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I am not a professional in the curvature. So I only think of it intuitively at the moment. So please understand, and please let me know how to think of this correctly.

[1] A circle should have the same curvature everywhere.

[2] And we introduce different 2 circles where they have different radius. Then, do they have different curvatures?

I had an applied subject which involves curvature. In this lecture, we didn't discuss the curvature itself, but we at least were supposed to know basic notion (even without the correct definition) of it where we have some mathematical process which depends on the curvature.

A velocity of process is inversely proportional to the curvature. If I have 2 circles of different radius but with the same center,

[!] As I believed the curvature will be the same, the velocity of the process per 2 circles are the same. Thus they don't collide each other under the process. (Actually this is called Mean Curvature Motion but as I am not professional in the curvature anyway...)

But I was wrong only because the curvature is different in circles of different radius, so their velocity is different, and the contours' distance vary. This was the correct answer. Why I was confused? Because I learned that they have 'Set-Inclusion property' where different sets doesn't collide each other.

So they move in different velocity but they don't collide. If the outer circle motion is faster at the beginning and if the curvature changes, it slows down but they never touch each other, which means there is the upper bound in the distance of contours over the process.

So this 'Set-Inclusion property' holds even though we don't have the same velocity for two circles. They don't collide.

[3] So.. any one can kindly teach me the curvature in moderate level? ( I am sorry... I want to know better this but I still have tons of exams... )

Thank you!!

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closed as unclear what you're asking by Yves Daoust, Shailesh, Lord Shark the Unknown, JonMark Perry, user99914 Feb 10 '18 at 6:15

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Sorry but it is impossible to find our way in your jungle. You are mixing a few unrelated concept and actually do not describe your problem. Problems start with "A velocity of process": ?????? $\endgroup$ – Yves Daoust Feb 9 '18 at 17:45
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Note that the curvature of a circle is defined to be the reciprocal of the radius

$$\kappa = \frac{1}{R}$$

therefore

  • a circle have the same curvature everywhere
  • if we introduce 2 circles with different radius, then they have different curvatures

Note also that for a generic curve the curvature in a point correspond to the curvature of the osculating circle in that point.

enter image description here

Because of the reciprocal relationship, greater is the radius of the osculating circle less is the curvature in that point. Thus, in the limit case, the curvature of a straight line is equal to 0.

enter image description here

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    $\begingroup$ Okay, this is what I wanted. I wanted to have general, and compact introduction to the curvature so that I can apply this to my problem of interest conceptually without too much in depth knowledge. Thanks a lot. $\endgroup$ – Robin Feb 12 '18 at 12:28
  • $\begingroup$ Well done, You are welcome, Bye. $\endgroup$ – gimusi Feb 12 '18 at 12:43
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The curvature of a circle is the reciprocal of its radius. For example if the radius is 5,the curvature is 1/5.

So different circles may have different curvatures based on their radii. The smaller the radius, the larger the curvature.

Basically you are measuring the rate of change of the unit tangent vector with respect to the arc-length.

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