How to find the polygon enclosing a point from a set of lines?

Question

I have a set of non-intersecting lines, some of which are connected at vertices. I'm trying to find the smallest polygon, if one exists, that encloses a given point. So, in the image below, out of the list of all the line segments, given the point in red, I want to get the blue ones only. I'm using Python, but could probably adapt an algorithm from other languages; I have no idea what this problem is called.

Answer 1

First, remove all line segments that have at least one free endpoint, not coincident with any other segment. Do that repeatedly until no such segment remains.

Now you have a plane nicely subdivided into polygonal areas.

Find a segment closest to your point. Not the closest endpoint but the closest segment, mind you. Find out which direction along the segment you need (your point should be to the right of the directed segment). Go to the endpoint, turn right (that is, take the segment next to the one you came from, counting counterclockwise). Continue going to the next endpoint and turning right until you hit the closest segment again.

Then, check if the polygon encloses the given point. If it is not, then you have found an "island"; remove that polygon and the entire connected component it belongs to, and start over by selecting the nearest segment again. Connected components can be found with a simple DFS.

This gives you a clockwise-oriented polygon. If you want counterclockwise, which is often the default "positive" direction in both the software an the literature, start with the point to your left, and turn left at each intersection.

It surely helps if, given an endpoint, you can quickly find all segments incident with it.