Determine non-convex hull of collection of line segments

Question

I have a computational geometry problem that I feel should have a relatively simple solution, but I can\'t quite figure it out.

I need to determine the non-convex outlin

Answer 1

1. Pick a safe starting point. Can be e.g. the endpoint with maximum x.
2. March along the line segment.
3. Upon encountering any intersection, always turn left and march along this new segment.
4. Upon encountering an endpoint, record it. Goto 2.
5. Stop when you have returned to your starting point. Your list of recorded endpoints now makes up the ordered list of vertices of your concave hull.

NB: This will fail if there is a "free-floating" outlying line segment that does not intersect any other line segment. However, you specify that "the bars uniquely define a solution", which rules out this fail condition. (Outlying segments make possible two distinct solutions.)

EDIT ... or rather, outlying segments can make two distinct solutions possible -- depending on the exact layout. Proof: Below is an example where the yellow segment that I added makes two solutions possible (blue and grey horribly hand-drawn lines). Were the yellow segment orientated perpendicular to the way it's drawn now, only one solution would be possible. Sounds like your problem is poorly defined.

EDIT Actually this can also fail if your segment collection is "very concave", i.e. if there are endpoints tucked away in recluse corners of your pile of segments. In the figure below I added a black segment. My algorithm would illegally join its endpoint to another endpoint (dashed grey line). I'll leave my answer be in case others are inclined to build upon it.

EDIT after giving this some more thought: Even in the "very concave" case, this solution will definitely give you all of the points of your concave hull in the proper order, but they may be interspersed with extra, inappropriate points such as the black one. So there may be too many points.

The answer is then, of course, to do some pruning. It would be fairly complicated pruning especially if you can have multiple, consecutive "recluse points" like the black one, so I don't have a smart algorithm in mind. But even blind, brute force could be feasible. Each point can be either accepted or rejected (boolean), so if you have N properly ordered candidate points in your concave hull, then there are only 2^N possibilities to check. This is way, way fewer possibilities than brute force for your original problem of permutations, which would have SUM of (n!/(n-k)!) for k=1:(n-1) possibilities (pardon my notation). So this algorithm narrows down your problem significantly.

I think this is the way to go.