I am trying to implement a radial/angular sweep algorithm that finds all segments (segments are disjoint) that are visible from one particular point in $O(n \lg n)$. So I am trying to order vertices by their angle with point P that we are checking visibility from. But I am uncertain with which point to start first with. If I am selecting them by their angle I have the following situation:
These are the first two points that are being checked:
But the problem is that I can't check them with lines before and thus they seem to be visible by the algorithm I found online $O(n \log n)$ algorithm for disjoint segment visibility problem. Those lines will end before the first line that's covering them ends so all 4 of them should be visible by this algorithm. So which criteria should I use to start my angular sweep algorithm?
Algorithm so far: I sorted points by the polar angle they make with initial point P. The approach is to start with a minimum degree. I am adding them to the binary search tree which has a custom comparator that adds them by comparing that segment with all other segments in a binary tree. If the segment we're inserting is $T$ and $O$ is comparing the segment, I'm checking whether $PT_{start}$ and $O$ have intersections. If they don't intersect then $T$ should be before $O$ in BST. That segment is visible. The problem is when I encounter lines like in the pictures above. It marks right segments as visible because there is no first line in BST so it can't check them with the biggest one on the left. I have no idea what should be my starting point and how to differ those problems.
