Deforming star-shaped polygons in the plane

Convexifying star-shaped polygons

K-star-shaped Polygons

We introduce k-star-shaped polygons, polygons for which there exists at least one point x such that for any point y of the polygon, segment xy crosses the polygon’s boundary at most k times. The set of all such points x is called the k-kernel of the polygon. We show that the maximum complexity (number of vertices) of the k-kernel of an n-vertex polygon is Θ(n) if k = 2 and Θ(n) if k ≥ 4. We giv...

Generating random star-shaped polygons

In this paper we deal with two problems on star-shaped polygons. At rst, we present a Las-Vegas algorithm that uniformly at random creates a star-shaped polygon whose vertices are given by a point set S of n points in the plane that does not admit degenerated star-shaped polygons. The expected running time of the algorithm is O(n logn) and it uses O(n) memory. We call a starshaped polygon degen...

Generating Random Star-shaped Polygons (extended Abstract)

In this paper we deal with two problems on star-shaped polygons. First, we present a Las-Vegas algorithm that uniformly at random creates a star-shaped polygon whose vertices are given by a point set S of n points in the plane that does not admit degenerate star-shaped polygons. The expected running time of the algorithm is O(n 2 logn) and it uses O(n) memory. We call a star-shaped polygon dege...

On the number of star-shaped polygons and polyhedra

We show that the maximum number of strictly star-shaped polygons through a given set of n points in the plane is (n). Our proof is constructive, i.e. we supply a construction which yields the stated number of polygons. We further present lower and upper bounds for the case of unrestricted star-shaped polygons. Extending the subject into three dimensions, we give a tight bound of (n) on the numb...