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

    • Thread Starter

    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.


    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.

    Attached Images

    This post in a nutshell: If you try and calculate the angle directly then you don't get much success. However the angle does depend on the ratio between arc length and chord length so it's worth recording that.


    Ok, let's say that your value for the arc length, the average of A and B, is D.

    Let t be the central angle of the curve, and r be the radius, if the curve had been drawn on a circle. See this diagram:


    By the cosine law, http://en.wikipedia.org/wiki/Law_of_cosines C^2=r^2+r^2-2\times r\times r \cos t. Simplified, this gives C^2=2r^2(1-\cos t).

    For the next equation, we have to assume that t is measured in radians instead of degrees. Radians is just another unit of measurement for angles where pi radians = 180 degrees. Radians have a number of useful properties which are needed for this formula to work.

    Using some of the formulas on http://en.wikipedia.org/wiki/Arc_(geometry) we have D=r\times t.

    Now we put these formulas together:

    C^2=2r^2(1-\cos t) and D=rt

    Hence r^2=\frac{C^2}{2(1-\cos t)} and r^2=\frac{D^2}{t^2}

    Therefore \frac{C^2}{2(1-\cos t)}=\frac{D^2}{t^2} and so \frac{1-\cos t}{t^2}=\frac{C^2}{2D^2}

    It would be harder to get t from this. If you really wanted to, there's probably some numerical method that would do it but otherwise let's just work with this.

    If you plot the graph of (1-cos(t))/t^2 http://www.wolframalpha.com/input/?i...i%2C+0%3Cy%3C1 then you'll see that you're likely to get a value slightly smaller than 1/2, where smaller values imply a larger angle.

    This suggests that if C/D is smaller then the curve has a larger angle. So you would probably be best off just recording the ratio C/D for simplicity.
    • Thread Starter

    hi thanks for your reply. so you're saying, the measurements i have made are useful, but there is no use in calculating the angle? what do you recommend for an approximate measurement of curvature?

    The way A and B are measured do not make for good arc lengths! See how A is more curved/wiggly than B, this means when you stretch the distance out to make an arc you will get a longer distance than what you are trying to model.

    Even if you do figure out a good way to use A and B, you will probably get much lower values for curvature than you would expect/if you did it by by hand etc.. You might even get curvature in the wrong direction if A is not really straight between vertebrae. Using straight lines would give a better result (see pic).

    You can see by taking the average of A and B in the second pic approximates to the blue line in the middle. This is the same as what you did before. Just divide the arc length by the chord length to get a "how curved is this" parameter. The higher the ratio the more curved it is. As you can see a circle wouldn't really approximate this well.

    You have enough info to get the angles but not from the lengths alone. If you had the point coordinates (ie start and stop point of each line) you could find the angles not only for each vertebrae but for the whole thing.


    (Original post by wooja978)
    hi thanks for your reply. so you're saying, the measurements i have made are useful, but there is no use in calculating the angle?
    You can use them to get the angle - it's just that you wouldn't be able to calculate it directly. You would have to use trial and improvement or some other method that gives you a sequence of approximations to the angle.
    what do you recommend for an approximate measurement of curvature?
    If you calculate the arc length divided by the chord length then the larger the value you get, the more curvey it is.
    • Thread Starter

    thanks guys you both have been really helpful in explaining things! i just clicked on the spoiler bar and saw that equation. too complicated i think. i wish i had your skills then this would be so much easier. i could get co-ordinates and things but the idea is so foreign to me it'll take me months to get it. i'll stick to arc/chord length for now and see what i get. thanks!
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Updated: November 27, 2011

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