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Math Help - Circle / Trapezoid Tangent

  1. #1
    mdb
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    Circle / Trapezoid Tangent

    I'm trying to figure out an equations for the distance between the midpoints of a circle and an isosceles trapezoid while they are tangent to each other as the trapezoid rotates. The circle and the trapezoid are centered on the same vertical axis and can only move relative to each other along that axis. See attached picture.
    Attached Thumbnails Attached Thumbnails Circle / Trapezoid Tangent-circle-trapezoid.jpg  
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  2. #2
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    Quote Originally Posted by mdb View Post
    I'm trying to figure out an equations for the distance between the midpoints of a circle and an isosceles trapezoid while they are tangent to each other as the trapezoid rotates. The circle and the trapezoid are centered on the same vertical axis and can only move relative to each other along that axis. See attached picture.
    Since the radius does not vary, it is only required that the distance from the midpoint of the isosceles trapezoid to a side of the trapezoid be determined.

    Let v be the distance from the midpoint to the upper or lower side of the trapezoid and let s be the distance from the midpoint to the sloping sides of the trapezoid.
    Let t be the distance from the midpoint (center) of the trapezoid to any point on the sides of the trapezoid.

    Assume that theta is zero from the center of the trapezoid to the center of the circle, and increases in a clockwise direction.

    Label angle points through the corners of the trapezoid:
    UR, upper right : LR, lower right :
    LL , lower left : UL upper
    and RS as the perpendicular angle to the right side
    and LS as the perpendicular angle to the left side.


    while theta < UR
    t = v/cos(theta)

    while UR < theta < LR
    t = (s/(cos(theta-RS))

    while LT < theta < 180
    t = -v/cos(theta)

    for theta 180 to 360, it's a mirror image of the above.
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  3. #3
    MHF Contributor
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    Aidan, say we label (in the diagram shown) the bottom right corner of
    the trapezoid as point P.
    Would sure be interesting to see the path traced by point P as the
    trapezoid does a full revolution.
    Can you tell what the "shape" of that path will be? Thanks.

    Note: don't mean to hijack your thread mdb; just curious.
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  4. #4
    mdb
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    Aiden-

    It seems that your solution doesn't account for the fact that the point of tangency is probably not on the vertical axis between the center of the circle and center of the trapezoid.

    (See attached picture)

    Thanks
    Attached Thumbnails Attached Thumbnails Circle / Trapezoid Tangent-circle-trapezoid-2-.jpg  
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  5. #5
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    Quote Originally Posted by mdb View Post
    aidan
    It seems that your solution doesn't account for the fact that the point of tangency is probably not on the vertical axis between the center of the circle and center of the trapezoid.
    ...
    Thanks
    You are correct. Totally ignored it.
    The attached image has some additional points identified.
    Assume that the midpoint of the trapezoid is at the origin, and theta is the clockwise angle from the y-axis.
    We are concerned with the distance from the midpoint of the trapezoid to the center of the circle: the dimension "t". As the circle is moved around the trapeziod the dash-dot line is a trace of the center of the circle.

    Ignoring sign:
    when theta is between point 9 and point 2
    or when thete is between point 5 and point6:
    t = (v+r)/cos(theta)

    when theta is between point 3 and point 4:
    t= (s+r)/cos(theta-phi)

    same for point 8 and point 9

    When theta passes through sector 2-3 or sector 4-5 or sector 6-7 or sector 8-9, additional calc's are required: the intersection of a line with a circle.


    mdb: How did you get such a clean/clear image?
    My scan is a jpg at 200dpi. It is not very good.
    Attached Thumbnails Attached Thumbnails Circle / Trapezoid Tangent-scan0001.jpg  
    Last edited by aidan; July 20th 2009 at 03:05 PM. Reason: typo
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  6. #6
    mdb
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    Aidan-

    I think you got it! Thanks.

    I can save high def jpeg files directly from my CAD program.
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