For $2$ circles of an arbitrary size moving at an arbitrary constant speed which are approaching each other head-on, how would I formulate a function to describe the area of their overlap over time ?.
I have tried integrating sine waves, and was suggested by a friend to try a circular convolution ( although I do not know how even after using wikipedia ). Thank you !.
Look here for proof of the area formed by two circle segments $$A(t)=r^2\cos^{-1}(\frac{d^2+r^2-R^2}{2dr})+R^2\cos^{-1}(\frac{d^2+R^2-r^2}{2dR})-\frac{1}{2}\sqrt {(-d+r+R)(d+r-R)(d-r+R)(d+r+R)}$$ when $0<t<\frac{2r}{v_{relative}}$where $d=v_{relative}t$ and $r,R$ are the radii of the circles. The rest is easy.