I was going to write that it's almost impossible that there is a closed O(1) formula for this, but after reading that the egg of Fritz Hügelschäffer is a cubic, I think there is hope. If your yellow curve is also a cubic, then it's solvable, but the formula for the exact solution is horrible. https://en.wikipedia.org/wiki/Cubic_equation#Cardano's_formu...

I prefer the lazy approach :) . The only hard step is 5B. How are you computing the intersection? Are you using a linear search in the angle? Can you use binary search? Probably the secant method is faster https://en.wikipedia.org/wiki/Secant_method but binary search is more foolproof.