Siggraph2019 Geometric Algebra (2019) [video]

[ivansavz]

Here are some direct links to some of the PGA (Projective Geometric Algebra) resources mentioned in the talk:

1. The 2D and 3D cheat sheets are here: https://bivector.net/doc.html

2. The talk is also available in "paper form" on the arXiv: https://arxiv.org/abs/1901.05873

3. Ganja.js repo https://github.com/enkimute/ganja.js and the associated demos in the Ganja coffee shop https://enkimute.github.io/ganja.js/examples/coffeeshop.html

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For more background about the concepts like quaternions, projective geometry, and exterior algebra, etc. that are all united in PGA, you can check this nice book by Jean Gallier and Jocelyn Quaintance: https://www.cis.upenn.edu/~jean/gbooks/geomath.html direct link https://www.cis.upenn.edu/~jean/math-deep.pdf I haven't read, but sampled it on several topics and found the explanations to be very good.

(edit) first-added links specific to PGA, and typo fix s/external/exterior/

[skydog23]

As the author of the arxiv preprint referred to above, I would like to point out a newer, expanded version written for the Siggraph 2019 course. It's available here: https://www.researchgate.net/publication/334397782_Course_no...

[cygx]

*exterior algebra

[carapace]

FWIW, "Geometric Algebra for Electrical and Electronic Engineers" is short and sweet: https://ieeexplore.ieee.org/document/6876131?arnumber=687613...

> This tutorial paper provides a short introduction to geometric algebra, starting with its history and then presenting its benefits and exploring its applications.

[bananabiscuit]

Here is an informative critique of this video along with an improved application of geometric algebra to computer geometry:

http://terathon.com/blog/projective-geometric-algebra-done-r...

[ivansavz]

Thanks for sharing this blog post and the cheat sheet http://terathon.com/pga_lengyel.pdf

I agree the representation chosen in the SIGGRAPH lecture and the associated docs seems somewhat unnatural and counter-intuitive (why are points being represented as trivectors?).

Have you reached out to C. Gunn or S. De Keninck to ask what they think about the representation you propose in that blog post and the geometric product ⟑ and antiproduct ⟇ ? If the representations are equivalent and one is closer to concepts people already understand (e.g. points being (x,y,z,w), with w=1), then maybe adopting the simpler representation would lead to wider adoption of the ideas.

Alternatively, maybe they will point out some other reasons why points-as-trivectors representation is preferable.

[enkimute]

For those interested - In a few weeks, 26-27-28 February, there is a (free) workshop featuring talks and workshops from Leo Dorst, Anthony Lasenby, Charles Gunn, Steven De Keninck (the lecturer from this vid - disclaimer - thats me), Vincent Nozick, Dietmar Hldenbrand, Hugo Hadfield and Eric Wieser.

They'll be talking about their research on applications of GA in robotics, AI, CS, physics, graphics.

more info : https://bivector.net/game2020.html

Everyone welcome !

[skwog]

Near the end of your Siggraph 2019 talk (excellent, thanks), an audience member mentions an IEEE Spectrum article "Geometric Algebra for Electrical and Electronic Engineers". What is the author's surname that you mention?

Peter Yolks? Jolts? Holt? Oates?

For me, multiple searches have yielded no results for a similar author name so far.

Specifically, there does not appear to be any such article on IEEE Spectrum [1], but there is one on Proceedings of the IEEE [2] whose listed authors do not include anyone named Peter.

Curious because you go on to state that his work is readily available. That caught my attention, so I became interesting in finding out more.

[1] https://spectrum.ieee.org/searchContent?query=Geometric+Alge...

[2] https://proceedingsoftheieee.ieee.org/most-downloaded/geomet...

[crubier]

This video is the best thing I watched in the past few years. Last time I loved something that much was Geoffrey Hintons course about deep learning back in 2015.

It is eye opening, just like in the beginning of mainstream deep learning, it made me wonder “why didn’t we discover this before?” It elegantly solves so many annoyances about geometry, maths and physics.

I’ve been evangelizing Geometric Algebra like crazy since I found this video, including in hacker news comments.

[alexkcd]

Agreed. Besides Geometric Algebra, dual numbers also play a huge role in automatic differentiation -- the core building block of modern machine learning frameworks.

I really like the analogy in this talk about how Al-Khwarizmi's six quadratic equations simplify to just one, once we learn about negative numbers and zero.

In a lot of ways, geometric algebra (and dual numbers) are our discovery of "negative numbers and zero", but for the 21st century.

[nimish]

Great article from the same presenter here: https://www.groundai.com/project/projective-geometric-algebr...

And for those who know conformal GA, see https://www.researchgate.net/publication/290437718_Geometric...

[DreamScatter]

Check out my geometric algebra software

https://grassmann.crucialflow.com

[adamnemecek]

If this interests you, you should check out the bivector community https://bivector.net/ which is ran by the presenter of this lecture.

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

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

[banachtarski]

Highly recommended. Absolutely eye opening talk and set of resources.

[amelius]

Does this solve the problem of consistency in computational geometry? I.e. point lies inside a polygon, but due to rounding errors, the point seems to lie outside the polygon, which might trigger e.g. a program crash.

[nutjob2]

Yes, if you do it algebraically.

This is one of the things that this talk touches upon. BAsically if you use GA you eliminate precision issues (at least until you render or otherwise materialise), various special cases, limitations and unnecessary complexity.

[garmaine]

No, but there are other ways of solving that. Setting rounding modes and bias values appropriately would prevent what you describe.

[injidup]

No

[msla]

A good introduction:

https://www.av8n.com/physics/clifford-intro.htm

Previous discussion on this topic:

https://news.ycombinator.com/item?id=22200260

[deepaksurti]

A recent discussion on HN related to GA in the context of removing quaternions: https://news.ycombinator.com/item?id=22200260