Circular velocity and acceleration with geometric algebra

Подписаться 1,6 тыс.

Просмотров 7 тыс.

50% 1

In this video, the geometric algebra form for the circular unit vectors will be derived, and then used to compute expressions for velocity and acceleration in a circular coordinate system.
This video includes a very brief introduction to (2D) geometric algebra, including the exponential form for a 2D rotation. Specifically, this video includes:
* A reminder of what circular coordinates are.
* A brief outline of what is meant by each of the circular basis vectors.
* A derivation of those basis vectors (just basic geometry, and no GA.)
* A brief introduction to geometric algebra, and geometric algebra for a plane, including the "imaginary" i = e_1 e_2, and it's use for rotation and polar form.
* How to express the circular basis vectors in polar form.
* Application of all the ideas above to compute velocity and acceleration.
* Circular coordinate examples of velocity and acceleration.
Prerequisites: calculus (derivatives and chain rule), complex numbers (exponential polar form), and basic vector algebra (basis, vector space, dot product space, arrow representation of vectors, graphical vector addition, ...)
If you liked this video, you may be interested in my book, Geometric Algebra for Electrical Engineers, available in hardcover or softcover on amazon:
amzn.to/49hplMm
An electronic (PDF) version of the book is available on my blog here:
peeterjoot.com/gaee

Опубликовано:

1 июн 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 23

@OmnipotentJC 10 месяцев назад

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

@PeeterJoot 10 месяцев назад

You are welcome.

@SirTravelMuffin 6 месяцев назад

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

@rainbow-cl4rk 10 месяцев назад

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 10 месяцев назад

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.

@swag_designs5470 8 месяцев назад

This was amazing I loved it thank you❤

@pre-universitygeometricalg5862 10 месяцев назад

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

@FoughtAgaisntSisera 10 месяцев назад

He's Back!

@SillySussySally 8 месяцев назад

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

@PeeterJoot 8 месяцев назад

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.

@minimath5882 7 месяцев назад

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 7 месяцев назад

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

@spiderjerusalem4009 7 месяцев назад

what abstract algebra did u use to learn from?

@PeeterJoot 7 месяцев назад

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/

@5ty717 7 месяцев назад

Wow

@BongoFerno 6 месяцев назад

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

@PeeterJoot 6 месяцев назад

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

@dsgowo 5 месяцев назад

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

@PeeterJoot 5 месяцев назад

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

@pochotom2834 6 месяцев назад

Why is it not taught this way in engineering universities?

@PeeterJoot 6 месяцев назад

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.)

@hotbit7327 2 дня назад

"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 2 дня назад

"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.