Finding Velocity On a Sphere Using a 3D Euler's Formula

Using a generalized version of Euler's Formula, exponential functions can be used to algebraically represent rotations in any dimension. But what is this generalized formula, and what can we use this representation for?

=Chapters=
0:00 - Intro
2:41 - Tilt Product Powers
5:13 - Generalized Euler's Formula Derived
6:57 - Classic From General
9:35 - Polar and Spherical Coordinates
13:20 - Trouble With the Matrix Exponential
15:56 - Computing Velocity of a Sphere Point
22:02 - Interpreting the Result
23:03 - Wrap Up and a Look Ahead

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This video was generously supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor, Mfriend.

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CREDITS

The music tracks used in this video are (in order of first appearance): Rubix Cube, Checkmate, Ascending, Orient, Falling Snow.

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The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:

[morphocular]

Hey everyone. If you haven't heard, I'm planning a Q&A video to mark reaching 100k subscribers and I'm still collecting questions for it. Feel free to ask any in reply to this comment! I'm thinking I'll answer some right here right now and save others for the video.

[samuelthecamel]

17:08 Those equations are looking a little sus

[joaopedrodiniz]

After watching these videos, i have to admit, you are one of my favorite math youtubers, if not my favorite. Congratulations on your journey and i hope you keep making these advanced math videos, because they are amazing!

[KinuTheDragon]

Among us spotted at 17:13! 2nd row, last theta

[DoxxTheMathGeek]

This video is uploaded at the *perfect* time!
I was just learning about this, but it's very hard to learn just using the internet. This is awesome! Thanks a lot! ^w^

[dank.]

Beautiful video! (This probably happens a lot) but I was wondering how to do the topic of this (and the previous) video, and thrilled to see such a direct answer on youtube. I was lightly looking into analogues of Euler's formula in higher dimensions a couple months ago, and I think I stumbled upon Rodrigues' function but didn't feel like looking into the context. At the time I was hoping to find a way to generalize a fourier transform to 3d, (mostly to make a present for a friend's birthday, which has long since passed lol) and seeing this video (and the last one in the series) got me very excited. Cheers!

[keldwikchaldain]

The generalized euler's formula looks almost exactly like how you'd do this in projective geometric algebra.

u tilt v is effectively the same as u wedge v, producing a bivector that gets factors of sin theta, which produces a rotor that rotates 2 theta radians in the plane of the bivector u wedge v, then since this is automatically normalized you can then add one (equivalent to the identity matrix here) to get the square root of the rotor which rotates theta radians instead of 2 theta.

[gcostello]

Wish videos like this were out when I was first learning about exponential maps and Lie algebras in physics!

[ryansamuel]

I have been watching many different math YouTuber videos and this is particularly clear, sufficiently abstract and entertaining. Nice job!!!

[jacobwalker]

Given that i'm in the midst of learning about robotics, i'm incredibly lucky to have found your content, explanation is the best i've found. Could i be cheeky and ask if you could do anything on the basics of kinematics in the future? The combination of the fluctuating diagrams and amazing explanation in your videos is soo much easier to get into my head than another abstract maths book that only seem to explains things well to other experts. Honestly i truly wish every tutorial was like this one!

[jacoblojewski]

I learned about the matrix exponential way back in Differential Equations when deriving the Jordan Normal Form. It's nice to see it make a comeback in something else really cool!

[uwuzote]

Here's my comment about geometric algebra
Your "tilt" product is the exterior product of vectors in 3D VGA (vanilla geometric algebra, works with linear space of vectors (and other polyvectors too)) which results in a bivector
Except exterior product will work even if your vectors aren't orthgonal or aren't normalised (But before expontiating a bivector you will need to normalise it). Bivectors are independent of basis, so no memorizing weird cross product rules etc.

[isobarkley]

goD THIS IS SO INTERESTING !!! thank you for answering some questions i've had for years!

[linux_devs]

Please keep going in this series. It is excellent

[timh]

very nice video. have just been getting into geometric algebra, couldn't be better timed

[mbayanzongele]

This video helps me make a relationship between christoffel symbols, tangential acceleration and velocity on a spherical surface.. Thank you so much.

[KINGSLPK]

I'll get back to this to improve my understanding of the topic!

[HuyNguyen-yijf]

Man, I was struggling to find the general solution to exp(At), where A is a matrix. Now I've seen it, it's so short but elegant!

[quantumkya]

OMG IT'S HERE!!
Awesome video as always; I love when you cover little-known topics in maths, like wheels and stuff like that.
I rarely comment and I just wanted to inform you of my appreciation of your channel.

[evandrofilipe]

23:33 I guess a brief nod is better than nothing huh?

I'm partially joking of course but I can't wait to see what you do after this series' final episode?
Any insight or not so much?

No pressure of course, I love these videos for the most part and I think you should be proud