Definition talk:Arc Length

Good page, but I'm seeing this the explanation as a proof rather than as part of the definition. Unfortunately I'm up to my elbows in something else right now and my time is short so I can't address this now. --prime mover 09:21, 1 January 2012 (CST)
 * I'll look into it. Thanks for the compliment. --GFauxPas 09:23, 1 January 2012 (CST)
 * How 'bout now? --GFauxPas 10:24, 1 January 2012 (CST)
 * Very nice. Good job. --prime mover 10:31, 1 January 2012 (CST)

Intuitive bit
I'm looking at:
 * The arc length of a curve can be thought of as how long the graph of the curve would be if you cut it at the points $\left({a, f \left({a}\right)}\right)$ and $\left({b, f \left({b}\right)}\right)$ and then straightened it out. 

and I'm trying to see what's wrong with it, because it doesn't pass the smell test.

I think it's because of the assumption that until it has been straightened out it has no length. "... how long it would be if you ... straightened it out." It's the same length bent as straight, but this sentence seems to suggest differently.

Trouble is, this is something which is intuitively not universal. I remember at age 7 we were all given pieces of string and we were expected to use them to prove that they had the same length curved as they did bent. Stupid exercise but it kept the teacher happy for an afternoon. --prime mover 21:21, 2 August 2012 (UTC)


 * I was under the impression that until we defined arc length, curves don't have a meaningful defined length. If you're saying that's only my point of view and that I'm mistaken, I can cut that part out. --GFauxPas 04:24, 3 August 2012 (UTC)


 * Okay, hadn't thought of it like that. No worries - but I may go away and think about it and suggest a clearer way of putting it. --prime mover 06:17, 3 August 2012 (UTC)


 * Maybe something along the lines (no pun) of travelling along the graph while measuring speed and velocity, and then employ physical intuition to deduce the length. --Lord_Farin 08:56, 17 August 2012 (UTC)


 * Thinking about it, it's probably good as it is. --prime mover 19:45, 17 August 2012 (UTC)