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 $\displaystyle r$ and distance of centres separation $\displaystyle 2d$. I have attached a diagram of this shape as a PNG.

When angle $\displaystyle \alpha = \pi/2$ radians, the shape is a single circle, because the two circles intersect completely. When $\displaystyle \alpha = 0$ radians, the shape is two independent circles which are just touching each other.

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

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

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

Where

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

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

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

Any ideas on an analytical or numerical solution very welcome!

Thanks!