Measuring central angle of a curve given arc and chord length (approximate)

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November 26th 2011, 04:23 PM
wooja978

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Measuring central angle of a curve given arc and chord length (approximate)

Hello math experts, thanks for giving me the opportunity to post here.

My maths education is limited to a 16 yr old's so your help is much appreciated.

In my attachment, I have measured A, B and C in 112 different subjects. Using these measurements, I hope to come up with an approximate index that can be used to compare between subjects.

To approximate the arc length, I am going to average A and B. C will be the chord length of that curve. My question is: how can I measure the central angle of that curve given these values? It does not have to be exact, just an estimation that can be used to subjectively describe if one subject is more curved than the other.

In fact, it doesn't even have to be the central angle. Just a discrete value of some sort. Even something as simple as a ratio?

Please help! Thank you.

Jason
November 26th 2011, 06:14 PM
TKHunny

First, let's be particularly clear what A, B, and C are. Are A and B the blue irregular curved distances? Is C the distance between the midpoints of the two short straight segments?

Is B the outer blue trace or some linear distance related to that?

Is A the inner blue trace or some linear distance related to that?

C appears to be the linear distance between the midpoints of the two straight segments and the straight segments are included in neither A nor B?
November 26th 2011, 06:45 PM
wooja978

Re: Measuring central angle of a curve given arc and chord length (approximate)

hi TKHunny, thanks for replying! Sorry I should have been clearer, I just didn't think of it. All your assumptions are CORRECT. B is the outer trace and A the inner trace. C is the linear distance between the midpoints of the straight segments, and the straight segments are NOT included in neither A nor B. any ideas?
November 26th 2011, 10:43 PM
earboth

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Re: Measuring central angle of a curve given arc and chord length (approximate)

Quote:

Originally Posted by wooja978

hi TKHunny, thanks for replying! Sorry I should have been clearer, I just didn't think of it. All your assumptions are CORRECT. B is the outer trace and A the inner trace. C is the linear distance between the midpoints of the straight segments, and the straight segments are NOT included in neither A nor B. any ideas?

I've added a few lines to your X-ray image which I would use to calculate (an approximate value of) the central angles of the two arcs.

Could you provide us with the lengthes of

the two green lines?

the value of the $\angle(\alpha)$

or at leat with the lengthes of the two yellow lines?
November 27th 2011, 06:57 PM
wooja978

Re: Measuring central angle of a curve given arc and chord length (approximate)

hey thanks for the replies guys, i've decided to use A+B/ chord length I think it's good enough for now