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...

Simplification or decomposition is a common strategy to handle large geometric models, which otherwise require excessive computation to process. Star-shaped decomposition partitions a model into a set of star-shaped components. A model is star shaped if and only if there exists at least one point which can see all the points of the model. Due to this interesting property, decomposing a model in...

We present a new sampling-based algorithm for complete motion planning. Our algorithm relies on computing a star-shaped roadmap of the free space. We partition the free space into star-shaped regions such that a single point, called the guard, can see every point in the star-shaped region. The resulting set of guards capture the intraregion connectivity—the connectivity between points belonging...

We study the problem of on-line searching for a target inside a polygon. In particular, we propose a strategy for finding a target of unknown location in a star-shaped polygon with a competitive ratio of 11.52. We also provide a lower bound of 9 for the competitive ratio of searching in a star-shaped polygon which is close to the upper bound. A similar task is the on-line recognition of a star-...

Let K1,K2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a ∈ K1, b ∈ K2}. We show that the set K1K2 is star-shaped if K1 is a line segment or a circular disk. Examples for K1 and K2 are given so that K1 and K2 are triangles (including interior) and K1K2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product num...

In this paper we show how a theorem in plane geometry can be converted into an O(n log n) algorithm for decomposing a polygon into star-shaped subsets. The computational efficiency or this new decomposition contrasts with the heavy computational burden of existing methods. 1.0 Introduction The decomposition of a simple planar polygon into simpler components plays an important role in syntactic ...

We construct an open bounded star-shaped set Ω ⊂ R 4 whose cylindrical capacity is strictly bigger than its proper displacement energy.

An orthogonal drawing of a plane graph G is a planar drawing of G, denoted by D(G), such that each vertex of G is drawn as a point on the plane, and each edge of G is drawn as a sequence of horizontal and vertical line segments with no crossings. An orthogonal polygon P is called orthogonally convex if the intersection of any horizontal or vertical line L and P is either a single line segment o...