Two circles are separated by a distance d and grow radially outwards. Over time, this creates an ‘impinged’ boundary.
![[.gif]](https://i.stack.imgur.com/RFlw2.gif)
How do you calculate the length of the boundary without simulation?
I thought about using the circle-circle intersection formula:
$$a=\frac{1}{d}\sqrt{4 d^{2} R^{2}-\left (d^{2}-r^{2}+R^{2} \right )^2}$$
Then saying R and r are functions of time R(t) and r(t), then taking its partial derivative with respect to R and r. Then solving numerically when R(t) and r(t) are known. But I’m not sure if that is going anywhere?
I saw in this paper https://doi.org/10.1016/S1359-6454(99)00376-6 around eqn 4 a chord construction may be possible but I'm not sure it would work for the case of curved boundary?
That boundary is described by the intersection points of the circles, while their radii grow. Set up a coordinate system such that the center of the left circle is at $O=(0,0)$, while the center of the right circle is at $C=(d,0)$, and let $P(t)$ be the intersection point having $y\ge0$. Then $OP=R(t)$ and $CP=r(t)$ are given and $\angle COP=\theta(t)$ can be computed with the cosine law: $$ \cos\theta(t)={R^2(t)-r^2(t)+d^2\over2dR(t)}. $$ The length of the boundary is then: $$ L(t)=2\int_{t_0}^t\sqrt{R^2(s)\,\dot{\theta\hspace{0pt}}^2(s)+\dot{R}^2(s)}\,ds, $$ where $t_0$ is the time the circles meet for the first time.