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
@@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 :-)
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
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.
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
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.
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
Thanks for the feedback: in case you're curious I've made another video proving the same thing in a more intuitive way: ua-cam.com/video/vyBkvGnPwJk/v-deo.html
Now I wonder what curve the centre of the approximating circles trace out?
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
@@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 :-)
@@Gauteamus Maybe hypercycloids? They seem very similar to hyperbolae to me
@Gauteamus That'd be the so-called evolute, see e.g. en.wikipedia.org/wiki/Evolute
Thank you so much ❤. Amazing explanation
wow... this was awesome
Wow, I see we make similar videos, but you've been at it for much longer than I have. Keep up the good work :)
Oh my god the music the feel... Everything makes a fresher cry
Great video! I love doing little diff eq curiosities like this
Me too! :)
Amazing video!
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
I agree. That was the motivation for making this video :)
dhanyavaad mere bade bhai, bahut hi sundar tarke se apne iss topic ko ujagar kiya "mai apka dilataht se thankwaad krta hu "
Came from your previous video on this glad to come thank you sir for amazing explanation
Vert interesting. Good job man.
Brillant. I did not make efforts to deduce this formula in exercises.
Thanks so much for the amazing explaination!
great video
thanks
I am very much satisfied by this now
Nicely explained 😊
Thanks a lot 😊
Amazing video
Glad you think so!
just wow man
thanks man it really helps.
Glad to hear it!
Very cleaver derivation, must I say!
Much to learn I sill have!
beautiful :3
I'm glad you like it :3
Taking differentiation assumes dr/dx=0 but r=radius of curvature and is dependent on how you move on the number line...
thanks!
great man! helping me with my homework!
Happy to help!
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.
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
Thank you 😊😊👍🏿
You’re welcome 😊
what application are you using for the visualization. I would like to do the same for a catenary curve
In this case I used desmos and and some editing in filmora. Nowadays I'd recommend manimCE using python
@@virtually_passed thanks for the feedback
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.
And if y-- = 0? How I can continue?
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
I solved this problem when I first discovered derivatives, 11th grade in highschool
🤓
@@juhanjames2653 jealous
nice
Thanks 🙏
I am surprised that this is not about April Fools
Or is it?
It's not an April fools joke :)
@@virtually_passed yeah I watched it entirely. Good video.
You're pulling rabits out of a hat with no consequential or motivational thought whatsoever. This is robotic, mechanical and unintuitive.
Thanks for the feedback: in case you're curious I've made another video proving the same thing in a more intuitive way: ua-cam.com/video/vyBkvGnPwJk/v-deo.html
@@virtually_passed Oh, thank you! That's what I meant, Very intuitive, clear and convincing! beatuful, thank you
great video