[Math] Curvature of Circles in different Radius

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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!!

Answer 1

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.

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.