where has this been all my life!?
on the plus side I found it now...
What is the inverse of a vector?
191–198 of 198 posts
Re: What is the inverse of a vector?
#192This article is really about geometric algebra, check out my geometric algebra software https://github.com/chakravala/Grassmann.jl
Re: What is the inverse of a vector?
#193Earlier quoted context omitted.
Yes, this. Though Zero to Geo is one of the links at the bottom of the article. It is really a shame that article does not clarify that, btw, what we've just derived is a re-derivation of a thing that has already been expressed and named, by Clifford, and well-characterized: https://en.wikipedia.org/wiki/Geometric_algebra Such a bummer to see very slick but very ahistorical articles.
Hi. I wrote whole section on the history of GA and what happened and why it isn't already the norm, but I chose to remove it because the article is already far too long, and I don't think that my intended audience (engineers, compsci people, university undergrads) would care about the history. Apologies that wasn't what you would have preferred.
I think the article is great, and I think the interactive illustrations are sweet. Thanks for the taking the time to write it.
As an educator, though, when I see presentations of existing ideas that present them as if they were new, I die a little. You're standing on the shoulders of giants whether or not you think so, and whether or not you say so. It's best to figure out who the giants are, and how you're standing on them. When you are up front about the connections to past scholars, you are giving the credit to those scholars that they deserve, and you are strengthening the storyline, and you are setting a good example for the people that look up to you.
You can add a few sentences at the bottom saying, if you've gotten this far, congrats, you understand some basics of GA, and then link to other resources about the history and current applications of it.
Re: What is the inverse of a vector?
#194Re: What is the inverse of a vector?
#195Re: What is the inverse of a vector?
#196Earlier quoted context omitted.
From my view, it goes both ways: geometric algebra/calculus is a more transparent version of the standard approach and the translation back to it is also a relatively small delta to pick up. Either way of going about what is in essence the same material entails becoming familiar with multivectors, the wedge product, and multilinear algebra, whether you do it through geometric algebra or the standard approach.
That makes sense but my argument is since further material (some examples which I listed) assumes and builds upon the standard approach, you'll likely be better off taking that path.
?? You'll likely be better off seeing different perspectives of the same thing.
Re: What is the inverse of a vector?
#197Earlier quoted context omitted.
German and french engineering school are pretty rough on math theory, for the better or the worse. Mostly because a lot of theory was born in these two countries.
Can confirm, before I was anywhere close to serious computer science, I basically did an undergraduate degree in mathematics / physics. And it goes through some very advanced subjects in both. It basically prepares you to be an engineer in whatever field you choose, be it a structural engineer, or a computer scientist.
Can you tell me the topics they teach you there?
Re: What is the inverse of a vector?
#198Earlier quoted context omitted.
The nicest written-up example I know is https://www.shapeoperator.com/2016/12/12/sunset-geometry/ * * * As a relatively recent personal example I spent a few months (in bits and pieces) working out a bunch of metrical spherical geometry for myself without reference to past work, with points represented as displacement vectors to stereographically projected points at https://observablehq.com/@jrus/planisphere with the…
That is a strange and unconvincing article. In outline, it goes: 1) Look at this slick solution using geometric algebra. 2) Look at how ugly the trigonometric solution is. GA is so great! 3) And, by the way, one can mechanically translate the GA solution into the usual vector notation. Point 3 is even a bit understated: GA concepts are really only used for a few lines under "Solving for the Earth's radius." Once you…
>It's really the same method in different notation. You take the cross product and separate into parallel and perpendicular components, and then you reach the epsilon^2 equation, and it's the same from there.
Bivectors and cross-product aren't just different notation for the same thing if that's what you meant. They're distinct (but very much so related) mathematical structures. For one thing, one's associative while cross products break associativity.
As far as GA sharing a lot with the more comment vector algebra/calc methods. Personally, I'm happy that GA has an attitude of "if it's not broke, don't fix it". It also means there's really not a lot of time lost in the transition due to the compatibility. Hell it's even backwards compatible in the sense that you can still easily retrieve your axial vectors the cross product gave you if you so wish (which cleared up instantly what the exterior algebra folks were doing with their hodge star business when I decided I wanted to explore that perspective later on).