The zoo of Geometric objects and transforms in Conformal Geometric Algebra (CGA)
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- เผยแพร่เมื่อ 5 ต.ค. 2024
- I don't recommend using conformal GA for computer graphics. The space humans live in, basically, euclidean, meaning that when you look at a line you instantly know whether or not it is straight - in conformal geometry you don't have that, the idea of a "straight line" is defined only with reference to a chosen point/plane at infinity. If someone says they're doing conformal GA and then makes excessive use of the point/plane at infinity, in their hearts they've actually given up on CGA and are using PGA.
But, the geometry is interesting! So I felt like infodumping when I got sick. -- Watch live at / hamish_todd - เกม
I laughed when you called the area outside the Poincare disk model "the Shadow Realm". But you actually got me thinking near the end, if you can "unwrap" the disk to a half plane and it's still the same model, then you can "unwrap" even further and the outside of the disk can serve as a hyperbolic model just as good as the inside of the disk.
Anyway, "Shadow Realm" and "Kebab" are the new additions to my extremely formal mathematical vocabulary today haha
I had to watch this again to try to absorb it more thoroughly after recent discussions. I can feel my brain stretching and wrinkling as it becomes more hyperbolic hahaha. But no seriously it is hard to fit all of this in my brain at once. I need to make myself a cheat sheet.
3:07:40 yeah as the resident Lengyel practitioner getting used to "Circle vs AntiCircle" or "AntiDipole vs Dipole" etc has REALLY helped me quickly catch on if there are situations where "two versions of an object but they act the same even though you got them by different means", or "two versions of an object that act different even though you got them by the same means". 3:07:56 I'm not sure what the reservations are, but at least I feel like I'm following along. I'm sure all the details will shake out as I get further with my library
The term that is established in the literature, it turns out, is "intrinsic" and "extrinsic"!
The moment you suggested transforming with e-, I thought it sounded so strange and unique I had to look into it. But yeah as far as I can discern too I agree it looks like it has no invariant. Which is so weird
I know right! You can read some about them here th-cam.com/video/YpKWa8vrULk/w-d-xo.html - at 14:35 he shows "hyperideal" vertices. e- is a hyperideal reflection - it has no appearance in the conformal geometry, though it does have an appearance in the klein model
As far as I know, this is the only explanation of CGA on TH-cam. Have you considered cutting it up into something digestible? For example, a lot of drawing and explanation is dropped or rescinded pretty fast, so unless the viewer is at least as patient and motivated as I am, they are not going to get a lot out of it. I realize you're streaming, off the cuff, with no preparation, so I'm not making a critique. I'm just saying, if there's only going to be one CGA video on TH-cam, maybe let's think about making it more accessible.
Unfortunately I'm very busy with other things, so if there's going to be an intro to CGA it's unlikely to come from me (unless someone pays me to make a big intro-to-GA-in-physics course, then I'd do it). You're quite right that people are only going to both with this if they are very motivated.
Aside from being intrinsically simpler/more learnable, PGA is just a lot more useful than CGA. This is because the world humans work in is (approximately) euclidean, so the euclidean group and euclidean objects and transformations are important - one cannot say the same about conformal objects or transformations.
I like CGA a lot. If I had the money, I'd be working on a GA-for-physics course/set of teaching materials that would use it. But, in a resource-constrained situation it makes wildly more sense to focus on PGA. One reason for this is that in my opinion, **one should definitely learn PGA before learning CGA**. There are some CGA fans who would disagree with that but they are just wrong. They might also tell you "1-vectors of PGA are like 4-vectors of CGA" - this is also wrong; CGA contains 1-vectors that behave precisely the same as the 1-vectors of PGA, with the same geometric interpretation, algebraic behaviour etc.
There is quite a lot of PGA content out there... although I still think we need more. If I was asked to teach a CGA, my first two lectures would be almost exactly my two PGA lectures I delivered at the Game Developer Conference (which are on yt).
Hey man, amazing video! I’m trying to use the clifford python library but keep throwing errors because of ‘numba’ incompatibility. I was wondering if you still use clifford and if so what versions you use to make it work. Any help is appreciated!
Never used Clifford! Good luck
No e0 for the line at infinity?
e0 = ePlus + eMinus
Sorry I didn't say that! It's very important and is how PGA is constructed inside CGA