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Math Help - Line between circles in space?

  1. #1
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    Line between circles in space?

    Say I have 2 circles which are rotating in space, sometimes parallel, sometimes perpendicular to each other. I have a line segment which is attached to a point on one circle, and must remain of a constant length while also touching the other circle. This is better illustrated by the following animation:

    www.versatileartist.com/temp/circles.avi

    I need to find a way to describe the point shown by the yellow cube in terms of the point shown by the green one.

    Can anyone show me how to do this??
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  2. #2
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    Quote Originally Posted by Malkalypse View Post
    Say I have 2 circles which are rotating in space, sometimes parallel, sometimes perpendicular to each other. I have a line segment which is attached to a point on one circle, and must remain of a constant length while also touching the other circle. This is better illustrated by the following animation:

    www.versatileartist.com/temp/circles.avi

    I need to find a way to describe the point shown by the yellow cube in terms of the point shown by the green one.

    Can anyone show me how to do this??
    Your description of the problem leaves a lot of freedom in how to set it up. Here is one of many ways to do it.

    Let the first circle with the yellow box be the set of points

    C_1 = \{ x \in R^3 |\ x_1^2 + x_2^2 = r^2,\ x_3 = 0 \}.

    Let the second circle with the green box be the set of points

    C_2 = \{x' \in R^3 |\ x' = M\bar{x} + b,\ \bar{x}_1^2 + \bar{x}_2^2 = s^2,\ \bar{x}_3 = 0 \}

    where M is a rotation matrix (orthogonal, determinant = 1).

    Let L be the length of the line connecting the boxes and let x be the coordinates of the yellow box and x' be the coordinates of the the green box. Then for a given x' \in C_2, the solution(s) x to

    || x' - x || = L,\ x \in C_1

    "describes" the yellow box x in terms of the green box x'.
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  3. #3
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    Hm, this is definitely beyond anything I've ever studied. The furthest I ever got was high school trigonometry, and that was over 10 years ago.. I don't even know what field of math you are using, so that makes it a bit harder for me to understand!

    Beyond that, I not only have to understand this myself, but understand it well enough to to write in terms my program (3DS Max) will understand.

    What resources can you point me to that might help me understand your notation?
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  4. #4
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    Okay, here is maybe a clearer description:

    www.versatileartist.com/temp/circles.avi

    (program coordinate system: x = right, y = back, z = up)

    I need to find either

    1) The position of point A or
    2) angle B


    static variables:
    length of side b = 20.485
    length of side c = 10.887

    radius of circle 1 = 10.877
    radius of circle 2 = 12.863

    position of point B = [0, 0, 7.384]
    center of circle 2 = [0, -14.737, 0]

    known, changing variables:
    The position of point C is known [x,y,z]
    The length of side a is known
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  5. #5
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    Quote Originally Posted by Malkalypse View Post
    Okay, here is maybe a clearer description:

    www.versatileartist.com/temp/circles.avi

    (program coordinate system: x = right, y = back, z = up)

    I need to find either

    1) The position of point A or
    2) angle B


    static variables:
    length of side b = 20.485
    length of side c = 10.887

    radius of circle 1 = 10.877
    radius of circle 2 = 12.863

    position of point B = [0, 0, 7.384]
    center of circle 2 = [0, -14.737, 0]

    known, changing variables:
    The position of point C is known [x,y,z]
    The length of side a is known
    Three sides a, b and c of a triangle are given. Angle B can be calculated using the Law of sines.
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  6. #6
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    I just realized I made a mistake in describing what I needed, but as a result I am now a bit closer.

    I don't even need the circles, and I can show what I need in a single picture.



    I need the angle ABD.

    I have the lengths for all sides described by triangles ABC and BCD. The only length I do not know is AD.

    Lengths:
    AB = 10.887, AC = 20.485
    BC changes, but is known at any time

    Right Angles:
    ADC and ADB are right angles

    Angles which change but are known
    BCD, BDC, CDB, and CBD

    unknown angles:
    all others
    Last edited by Malkalypse; June 11th 2007 at 10:48 PM.
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  7. #7
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    Quote Originally Posted by Malkalypse View Post
    I just realized I made a mistake in describing what I needed, but as a result I am now a bit closer.

    I don't even need the circles, and I can show what I need in a single picture.



    I need the angle ABD.

    I have the lengths for all sides described by triangles ABC and BCD. The only length I do not know is AD.

    Lengths:
    AB = 10.887, AC = 20.485
    BC changes, but is known at any time

    Angles:
    ADC, ADB, and BDC are right angles

    Angles which change but are known
    BCD, CDB, and CBD

    unknown angles:
    all others
    Only very basic trig is needed here. When you have right angles, use them. ADB is a right triangle, so you can use the Pythagorean Theorem to calculate the unknown side. Then use sin(ABD) = ratio of two of the sides.
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