# Detecting polygon self intersection

I'm looking for the algorithm that determines the fact that a polygon has self intersection or hasn't. I'm not needed in calculation of the intersection points coordinates or how many intersection points there are.

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The naïve algorithm is to compare each pair of edges to see whether they overlap. Do you want an optimal algorithm or have any interesting constraints (e.g. polygon endpoints on lattice points)? –  Peter Taylor Nov 10 '11 at 9:10
No special constraints. I only need a very fast algorithm for this task. –  MaxFX Nov 10 '11 at 10:05
Posssible duplicate of math.stackexchange.com/questions/52733/… –  lhf Nov 10 '11 at 10:16

There is the obvious algorithm of comparing all pairs of edges, which is $O(n^2)$ but probably is ok for small polygons. There is the Bentley–Ottmann algorithm sweep algorithm, which is $O(n \log n)$ but is harder to implement and probably only needed if $n$ is large. I think there is a $O(n)$ algorithm by Chazelle, which is very likely to be impractical.

In any case, note that you can test whether two line segments intersect without finding the intersection point. It's a simple matter of comparing signs of a few determinants. See http://algs4.cs.princeton.edu/91primitives/.

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