# Thread: A puzzle of 2 intersecting circles

1. ## 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!

2. ## 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 $\theta$.

3. ## 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.

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.

4. ## 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.