 # Does the intersection of two concave polygons without holes result in a set of several polygons without holes? Ilnaus5 2022-09-16 Answered
Does the intersection of two concave polygons without holes result in a set of several polygons without holes?
I did not find a proof anywhere, so here is my assumption :
If I have 2 concave and simple polygons without holes, then the intersection yields a set of different polygons without any holes.
If it can result to a polygon with a hole inside, could you provide an example?
Edit: the polygons are simple. edges of one polygone don't intersect, except consecutive edges, which intersect in their common vertex. The polygons in input do not have holes. I consider the intersection of the interiors of the polygons. The resulting set is all the points that are included both in interior of polygon P1 and the interior of the polygon P2.
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Explanation:
Call the two polygons ${P}_{1}$ and ${P}_{2}$. Suppose that some point $x\notin {P}_{1}\cap {P}_{2}$, but there is a curve inside ${P}_{1}\cap {P}_{2}$ encircling x. Since $x\notin {P}_{1}\cap {P}_{2}$ then $x\notin {P}_{1}$ wlog. But there is a curve encircling x which is inside ${P}_{1}$ and therefore ${P}_{1}$ has a hole.

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