Say we have a circle of infinite radius. If we zoom in infinitely on its perimeter, we should end up looking at a straight line. Intuitively. But, such a line, I believe, should have a curvature of $1/∞$. But a straight line is defined to have $0$ curvature in Euclidean geometry. My interpretation also has problems if we consider parallel lines. Can someone guide me on this?
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1What does "circle of infinite radius" even mean? – hmakholm left over Monica Nov 04 '17 at 04:56
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"Zooming in infinitely" : you really need to make things more precise. – Sarvesh Ravichandran Iyer Nov 04 '17 at 04:59
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Um, if I was looking at a part of the perimeter of a circle on a flat plane, and then kept increasing the radius such that it tended to infinity, while keeping my view fixed on the portion I was observing, wouldn't it converge to a straight line? – user406287 Nov 04 '17 at 05:11
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"tending to infinity" is not "equal to infinity". It will "tend" to a straight line. But it will never achieve it. And it will never achieve infinite radius either. – fleablood Nov 04 '17 at 05:31
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"circle of infinite radius"
Meaningless.
"If we zoom in infinitely"
Meaningless.
"we should end up looking at a straight line"
A straight line that is a magnification of a single point. Which is meaningless.
$\infty$ is not a number or a value. It is a concept to describe values that may be arbitrarily large and have no upper bound.
All circles have finite radius. But for any value $M$ we can have a circle with radius $> M$. So the radius becomes large the curvature becomes smaller. We state that as $\lim_{r\to \infty} Curvature (r) = 0$ (and similarly $\lim_{x \to \infty} \frac 1x = 0$). But that does not mean infinite radius or $0$ curvature or $\frac 1 {\infty}$ ever occur. They don't and can't.
But yes for a circle with extremely huge radius will have extremely small curvature and the arcs will become arcs that differ from a line segments only very slightly. But it is never infinite radius or zero curvature. And although the arc will differ from straight lines only minisculely, they will differ.
fleablood
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No, it can't. A straight line can't be considered as part of a circle.
I can remember from my algebra class (long time ago), that at some point we had to take the limit of $x$ to zero. And the teacher was screaming: we never get to zero (my ears are still ringing). To give you an example, for $f(x)=\frac{1}{x}$ we have: $$ \lim_{x\rightarrow0}f(x)=\lim_{x\rightarrow0}\frac{1}{x}=\infty, $$ but $f(x)$ does not exist for $x=0$.
EdG
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