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Math Help - Determining the diameter of a hole within a circle

  1. #1
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    Determining the diameter of a hole within a circle

    Hi,

    Can someone nudge me in the right direction to solving this problem?
    I have a circle with a circular hole in it. I keed to find the diameter of the hole (d)

    1. The circle diameter is D
    2. The center coordinates of the hole is unknown
    3. The hole can be anywhere within the circle
    4. The hole is not in the center of the circle

    Do you think it is possible to calculate the diameter of the hole by
    creating two chords from a fixed coordinate on the circumference of
    the circle that run tangental to, and on either side of the hole?

    Would it be correct to suggest that
    c1 + c2 <= 2D
    Where c1 and c2 are the lengths of the two chords
    D is the diamater of the circle

    Am I on the right track? Where do I go from here?

    Any help is appreciated.
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  2. #2
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    Dear clearfix,

    Well I don't know if my suggestion would help you. But I thought to give a try.
    First you could draw a tangent to the hole from any point in the circumference. Then you could draw another tangent to the hole parallel to the tangent you had drawn. Calculating the perpendicular distance between the tangents will give you the diameter of the hole.

    Hope this helps.
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  3. #3
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    Hi Sudharaka,

    Thanks for your reply. I did think about it, but the problem with parallel lines is practicality. I have a practical need to work out this problem, and all I can work with is the diameter of the circle, and points about it's circumference.

    I'm still thinking along the lines of the chords. I do hope someone can shed some light.

    Thx again.
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  4. #4
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    Here is a way; maybe you can apply it to your practical case.

    Look at the enclosed picture. In red are chords you can find, if I understood well: starting from A, draw the two chords tangent to the hole, they hit the circumference at B and C. From B, draw the second chord tangent to the hole, it hits the circumference at D.

    You can find the angle \widehat{BAC} by measuring the distance along the circle from B to C: the length \stackrel \frown {BC} of the arc from B to C satisfies \frac{D}{2}\widehat{BAC}=\stackrel \frown {BC}. Then you deduce the value of \alpha (on the picture): 4\alpha=\widehat{BAC} hence \alpha=\frac{\stackrel\frown {BC}}{2D}. Similarly, \beta=\frac{\stackrel\frown {AD}}{2D}.

    Then, if \ell_A=AI, \ell_B=BI (cf. picture), you have \tan\alpha=\frac{d}{2\ell_A} and \tan\beta=\frac{d}{2\ell_B}. And you can probably find the length AB=\ell_A+\ell_B in your practical case? Then AB=\ell_A+\ell_B=d\left(\frac{1}{2\tan\alpha}+\fra  c{1}{2\tan\beta}\right), and you get d from there.
    Attached Thumbnails Attached Thumbnails Determining the diameter of a hole within a circle-mhf.jpg  
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  5. #5
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    Hi Laurent,

    Thank you for your reply. I beleive this is what I am looking for. I was reading an article about Non Right-angled Triangles and the proof of The Sine Rule, but I couldn't see a connection to what I needed.

    To further my interest, I will "try" to expand on your diagram by creating another vertex, continuing from D. I'd like to know if at some point, a conection to your point C will eventuate. And if so, whether or not the number of vertices can be mathematically determined. And finally whether or not there is any connection to the to radius of the hole.

    Ultimately, what I'd like to see is something like d = D/n ; where d is the diameter of the hole, D is the diameter of the circle, and n is the number of vertices. I ca't beleive that it would be this simple, but it would be nice to see something along these lines.

    It seems like a task and a half, but I'm sure it will be interesting never-the-less.

    Thanks again.
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  6. #6
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    Quote Originally Posted by clearfix View Post
    To further my interest, I will "try" to expand on your diagram by creating another vertex, continuing from D. I'd like to know if at some point, a conection to your point C will eventuate. And if so, whether or not the number of vertices can be mathematically determined.
    In general, the polygon you're drawing won't ever close, i.e. eventually come back to C; I don't know if a criterion is known for that to happen. However, there is a very nice (and nontrivial at all) theorem called "Poncelet's porism" telling you that if the polygon closes (starting from point A), then you get the same result when you start from any other point, and the number of sides is always the same. This theorem even generalizes to conics.
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