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Math Help - equation of a circle from 2 circles and a point?

  1. #1
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    equation of a circle from 2 circles and a point?

    My husband is trying to build a robot that follows you around. We are trying to triangulate the position of a point with three sensors, and we have only gotten this far.

    This image is not at all to scale. In fact, circle C should be much much larger than circle D. Oops.
    B is the person.
    The origin is the lower left corner of the image.
    A, C, and D are sensors on the robot.
    We know the points A, C, and D.
    We know the radii of circles C and D by measuring the difference in time it took our ultrasonic signal to travel from B to A, C, and D.
    We know the equations of circles C and D because we know the centerpoints and the radii.

    We do not know the equation of circle B.
    We do not know the points of intersection of circle B and circles C and D.

    Can we calculate the equation of circle B given the equations of C and D and the point A? I suspect there is some way to do this, perhaps by solving these equations simultaneously, perhaps using the equation of the curvature of the arc made by A and the intersections of circle B and circles C and D.

    Any ideas? I know there is somebody out there smarter than me who can figure this out!

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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Ok

    Quote Originally Posted by erinspice View Post
    My husband is trying to build a robot that follows you around. We are trying to triangulate the position of a point with three sensors, and we have only gotten this far.

    This image is not at all to scale. In fact, circle C should be much much larger than circle D. Oops.
    B is the person.
    The origin is the lower left corner of the image.
    A, C, and D are sensors on the robot.
    We know the points A, C, and D.
    We know the radii of circles C and D by measuring the difference in time it took our ultrasonic signal to travel from B to A, C, and D.
    We know the equations of circles C and D because we know the centerpoints and the radii.

    We do not know the equation of circle B.
    We do not know the points of intersection of circle B and circles C and D.

    Can we calculate the equation of circle B given the equations of C and D and the point A? I suspect there is some way to do this, perhaps by solving these equations simultaneously, perhaps using the equation of the curvature of the arc made by A and the intersections of circle B and circles C and D.

    Any ideas? I know there is somebody out there smarter than me who can figure this out!



    Unless I am misunderstanding your question if you know the length from the center of circe B to the center of circle D/C ...and you know the centers of either of those circles then just do this d=\sqrt{(x-x_0)^2+(y-y_0)^2}....letting (x_0,y_0) be the center of the circle you know...then solving this simultaneously with the fact that you know the radius...now here is where it gets tricky...since you know that circle C and circle D intersect but you dont know the intersections make a very good guess then with that guess say its (x_{00},y_{00}) then r=\sqrt{(x-x_{00})^2+(y-y_{00})^2} where r is the radius
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  3. #3
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    Quote Originally Posted by Mathstud28 View Post
    Unless I am misunderstanding your question if you know the length from the center of circe B to the center of circle D/C ...and you know the centers of either of those circles then just do this d=\sqrt{(x-x_0)^2+(y-y_0)^2}....letting (x_0,y_0) be the center of the circle you know...then solving this simultaneously with the fact that you know the radius...now here is where it gets tricky...since you know that circle C and circle D intersect but you dont know the intersections make a very good guess then with that guess say its (x_{00},y_{00}) then r=\sqrt{(x-x_{00})^2+(y-y_{00})^2} where r is the radius
    I made a typo in the image -- we don't know anything about circle B including the center and radius. However, I think we can apply the solution to Apollonius' Tangency Problem to solve this. I haven't tried it yet though. This solution was suggested to me by someone on another forum.

    Apollonius' Tangency Problem
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Yeah

    Quote Originally Posted by erinspice View Post
    I made a typo in the image -- we don't know anything about circle B including the center and radius. However, I think we can apply the solution to Apollonius' Tangency Problem to solve this. I haven't tried it yet though. This solution was suggested to me by someone on another forum.

    Apollonius' Tangency Problem
    that would work but that is a lot of work
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  5. #5
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    Quote Originally Posted by Mathstud28 View Post
    that would work but that is a lot of work
    Do you think that there is an easier solution?
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