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Math Help - A puzzle of 2 intersecting circles

  1. #1
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    A puzzle of 2 intersecting circles

    Dear community, this is my first post on mathhelpforum and I would be really very grateful for some assistance with this problem. I am going quite crazy!

    It concerns a simple 2D shape which is made of 2 intersecting circles of equal radius r and distance of centres separation 2d. I have attached a diagram of this shape as a PNG.
    When angle \alpha = \pi/2 radians, the shape is a single circle, because the two circles intersect completely. When \alpha = 0 radians, the shape is two independent circles which are just touching each other.

    I have come up with expressions for the total area A, and total circumference C of this object:

    A=2(A_{1}+A_{2})=2r^{2}(\pi-\alpha)+2d\sqrt{r^{2}-d^{2}}

    C=4\pi r(1-\frac{\alpha}{\pi})

    Where

    cos(\alpha)=\frac{d}{r}

    So, given radius r, and the distance 2d separating the centres of the circles, using the above expressions, you can easily calculate the total area A, and the circumference C.

    I however need to solve the inverse problem. That is, I know the total circumference C, and the total area A, and I instead would like to calculate value pairs of radius r and separation distance d which fulfill these constraints.

    Any ideas on an analytical or numerical solution very welcome!
    Thanks!
    Attached Thumbnails Attached Thumbnails A puzzle of 2 intersecting circles-mhf.png  
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  2. #2
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    Re: A puzzle of 2 intersecting circles

    If you think about a different angle things can look a lot simpler.

    I get

    P=2r\theta

    and

    A=r^2(\theta+\sin\theta)


    The B in my diagram is \theta in the above formulae.

    Even if you know P, A and r I think you'll need a numerical method to find \theta.
    Attached Thumbnails Attached Thumbnails A puzzle of 2 intersecting circles-mhf2.png  
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  3. #3
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    Re: A puzzle of 2 intersecting circles

    Thanks tutor for taking the time to respond to my post.

    However, I think taking the path you propose ultimately leads to the same problem I have.
    I will illustrate the problem with a graph.

    A puzzle of 2 intersecting circles-ca.png

    The graph shows two circles which are separated by a fixed distance of d=1. It shows what happens to the total area A (blue line) and circumference C (green line) of the circle pair as the radius r is increased from 1.
    You can see that by fixing d, their are only certain distinct (C, A) value pairs which lead back to a value of the radius r.

    It is thus not possible to get a full reverse mapping from an arbitrary (C, A) pair to a value for r for a fixed d, because only 1 distinct (C,A) pair will lead back to a value of the radius r.

    You could relax the constraint that you are fixing d, but then you have to numerically solve a 3d function, and this does not appear straightforward to me.
    In the case you propose, I think it is the same: you are trying to solve in terms of r and theta, whereas I am trying to solve in terms of r and d. Both cases require numeric solution in 3d.

    It is amazing that a task seemingly simple can turn out to be so complex.
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  4. #4
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    Re: A puzzle of 2 intersecting circles

    Actually I had a sign error but other than that, yes it's the same problem. I just thought my small change made the area and perimeter formulae a little simpler.
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