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I am using the following definition for a monotone polygon:

A simple polygon is called monotone with respect to a line l if for any line l' perpendicular to l the intersection of the polygon with l' is connected. In other words, the intersection should be a line segment, a point, or empty. A polygon that is monotone with respect to the y-axis is called y-monotone.

It is clear that the union of two monotone polygons is not always a monotone polygon. But what about the intersection of two monotone polygons?

I feel that it is not always the case(the intersection of two monotone polygons is not always a monotone polygon), but I am stuck trying to come up with an example which would demonstrate it.

I am not interested in the number of resulting polygons, I am only interested in all of them being monotone.

mathematics2x2life
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hellomates
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  • Hint: for any point $p$ on $l$, consider the perpendicular through $p$; what can you say about its intersection with the intersection of the polygons? – platty Sep 11 '17 at 16:35
  • @platty, I feel it intuitively, but I do not have a rigorous proof. – hellomates Sep 11 '17 at 16:47

1 Answers1

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The intersection of the polygons is the union of the intersections with a sweeping line. The latter are the intersections of two points or line segments or empty, giving a point or a line segment or empty.

In general, the intersection is a set of disconnected monotone polygons.

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  • I do not get what you mean. ) : – hellomates Sep 11 '17 at 16:43
  • Oh! Sorry! I will add a bit more info by updating the question. – hellomates Sep 11 '17 at 16:45
  • The latter are the intersections of two points what does it mean? – hellomates Sep 11 '17 at 16:49
  • May I say that in general, the intersection is a set of disconnected monotone polygons. Because, the intersection will be formed from union of two points or two line segments, from different sets which will be covered be sweeping line while it is going through the sets perpendicular to l. And if there is a disjoint set intersected by sweeping line that means that the disjoint set consists of points and line segments, but that means that the input sets are not monotone, which contradicts. – hellomates Sep 11 '17 at 17:03
  • @ЯрикТроф: the key point is that the intersection of two segments is a single segment (not more). –  Sep 11 '17 at 17:13