# Overlapping circles

#### Question

Consider two circles of radii $R$ and $r$ whose centres are separated by a distance $d$. The area of overlap between them (assuming $|R-r| \le d \le R+r$) is given by the following formula, derived below.

$$A(d; R, r) = \beta R^2 + \alpha r^2 - \frac{1}{2}r^2\sin 2\alpha - \frac{1}{2}R^2\sin 2\beta,$$ where $$\cos \alpha = \frac{r^2 + d^2 - R^2}{2rd} \quad \mathrm{and} \quad \cos \beta = \frac{R^2 + d^2 - r^2}{2Rd}.$$

Inverting this formula to find $d$ for a given overlap area $A=A_0$ cannot be achieved analytically, but a numerical solution can be found as the root of the function $A(d; R, r) - A_0$.

Write a Python function which takes arguments A, the target overlap area, and R and r the two circle radii and returns d, the distance between the circle centres giving overlap area A. Use, for example, brentq.

#### Derivation

First note that $A=0$ if $d \ge R+r$: the circles do not intersect at all in this case. Also, $A = \pi (\mathrm{min}(R,r))^2$ if $d \le |R-r|$: the smaller circle is entirely enclosed in the larger in this case. For the case of partial overlap, $|R-r| < d < R+r$ the area to find is shaded in the diagram below. The cosine formula gives the angles $\alpha$ and $\beta$:

$$\cos \alpha = \frac{r^2 + d^2 - R^2}{2rd} \quad \mathrm{and} \quad \cos \beta = \frac{R^2 + d^2 - r^2}{2Rd}.$$

The required area may be found as the sum of the two circle segments cut off by the chord CD. For the circle centred at A in the diagram above, its segment is the area of the circular sector ACD minus the area of the triangle ACD:

With $x=\mathrm{AE} = R\cos\beta$ and $h=\mathrm{CE}=R\sin\beta$, triangle ACD has area $xh=R^2\cos\beta\sin\beta=\frac{1}{2}R^2\sin2\beta$. The area of the circular sector ACD is simply $\beta R^2$ with $\beta$ measured in radians (the entire circle has area $\pi R^2$ and we want the fraction $\beta/2\pi$ of it).

Therefore, the shaded circular segment has area $\beta R^2 - \frac{1}{2}R^2\sin2\beta$.

Similarly, the area of the segment of the circle centred at B cut off by chord CD is $\alpha r^2 - \frac{1}{2}r^2\sin2\alpha$.

The total intersection area is therefore

$$A = \alpha r^2 + \beta R^2 - \frac{1}{2}r^2\sin2\alpha - \frac{1}{2}R^2\sin2\beta.$$