Radius of Curvature Proof - approximating a curve with a circle!

[@Gauteamus]

Now I wonder what curve the centre of the approximating circles trace out?

[@virtually_passed]

What a great question! I plotted the curve in desmos for you so you can see for yourself what it looks like. All I did was plot the (x0,y0) parametric equation from the formulas I derived at 2:00
www.desmos.com/calculator/lbvkjztsmd

[@Gauteamus]

@@virtually_passed Fantastic! I think we should call those curves megacycloids, as they seem to describe the trajectory of bouncing balls thrown at earth from deep space :-)

[@Celastrous]

@@Gauteamus Maybe hypercycloids? They seem very similar to hyperbolae to me

[@pjbarendrecht]

@Gauteamus That'd be the so-called evolute, see e.g. en.wikipedia.org/wiki/Evolute

[@avyakthaachar2.718]

Thank you so much ❤. Amazing explanation

[@mathisalwaysright4048]

Thanks for this. Most videos on this topic just defined curvature by the formula and as 1/R without showing the connection. This video is the only one that explains why does it work, and not only shows 2 strange definitions with no connection between them

[@virtually_passed]

I agree. That was the motivation for making this video :)

[@dorinep2037]

Oh my god the music the feel... Everything makes a fresher cry

[@PenandPaperScience]

Wow, I see we make similar videos, but you've been at it for much longer than I have. Keep up the good work :)

[@gynemedee]

dhanyavaad mere bade bhai, bahut hi sundar tarke se apne iss topic ko ujagar kiya "mai apka dilataht se thankwaad krta hu "

[@vishalpratap244]

Came from your previous video on this glad to come thank you sir for amazing explanation

[@Celastrous]

Great video! I love doing little diff eq curiosities like this

[@virtually_passed]

Me too! :)

[@SmallWave80]

Brillant. I did not make efforts to deduce this formula in exercises.

[@goldfishdoc1999]

wow... this was awesome

[@crossfadez5521]

Vert interesting. Good job man.

[@NoNTr1v1aL]

Amazing video!

[@ryanguo13]

Thanks so much for the amazing explaination!

[@ShaikhMusaddiq-hw8lr]

Nicely explained 😊

[@virtually_passed]

Thanks a lot 😊

[@rachitchauhan8164]

Taking differentiation assumes dr/dx=0 but r=radius of curvature and is dependent on how you move on the number line...

[@SimchaWaldman]

Very cleaver derivation, must I say!
Much to learn I sill have!

[@Dr.1.]

great video

[@bisheshwarraj3225]

thanks man it really helps.

[@virtually_passed]

Glad to hear it!

[@theanonymousgamer8394]

Amazing video

[@virtually_passed]

Glad you think so!

[@Aman_iitbh]

thanks

[@Aman_iitbh]

I am very much satisfied by this now

[@carloscabanilla9685]

beautiful :3

[@virtually_passed]

I'm glad you like it :3

[@30vaibhavdhaygondexdr.v.sa22]

great man! helping me with my homework!

[@virtually_passed]

Happy to help!

[@funhouse7463]

just wow man

[@sardauchiha4690]

Thank you 😊😊👍🏿

[@virtually_passed]

You’re welcome 😊

[@ichigo_nyanko]

I was trying to do this myself a few months ago (though I only tried for a few hours), I didn't get too far. I believe I tried to find the circle with the lowest area difference when putting it over the curve, and then I tried with the lowest derivative difference - the problem I ran into was I had no idea where to set the bounds of the function when taking the derivative/integral (between which points do I want to minimise the difference in derivative) - but I left it soon after and forgot to ever go back to it.

[@ashishsam26]

thanks!

[@kodapasy]

thanks man, how did you animate the last part where the curve is tangent to the circle? I cant seem to configure how to make the center of the circle move in a why that would ensure tangential interactions.

[@kodapasy]

found it, I make a line perpendicular to the first derivative. Then use the radius to find the center of the circle. I just need to plot the center now, Still dont know how

[@virtually_passed]

I coded this up for you. Hopefully it helps:
www.desmos.com/calculator/lbvkjztsmd

[@kodapasy]

@@virtually_passed thanks buds

[@كرتونزمان-ذ9ق]

And if y-- = 0? How I can continue?

[@virtually_passed]

If the double derivative of y is zero then the radius is infinite. This makes sense intuitively because y''=0 has the solution y=ax+b which is a line. The radius of curvature of a line is +- infinity

[@giftcp82]

what application are you using for the visualization. I would like to do the same for a catenary curve

[@virtually_passed]

In this case I used desmos and and some editing in filmora. Nowadays I'd recommend manimCE using python

[@giftcp82]

@@virtually_passed thanks for the feedback

[@rescyy2235]

I solved this problem when I first discovered derivatives, 11th grade in highschool

[@juhanjames2653]

🤓

[@rescyy2235]

@@juhanjames2653 jealous

[@marcelomarinos845]

nice

[@virtually_passed]

Thanks 🙏

[@SDc-e3s]

only mathematical proof given

[@aaaaaa-rr8xm]

I am surprised that this is not about April Fools
Or is it?

[@virtually_passed]

It's not an April fools joke :)

[@aaaaaa-rr8xm]

@@virtually_passed yeah I watched it entirely. Good video.

[@solcarzemog5232]

You're pulling rabits out of a hat with no consequential or motivational thought whatsoever. This is robotic, mechanical and unintuitive.

[@virtually_passed]

Thanks for the feedback: in case you're curious I've made another video proving the same thing in a more intuitive way: th-cam.com/video/vyBkvGnPwJk/w-d-xo.html

[@solcarzemog5232]

@@virtually_passed Oh, thank you! That's what I meant, Very intuitive, clear and convincing! beatuful, thank you

[@Apple-sq4wr]

great video