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Math Help - how to calculate area of horizontal band of a torus surface?

  1. #1
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    how to calculate area of horizontal band of a torus surface?

    I have a ring torus of the following proportions illustrated in the diagram below.
    The diagram shows a cross section of the torus (it is not proportional)

    Inner hole radius = a = 5.920118575210524
    tube radius = r = 82.983609982018226

    I am interested in surface areas.
    I know how to calculate the total surface area of the torus.
    But how can I calculate the area of any horizontal strip of the surface?

    I have a specific example in mind, although I would like to know the methodology too.
    In the diagram there is a strip labelled d, which runs from the inner equator to a point which is c degrees above.
    I have worked out that:
    d=15
    Therefore c = 10.3567040
    How can I calculate the surface area of the torus that this strip d covers as it rotates 360 degrees around the y axis?

    I will appreciate any help anyone can give.
    Thanks,
    Tom
    how to calculate area of horizontal band of a torus surface?-math-q.jpg

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  2. #2
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    Re: how to calculate area of horizontal band of a torus surface?

    Quote Originally Posted by tombarnett View Post
    I have a ring torus of the following proportions illustrated in the diagram below.
    The diagram shows a cross section of the torus (it is not proportional)

    Inner hole radius = a = 5.920118575210524
    tube radius = r = 82.983609982018226

    I am interested in surface areas.
    I know how to calculate the total surface area of the torus.
    But how can I calculate the area of any horizontal strip of the surface?

    I have a specific example in mind, although I would like to know the methodology too.
    In the diagram there is a strip labelled d, which runs from the inner equator to a point which is c degrees above.
    I have worked out that:
    d=15
    Therefore c = 10.3567040
    How can I calculate the surface area of the torus that this strip d covers as it rotates 360 degrees around the y axis?

    I will appreciate any help anyone can give.
    Thanks,
    Tom
    Have a look here: Pappus's centroid theorem - Wikipedia, the free encyclopedia

    (Remark: I'm not sure if this method works in your case properly ... )
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  3. #3
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    Re: how to calculate area of horizontal band of a torus surface?

    Thanks for this - it has been very useful.
    I now understand I could follow the first theorem. Now the question is though - how can I find the geometric centroid of the curve I have described?
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  4. #4
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    Re: how to calculate area of horizontal band of a torus surface?

    Quote Originally Posted by tombarnett View Post
    Thanks for this - it has been very useful.
    I now understand I could follow the first theorem. Now the question is though - how can I find the geometric centroid of the curve I have described?
    1. Let \rho denotes the radius of the circle which is described by the centroid of d during the rotation. Then

    \rho = a + \left(r - r \cdot  \cos \left(\frac c2 \right) \right)

    3. The area is calculated by:

    A = 2 \pi \rho \cdot d

    4. I've got A \approx 589.87897
    Attached Thumbnails Attached Thumbnails how to calculate area of horizontal band of a torus surface?-guldinbsp.png  
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  5. #5
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    Re: how to calculate area of horizontal band of a torus surface?

    Quote Originally Posted by earboth View Post
    1. Let \rho denotes the radius of the circle which is described by the centroid of d during the rotation. Then

    \rho = a + \left(r - r \cdot  \cos \left(\frac c2 \right) \right)

    3. The area is calculated by:

    A = 2 \pi \rho \cdot d

    4. I've got A \approx 589.87897
    Brilliant!
    So the centroid is located at the point intersected by half the angle. That was the part that was confusing me. and so I got the same answer.
    Thank you so much for taking the time to help me out here. Its very kind of you. I am now out of my mathematical impass and can continue on the journey!

    Respect.

    Tom
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