All,
I'm not sure how to phrase this. I've calculated the area of overlap of two circles of radius 1, with the first circle centered at $x=0$ and the second at $x=2d$ as $$A = 2\arccos(d) - 2d\sqrt{1-d^2}$$
I'm pretty confident that this is correct. The circles touch at $x=d$, the midpoint of the centers. This is a convex function for $d\in[0,1]$, which I'm struggling to understand. That means that if I slide a circle from $x=2d$ to $x=-2d$ there will be a cusp in the graph of area, which seems unintuitive?
Is this true? I feel like it should be a smooth curve...
Here is a graph I've generated of this case in MATLAB GRAPH