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Math Help - Length of a line displaced by a circle

  1. #1
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    Question Length of a line displaced by a circle

    Hi,

    I hope I have posted this in the correct forum.

    I am a bit of a maths novice and I am having some difficulty calculating the final length of a line fixed between points A and B that is displaced by a circle of radius R.

    The circle is at midpoint line AB (width W) and displaces line upwards a distance (D)

    I need to calculate the minor arc length from D, w and R. I don't know the angle between the radii of the points of tangency, otherwise this would be easy

    I have figured out how to calculate the minor arc length if I know the chord length and the sagitta (Hc) but these will depend on lengths D, w and R which will be known.

    See the attched file for basic diagrams of the problem.

    There must be some expression to calculate the minor arc length or even the chord length in this problem but I just can't figure it out.

    Any help would be much appreciated.

    Thank you.
    Mark
    Attached Thumbnails Attached Thumbnails Length of a line displaced by a circle-line-length-problem.jpg  
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    Last edited by Cartermgg; June 29th 2011 at 04:14 AM.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Length of a line displaced by a circle

    Quote Originally Posted by Cartermgg View Post
    Hi,

    I hope I have posted this in the correct forum.

    I am a bit of a maths novice and I am having some difficulty calculating the final length of a line fixed between points A and B that is displaced by a circle of radius R.

    The circle is at midpoint line AB (width W) and displaces line upwards a distance (D)

    I need to calculate the minor arc length from D, w and R. I don't know the angle between the radii of the points of tangency, otherwise this would be easy

    I have figured out how to calculate the minor arc length if I know the chord length and the sagitta (Hc) but these will depend on lengths D, w and R which will be known.

    See the attched file for basic diagrams of the problem.

    There must be some expression to calculate the minor arc length or even the chord length in this problem but I just can't figure it out.

    Any help would be much appreciated.

    Thank you.
    Mark
    Convert the data on doc file to some image file. Could you do that?
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  3. #3
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    Re: Length of a line displaced by a circle

    Sorry I couldn't figure out how to post the jpeg file in the reply so I have edited the original post.

    Thanks.

    Doh, I've just seen the 'go advanced' button. Forgive me, I'm new to the forum.
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  4. #4
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    Re: Length of a line displaced by a circle

    Quote Originally Posted by Cartermgg View Post
    Sorry I couldn't figure out how to post the jpeg file in the reply so I have edited the original post.

    Thanks.

    Doh, I've just seen the 'go advanced' button. Forgive me, I'm new to the forum.
    1. I've drawn a sketch of the situation as far as I understand it. (see attachment).
    If I'm right you want to know the length of the chord z, right?

    If so:

    2. You are dealing with 2 similar right triangles. Use Pythagorean theorem (eq. #1) and proportions (eq. #2):

    x^2+r^2=y^2

    \frac xr = \frac{y+d}w~\implies~x=\frac rw \cdot (y+d)

    3. Plug in this term into eq. #1 and solve for y. y depends only on r, w, d. Resubstitute afterwards to get x.

    4. The length of z is twice as large as the height in the smaller right triangle. Therefore

    z = 2\cdot \frac{x \cdot r}y
    Attached Thumbnails Attached Thumbnails Length of a line displaced by a circle-sehneaustangenten.png  
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  5. #5
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    Thumbs up Re: Length of a line displaced by a circle

    Quote Originally Posted by earboth View Post
    1. I've drawn a sketch of the situation as far as I understand it. (see attachment).
    If I'm right you want to know the length of the chord z, right?

    If so:

    2. You are dealing with 2 similar right triangles. Use Pythagorean theorem (eq. #1) and proportions (eq. #2):

    x^2+r^2=y^2

    \frac xr = \frac{y+d}w~\implies~x=\frac rw \cdot (y+d)

    3. Plug in this term into eq. #1 and solve for y. y depends only on r, w, d. Resubstitute afterwards to get x.

    4. The length of z is twice as large as the height in the smaller right triangle. Therefore

    z = 2\cdot \frac{x \cdot r}y
    Thanks earboth, I didn't think about proportions.
    I shall have a go and then let you know later how I got on.
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