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Comment by Lichtso | Hacker News Reader

Not sure if mathematics counts as software:

Geometric / Clifford Algebra.

It makes just about every aspect of handling geometry in computer graphics and robotics so much easier. It comes with a ton of upsides and only two downsides: You probably were not taught it yet (so you have to learn it) and if you do, others will have trouble following you unless they learn it too.

I somehow feel like coming from a tribe where we only count up to three, and then being introduced to the concept of natural numbers (and other number classes) by outsiders. As I stared to use it myself, it changed the way I think about things, but now I can't communicate with the rest of my tribe anymore. Yet, I think it is worthwhile and about the only silver bullet I have seen.

If you are interested, here [0] is a nice introduction to one class of clifford algebras.

[0] https://arxiv.org/abs/2002.04509

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Do you think it is all of GA that helps, or really just the concept of bivectors / multivectors?

(I have a lot of skepticism about the geometric product as a useful object, but I think that multivectors are clearly fantastically useful everywhere. My exposure is mostly in physics where multivectors are basically necessary to make any sense of a bunch of concepts. But I don't work in robotics/graphics so maybe don't know how much good the geometric product does in practice?)

>> Do you think it is all of GA that helps, or really just the concept of bivectors / multivectors?

Yes, definitely all of it. Especially in physics, where every well known equation comes with its own, not so well known algebra. There are dozens (maybe even hundreds?) of them, all with their own notations and shenanigans. The thing is, all of them can be covered by GA. And you only need three instances of GA to cover everything [0]: Classical mechanics, quantum mechanics and relativistic mechanics / space time.

>> I have a lot of skepticism about the geometric product as a useful object

The geometric product is kind of like: "Here is everything you could possibly want to do, all at once!". We usually pick out specific parts of it (called grade projections) to operate on sub-algebras matching the specific problem domain at hand. Yet, all of them are still connected in one über-algebra so to speak. So, we don't need to switch notations.

>> bivectors / multivectors

Multivectors, the vectors of GA, are like "normal" vectors, from linear algebra, just tuples. There is nothing special about them. The genius lies in the idea that unlike "normal" vector algebra, where you have one element in your tuple for every dimension, you also have elements for every combination of dimensions in a multivector. The multivector is usually sorted by the grade (number of dimensions an element combines). That way a multivector has a 0D scalar, 1D vectors, 2D bivectors, 3D trivectors, etc.

[0]: https://www.scirp.org/journal/PaperInformation.aspx?PaperID=...

I'm very aware of GA and multivectors and the usual equations that are included in GA texts, like the one-term form of Maxwell's equations. I'm specifically interested in whether you know of a good reason to include the geometric product in all this. It seems to me that for the most part you can do all of the useful stuff without ever mentioning it, and the equations it produces tend to be very awkward (compute something, grade project, compute something else, project again...) -- it's very 'just so', and kinda feels like it's all using the wrong tool. But I haven't figured out what the better tool is yet.

Nice. I’ve been always curious but also worried about alienating myself. Any communities of GA users out there to feel connected?

I think the biggest gathering is: bivector.net

It has a discourse and a discord.

Do you know 3D computer graphics libraries that use GA?

You are right, while there are already some libraries which provide GA, there are almost non which use it. I know that it is used in some photogrammetry pose estimation / frame registration code and I am currently porting my vector / path rendering library to GA.

It is still really niche, but gaining traction in the last 3 years or so (considering it was invented in 1878).

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