Intersection of two monotone polygons.

Question

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.

Answer 1

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.