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Curvature of a circle approaches zero as radius goes to infinity

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BOAS
#1
Apr15-14, 06:33 AM
P: 271
Hello,

this isn't a homework problem, so i'm hoping it's okay to post here.

I would like to know the correct way to mathematically express the idea in my title. It is intuitively obvious that as the radius of a circle increases, it's curvature decreases.

I looked it up and found that the curvature of a circle is equal to the reciprocal of it's radius. Certain assumptions are often made when looking at lenses, i.e the wave fronts reaching the lens are parallel, or have 0 curvature - In other words, the object distance is infinitely far away.

But, 1/∞ ≠ 0

So how do I express it properly?

In words, I think it goes something like this - As the radius tends towards infinity, the curvature of the circle tends towards zero.
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jedishrfu
#2
Apr15-14, 06:53 AM
P: 2,984
Wouldn't you just use the lim 1/r expressions with r-> infinity to express it?
BOAS
#3
Apr15-14, 07:01 AM
P: 271
Quote Quote by jedishrfu View Post
Wouldn't you just use the lim 1/r expressions with r-> infinity to express it?
That would be my guess but i'm unsure of how to formulate that.

[itex]lim_{r \rightarrow ∞} \frac{1}{r} = 0[/itex]

Like that?

jedishrfu
#4
Apr15-14, 07:20 AM
P: 2,984
Curvature of a circle approaches zero as radius goes to infinity

Yes thats the way I'd express it.
7777777
#5
Apr16-14, 08:37 AM
P: 18
If you imagine a circle with infinite radius, then its circumference is also infinite.
Then what would be the value of pi be? Infinite divided by infinite. Can you say what
it is?
I think the real projective line may be a picture of this kind of "circle":
http://en.wikipedia.org/wiki/Real_projective_line
Mark44
#6
Apr16-14, 09:02 AM
Mentor
P: 21,283
Quote Quote by 7777777 View Post
If you imagine a circle with infinite radius, then its circumference is also infinite.
Then what would be the value of pi be?
The same as always. ##\pi## is a constant (its value never changes).
Quote Quote by 7777777 View Post
Infinite divided by infinite. Can you say what
it is?
No. There are several indeterminate forms, including [∞/∞], [0/0], [∞ - ∞], and a few others. These are indeterminate, because you can't determine a value for them.

They usually come up when we are evaluating limits of functions.
Quote Quote by 7777777 View Post
I think the real projective line may be a picture of this kind of "circle":
http://en.wikipedia.org/wiki/Real_projective_line


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