I am interested in finding a closed-form formula to calculate the area of intersection between two circles with known radii, $r_1$ and $r_2$, where $r_1 \geq r_2$, and a known distance $d$ between their centers. Importantly, I need this formula to work even in scenarios where the distance $d$ exceeds the sum of the radii $r_1 + r_2$, or is less than their difference $r_1 - r_2$.
I have searched online for solutions, but most of them do not account for cases where $d$ exceeds the sum of the radii ($d \geq r_1 + r_2$) or is less than their difference ($d \leq r_1 - r_2$). For example, solutions like the one shown in https://dassencio.org/102 do not work in these scenarios.
Can someone provide such a formula or suggest an approach to derive it? Any insights or references to existing solutions would be greatly appreciated. Thank you.