is the same as in group theory. For example in 3DPGA, rotors are elements of the (double cover of) the Special Euclidean group SE(3).
Gram-Schmidt in GA is simply replaced by renormalizing your rotor, so there isn’t even a need for the projections done in the GM process. This has the advantage of keeping the bias smaller compared to GM since we don’t have to pick a favorite vector as our reference.
Great video! I’ve noticed a ton of videos/lecture series on CZcams are more focused on PGA than VGA. At some point could you do a comparison of VGA and PGA and what the benefits and downsides for each one are (and why you’d might want to use one over the other)?
I'll probably get there eventually in From Zero to Geo. However, here's a simple rule for this that works most of the time for PGA vs. VGA: If you care just about orientations and rotations around a single point, use VGA. If you want arbitrary orientations, rotations, and translations, use PGA.
Also I find it funny how you say a ton of things on CZcams are more focused on PGA when my whole channel up to this point has been almost VGA-exclusive.
@@sudgylacmoe haha yes I meant to say ‘other’ videos!
The biggest downsides of PGA are that the algebra gets bigger, and the fact that it requires you to abandon the notion of directions or points being grade-1 objects. The plane based view of PGA can actually be applied to VGA, giving mirrors and axes through the origin, but attempting to use the more traditional model for PGA requires a lot of extra, arguably unnecessary, structure that just complicates things further.
Great content as always!
Will you ever talk about Conformal Geometric Algebra in a video?
I want to, but it will take a while since I currently don't know much.
@@sudgylacmoe That's fair, I was hoping you could make a video because I don't know much of that either 😅
Has anyone figured out how to get the code towards the end to work?
I can tell that this is good, but I've only watched a bit and im kinda confused, gonna just save this to a playlist and review it later
which software?.
But how do you calculate the regressive product? How does it relate to the geometric product?
I originally was going to go into all of the details of how exactly to calculate all of the products in this video but it ended up getting way too long and split into its own video: czcams.com/video/2AKt6adG_OI/video.html (this is timestamped to the part about the regressive product).
@@sudgylacmoe thanks!
Cool!❤
I think it could be interesting if you could do a video about complexification of real vector spaces and its properties with hermitian product
So geometric algebra does a lot of the same things as linear algebra does. Is there a way that geometric algebra carries over to non vector linear spaces? Like linear operators on functions. Does it work and is there something fun there?
All linear spaces contain vectors by definition, so I'm not quite sure what line you're trying to draw. In general, you can define a geometric algebra on any module over a commutative ring with a symmetric bilinear form. However, most GA research is done with finite-dimensional spaces over the reals, because they are much better-behaved and useful in GA. Really, the point of GA is not to do new things, but to do old things in a much better way.
Yeah I guess I just mean infinite dimensional vector spaces!
Functional analysis also contains a lot of projection, subspaces, least distances to planes, linear maps and so on and so on so I was mostly thinking about that
4:09 - Why does (2e2 + 2e0) represent the same line as (e2 + e0)? Doesn’t e0 shift the line? I think they should be two parallel lines, with the same vector being perpendicular to both of them.
Never mind, got it. You have to convert it to y + 1 = 0 to get it.
Any ideas about how to go about extending PGA into spacetime a la STA? It's easy enough to think of geometrically, with time-like lines forming object paths and space-like planes being full 3D volumes. However in implementation, there are a number of stumbling blocks I can anticipate, starting with how to blend the two concepts of vector. VGA/STA have vectors that indicate points/events with the coordinates equal to their components, whereas in PGA, vectors are equations that involve the coordinates, and the objects are formed from all points for which the equation equals zero. Very different concepts, and I'm not quite sure how to bridge them. Where does the metric fit, i.e. which basis vector(s) need to square negative in order to make all the transformations work?
If I could fill in some of the blanks, I think it could be really interesting. You would be able to capture an object's motion through spacetime as an exponential rotor/motor, and maybe formulate forces that way.
This is an active area of research! I don't know much about it, but I do know that spacetime events are represented using pseudovectors, similar to how points are represented using pseudovectors in normal PGA.
HYPE!!!!!!
"You're a wizard, sudgy!"
I remember the Rotor as: (RO)T(OR) what rotates T in R & O
Probably, our ancestors had all this secret knowledge. 🤔
18:15 How did you get this result?
czcams.com/video/2AKt6adG_OI/video.html
PGA is love for 3D graphics. There needs to be a tool that compiles PGA to VLSL.
this is a great video and demonstration of the basis for PGA, but i was wondering how this framework applies to real world problems, and the demo in the end of the video is a great example of that, but i would like to see more examples like maybe a textbook problems what are hard to solve using classic tools 🤔
A tutorial at SIBGRAPI covered n-dimensional rigid body kinematics and dynamics in PGA : czcams.com/play/PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc.html
what does the pseudoscalar represent?
This is most easily understood by going one dimension up. For example, the pss of 2DPGA is e₀₁₂ which in 3DPGA is the infinite point in the direction of the 'z' axis. Hence, the infinite direction orthogonal the xy plane, the space 2DPGA models. In general it always captures the direction orthogonal to the space you are modelling.
Finally the tease is over :)
How do you calculate a dual?
I made a video talking about many of the operations in geometric algebra. Here it is timestamped to the section about duals: czcams.com/video/2AKt6adG_OI/video.html
Hey do you have any pointers to videos about hyperbolic and elliptic PGA too? I’m lazy to read about them… 😁
Or do we just change the square of e₀ and that’s all? This seems like a logical place for it: 0 is the curvature of euclidean space, >0 means elliptic and
Now I wonder: is there an algebra to encompass both multivectors and linear operators acting on them. Like Clifford algebras allow to internalize orthogonal transformations, something could really internalize _all_ linear operators. But how even to approach this? The algebra product should allow both to raise degrees of multivectors and lower them, or we won’t be able to remain in the same subspace by acting with sandwich products. For example, bare exterior algebra is unfit: we can only raise degrees until we suddenly annihilate what we acted upon, and can’t ever go back to 1-vectors.
I don't really have an answer to your first question, but to your second question, this is technically possible in geometric algebra! You have to use a space with double the dimension (so the dimension of the algebra gets squared). I don't know the details though. But to be honest, I wouldn't suggest doing this in applications because the dimension of the algebra gets high really quickly, and if you really want arbitrary linear transformations it's better to just use matrices.
16:10 @sudgylacmoe
What if the lines have opposite directions?
It will be the point at infinity in the other direction. (In fact, because these two points at infinity are scalar multiples of each other, in the homogeneous sense they represent the same point, so the meet here is the point at infinity in both directions.)
I've wondered how PGA would look if you stopped normalizing everything. I know in Lengyel's version, non-normalized points act like point masses, and addition of non-normalized points describes the center of mass
That works in this version as well, even for other objects like lines and planes.
Im curious about how hyperbolic and elliptic PGA work.
Aside from "e0 doesn't contribute to the magnitude/inner product", pretty much everything applies as-is. Rotations are still rotations, and translations become translations in hyperbolic and elliptic space, which, like how Euclidean translations can be viewed as shears in the higher dimension, hyperbolic and elliptic translations become hyperbolic rotations and traditional rotations in the higher dimension.
There are a few quirks of the geometry that can give potentially surprising results, like translating in elliptic geometry eventually getting you back to where you started, and 2D hyperbolic PGA having a noticeably finite "line at infinity", and points that can be said to be "beyond infinity". Also, while hyperbolic and elliptic geometry are more often visualized with stereographic projections, PGA actually favors the Beltrami-Klein model and gnomonic projection, which would get used automatically if using and existing plot intended for Euclidean PGA, so they may look different from how you might expect because of that, though they'd also look more like Euclidean PGA at the same time.
sudgy can u show us conic sections
Think I found a small caveat with something you said:
9:29
You say that the regressive product can be defined in terms of the geometric product, and suggest that “everything on the screen” is enough for this. I suppose there is a caveat which is that you also need a duality operator, which you understandably didn’t mention since it’s out of the scope of the video.
As you said in your other very helpful video on the operations of GA, when a basis vector squares to 0, you have to use the hodge dual instead of the simpler pseudoscalar dual, if I understand correctly.
In PGA, e0 squares to 0. So really you would have to also come up with a hodge dual table to use the regressive product in PGA, right?
This whole thing with the dual and the regressive product is pretty messy in PGA. Yes, in this particular circumstance, you can't define the regressive product with purely the geometric product. However, once you have picked a basis (such as e1, e2, and e0 here), you can define the dual in terms of that basis and the geometric product, so "You can do everything with what's on the screen right here" is still true, because I showed the basis on screen as well. I guess I did cheat a little bit with including the regressive product in the statement about the geometric product, but I wanted to mention all of the other products being used without going into a long shpeal about basis-dependence right then. I probably should have put a note there.
@@sudgylacmoeThat’s totally fair! I think it could be a good topic for a short. Someone who watches this video and your other general video on the operations of GA would have enough info to make their own PGA implementation, which is great.
Yeah that other video was actually originally going to be a quick sidenote at the beginning of this video but then I realized it was getting way too long so I split it off into its own thing.
@@sudgylacmoe I actually find it to be the one I end up going back to most often because it’s so general! I think it’s good that it got its own video
How come the projection formula is given as (a • b)b but in ganja.js it's (a | b)/b
a | b is just programmer speak for a · b because "·" isn't ASCII. As for the inverse, I explain it at 20:26.
Cool