Circular velocity and acceleration with geometric algebra

[@SirTravelMuffin]

Thank you for such a great demonstration of the power of GA!

[@minimath5882]

Awesome video! I find unfortunate that the convention for unit vectors are e1 and e2 because e the transcendental numbers shows up next it. It can be a little confusing but you did distinguished between them italicizing the e.

[@PeeterJoot]

Yes, that is unfortunate, especially when you have rotations like e_1 e^{e_1 e_2 theta} -- too many e's!

[@rainbow-cl4rk]

Really good vidéo ! It's a little bit odd to use GA for this problem since you can do all the thing you have done with complexe numbers or matrices. I know it's equivalent since one can draw an isomorphism but it feels you use GA only for the sake of using it. The animation are really well done, congratulations 👏

[@PeeterJoot]

Absolutely, if you write r-hat = e^(i theta), and theta-hat = i e^(i theta), you do get the r-hat' = omega theta-hat and theta-hat' = - omega r-hat results very easily (which is what everything that follows depends on.) I admit that I used geometric algebra only because I like it. There isn't a good excuse for GA here over complex numbers -- probably need a 3D problem, perhaps doing the same thing in spherical coordinates, to provide a better justification.

[@OmnipotentJC]

This was a very informative and well put together video. Thank you!

[@PeeterJoot]

You are welcome.

[@swag_designs5470]

This was amazing I loved it thank you❤

[@spiderjerusalem4009]

what abstract algebra did u use to learn from?

[@PeeterJoot]

In this video I've made use of geometric algebra, as well as conventional linear and vector and complex algebras. The first chapter of my book, "Geometric Algebra for Electrical Engineers", has what I believe to be an accessible introduction to geometric algebra (assuming that you've studied high school level linear and vector algebra). You can find a free PDF version of the book here: peeterjoot.com/writing/geometric-algebra-for-electrical-engineers/

[@BongoFerno]

How do you use nilpotent versors ε²=0 to get the velocity and acceleration for free?

[@PeeterJoot]

I'm not sure, but it sounds like you might? If you know, perhaps you can demonstrate, or point to a reference.

[@dsgowo]

@@PeeterJoot This sounds like something you might be able to do in PGA? Not entirely sure.

[@PeeterJoot]

@@dsgowo Perhaps. I haven't spent much time on either PGA/CGA, so I'm not in a good position to comment on that.

[@pre-universitygeometricalg5862]

Very useful. Thanks. I'll add it to playlists on our channel.

[@pochotom2834]

Why is it not taught this way in engineering universities?

[@PeeterJoot]

Geometric algebra is currently still very obscure, and not known by many engineering instructors. What I've shown in this video would probably be taught using complex variables, or matrices in engineering classes, which can both be used very effectively for this planar material (but generalizing to the 3D spherical coordinate case is not as nice.)

[@5ty717]

Wow

[@hotbit7327]

"Let i = e1e2" ... WHY? What I'm looking for is understanding, instead, usually people juggle words (like i, e1 etc.) according to some syntax rules and that's it. Therefore, it's some playing within linguist8ics system. Also, in 10 min. video, talking fast, people pack a tonne of information, as if they (and viewers) are so smart. But I don't see this cleverness in the real world. That's why I don't like your video.

[@PeeterJoot]

"Let i = e1e2". Why? Because i^2 = -1, just like complex numbers: e_1 e_2 e_1 e_2 = (e_1 e_2) (e_1 e_2) = (-e_2 e_1) (e_1 e_2) = - e_2 (e_1^2) e_2 = - e_2 (1) e_2 = - e_2^2 = -1.

[@SillySussySally]

You can tell how much effort he put in the video ❤ The mic threw me off but really nice video, man!

[@PeeterJoot]

Thanks. Re: mic. For my newer videos, I swiped my wife's mic for the later videos, instead of using my earpods, and think that you'll find the sound is improved.

[@SuperDeadparrot]

What frame is d/dt being taken in?

[@PeeterJoot]

This isn't a relativistic discussion, so any time derivatives do not depend on the motion of any observer frame... but please consider the origin fixed for this discussion.

[@FoughtAgaisntSisera]

He's Back!