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Curvature intuition

An introduction to curvature, the radius of curvature, and how you can think about each one geometrically. Created by Grant Sanderson.

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  • hopper jumping style avatar for user Nils Petter
    Is this the same type of curvature physicists mean, when they talk about the curvature of space (space-time)? If so, they talk about negative curvature, which seems impossible with the definition: k = 1 / R, with R positive (and also with k = || dT/ds ||). Great video! :)
    (12 votes)
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    • male robot hal style avatar for user Gal Barkai
      Not really. Without getting too much into it, "physical" curved space is modeled using a non euclidean, topological, metric space. In a non euclidean space the Pythagorean theorem does not hold - which intuitively could be described as a space where the shortest path between two points isn't a straight line, but a curved one. General relativity asserts that the only reason we think we live in a euclidean space is because the mass around us bends the very fabric of space (and time) around us.

      I guess you could look at it as some kind of generalization of this for Riemannian manifolds, but it's not exactly analogues. Sorry if this sounds a bit vague, but this is a really confusing concept as is (not to mention one I'm not overly familiar with!), so trying to describe it without diving too much into advanced math is a bit hard.
      (26 votes)
  • piceratops sapling style avatar for user James
    am I the only one who did not understand his car analogy at all? what does he mean the car is turning? if the road is the curve wouldn't he just be going straight on the road and its like he's going up or downhill? So confused
    (3 votes)
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    • leafers ultimate style avatar for user Charles Nalbandian
      The graph is like a bird's eye view, looking down from above. He's drawing curved line on the road and you are driving tracing that curved line. It's not uphill or downhill.

      He is driving and trying to trace out the curved line so he has to turn the steering wheel. Since the steering wheel is turned and if it get stuck at that turned position, the car would be curving and it wouldn't be going straight on the road. He mentions that steering wheel isn't straight and you get stuck in that state at .
      (6 votes)
  • leaf yellow style avatar for user Sunil Subramanian
    How do we know that the car traces out a circle?
    (4 votes)
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    • leafers ultimate style avatar for user Charles Nalbandian
      If you steer the handle and kept it at the same place and you push down the gas pedal and kept it at the same speed, the car will trace out a circle. You can try it at the big open place or by using a remote control car. The curve angle, or how much it curves, is going be be consistent because the steering wheel is fixed at certain steered angle. Constant speed and constant curve angle traces out a circle.
      (3 votes)
  • male robot hal style avatar for user GeniusKid88
    Can the curvature be calculated by the absolute value of the second derivative, or concavity?
    (4 votes)
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    • piceratops ultimate style avatar for user declanhoban
      I am definitely a bit late, but I looked it up and it seems one definition of curvature is that if you have a unit tangent vector on a curve, the derivative of that tangent vector with respect to time (as the vector moves along the curve) is the curvature. So in a way, I think the second derivative notion is correct.
      (1 vote)
  • blobby green style avatar for user HI
    How do you determine positive curvature and negative curvature?
    (3 votes)
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  • old spice man green style avatar for user Tushar
    Is he the person from the youtube channel 3Blue1Brown?
    (2 votes)
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  • purple pi purple style avatar for user Aneesh Vasudev
    What the? I didn't get this at all! Why is the center of the green circle moving? The blue curve is a circle, right? And at a certain point of the center of the green circle, the 2 circles (curves) seem to coincide...? somewhere around .
    (1 vote)
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    • leafers tree style avatar for user Erfinder99
      The blue curve is actually not a circle, otherwise the curvature would be the same everywhere and the green circle would stay the same.
      The two curves almost coincide when the red point ist at the top but there's still a noticable difference where the blue curve is close to the x-axis.

      The center of the green circle is moving because the curvature of the blue curve isn't the same everywhere, so the radius of the green circle changes. SInce its edge always touches the red point, its center has to move.
      (4 votes)
  • leaf green style avatar for user tinfungster
    Is the radius of curvature perpendicular to the tangent line of the curve at the point? Why is the formula of curvature the inverse of the radius of curvature?
    (1 vote)
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  • blobby green style avatar for user Tarang Janawalkar
    What is the equation of the curve he has pre graphed?
    (1 vote)
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    • starky sapling style avatar for user Chuck Cleason
      I think it is some sort of cycloid, which is a type of curve that is made by a point on the rim of a rotatinc circle above an axis (in this case, x). I guess it is a fitting example for curvature, because the curve is made out of rotating a steering wheel :)
      (1 vote)

Video transcript

- [Voiceover] Hello everyone. So what I would like to do here, is talk about curvature. And, I've drawn on the X Y plane here, a certain curve. So this is our X-axis. This is our Y-axis. And this is a curve running through space. And I'd like you to imagine that this is a road of some kind, and you are driving on it. And you are at a certain point, so let's say this point right here. And if you imagine, what it feels like to drive along this road, and where you need to have your steering wheel, you're turning it a little bit to the right. Not a lot, because it's kind of a gentle curve at this point, you're not curving a lot. But the steering wheel isn't straight, you are still turning on the road. And now, imagine that your steering wheel's stuck. That it's not gonna move. And however you're turning it, you're stuck in that situation. What's gonna happen, hopefully you're on an open field or something, because your car is going to trace out some kind of circle. Right? You know your steering wheel can't do anything different, you're just turning it a certain rate, and that's gonna have you tracing out some giant circle. And this depends on where you are, right? If you had been at a different point on the curve, where the curve was rotating a lot, let's say you were back a little bit towards the start. At the start here, you have to turn the steering wheel to the right, but you're turning it much more sharply to stay on this part of the curve, than you were to be on this relatively straight part. And the circle that you draw as a result, is much smaller. And this turns out to be a pretty nice way to think about a measure, for just how much the curve actually curves. And one way you could do this, is you can think, "Okay, what is the radius of that circle, the circle you would trace out if your steering wheel locked at any given point.". And if you kind of follow the point, along different parts of the curve, and see, "Oh, what's the different circle that my car would trace out if it was stuck at that point.". You would get circles of varying different radii. Right? And this radius actual has a very special name. I will call this "R". This is called the "Radius of Curvature". And you can kind of see how this is a good way to describe, how much you are turning. You may have heard with a car, descriptions of the turning radius. You know, if you have a car with a very good turning radius, it's very small, 'cause what that means if you turned it all the way you could trace out just a very small circle. But a car with a bad turning radius, you know, you don't turn very much at all, so you'd have to trace out a much larger circle. And, curvature itself isn't this "R". It's not the radius of curvature. But what it is, is it's the reciprocal of that, it's one over "R". And there is a special symbol for it. It's kind of a "K", and I'm not sure in handwriting I am going to distinguish it from an actual "K", maybe give it a little curly there. It's the Greek letter Kappa. And this is curvature. And I want you to think for a second, why we would take one over "R". "R" is a perfectly fine description of how much the road curves. But why is it that you would think one divided by "R", instead of "R" itself? And the reason basically, is you want curvature to be a measure of, how much it curves in the sense that, more sharp turns should give you a higher number. So, if you're at a point, where you are turning the steering wheel a lot, you want that to result in a much higher number. But, radius of curvature will be really small, when you are turning a lot. But if you are at a point that's basically a straight road, you know, there's some slight curve to it, but it's basically a straight road, you want the curvature to be a very small number. But in this case, the radius of curvature is very large. So it's pretty helpful to just have one divided by "R", as the measure of how much the road is turning. And, in the next video, I am going to go ahead and start describing a little bit more mathematically, how we capture this value. 'Cause as a loose description, if you are just kind of drawing pictures, it's perfectly fine to say, "Oh yeah, you imagine a circle that's kind of closely hugging the curve, it's what your steering wheel would do if you were locked.". But in math, we will describe this curve parametrically. It'll be the output of a certain vector valued function. And I wanna know how you could capture this idea. This, one over "R" curvature idea, in a certain formula. And that's what the next few videos are going to cover.