A Visual Intro to Curves and the Frenet Frame

Our submission for the Summer of Math Exposition 2 #some2.
Topics:
An introduction to the Mathematics of differential geometry of plane and space curves, leading up to the Frenet Frame, and Frenet-Serret Formulas and the Fundamental Theorem of Space Curves.

Content:
0:00 Introduction, Motivation, and Applications
0:45 Overview
1:23 Circles and the Idea Behind Curvature
4:19 Definition of Curvature and Examples
6:54 Moving into the Third Dimension and the Frenet Frame
10:16 Derivation of the Frenet-Serret Equations and tau
13:51 Visualization and Conceptualization of the Frenet Frame
16:23 Frenet Frame in Popular Culture
16:56 The Remarkable Fundamental Theorem of Space Curves

Prerequisites:
Derivatives, Dot Product, Cross Product, Basis of a Vector Space

Credits:
Dan Walsh: Narration and Animations
Franciscus Rebro: Concept and Script

Bibliography / Works Consulted:
Needham, Tristan - Visual Differential Geometry and Forms
Pressley, Andrew - Elementary Differential Geometry, 2nd Ed.
Tu, Loring W. - Differential Geometry
Hidden Figures, 2016, Directed by Theodore Melfi

[Sofialovesmath]

Can't expressed how much I was impressed by this. The ideas and presentation are simply flawless.

[unflexian]

i've never used the expression "blown away" in my life, but there's no other way to describe how clear, consice, and informative this video was. thank you dan walsh.

[jbofficial]

You're so underrated. Amazing and more high quality than official documentaries

[krabkrabkrab]

This is a really clear explanation. Thanks. In my field, accelerator physics, including particle accelerator design, we depend heavily on Frenet-Serret theory.

[pl]

immediately one of my favortie videos. I really struggled with space curves in school but now I have better visualization and I feel like I can go back and approach the material differently. Also, I will never confuse pitch yaw and roll again!

[debblez]

I think this is my favorite some2 entry so far. And I’ve seen like 20+

[JosueCastillo-fshb]

This is such a concise video. Thank you!

[Number_Cruncher]

Thank you for this clear and understandable introduction to this topic.

[NinjaAdvisor]

Nice one! Please keep making more of this kind of visualization! Instant subscribed!

[jorex]

The animations really helped me to understand these formulas, well done!!

[danieljulian]

Robert Ghrist introduced me to F-S frames in his Calculus Blue series. Your presentation (along with a nice series by BillCookMath - YT channel) solidifies what I have learned even more, so thanks. By the way, say "Hi" to Santa Barbara, where I resided for the last decade and a half of the 20th Century.

[JoseMuzikantas]

Congrats really a great video and explanation.

[symbolsforpangaea]

Thank you!!! One of best explanations I've seen!

[Quate]

fantastic entry. most of the other some2 entries are not good. this one is very good.

[georgeshillcock]

Great work, keep it up! Would love more diff geo

[zokru]

first intro ive herd that was top notch in logos

[jorgeabelmejiavenegas]

Thanks u. Can you help me with the manim documentation that you create for this video?

[rk]

Wonderful video. Helped a lot. BTW, how do you visualize the second equation, which is a combination of curvature and torsion?

[angeldude]

Those sounded like Euler angles near the end! Could the failure for curves that have 0 acceleration at a given point have something to do with gimbal lock?

P.S. I don't think the frame is 100% geometrically significant, because the Right Hand Rule is not intrinsically geometrically significant. Regardless of which direction you pick, all it's going to change is a minus sign in a few places. Speaking of, I'm curious if this would also work in 4D. Since 3D requires a non-zero acceleration everywhere so that we have a second derivative, and I think it would also break down in 2D if there was anywhere with 0 velocity, it would make sense if 4D required the curve to have non-zero "jerk" everywhere.

[filipo]

What about curves in 4th dimension? How many parameters do we need? Like 3?