Visibility methods are often based on the existence of large convex occluders. We present an algorithm that for a given simple non-convex polygon P finds an approximate inner-cover by large convex polygons. The algorithm is based on an initial partitioning of P into a set C of disjoint convex polygons which are an exact tessellation of P . The algorithm then builds a set of large convex polygon...

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affi...

We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an n-sided polygon in O(n2) time. Achieving this running time requires proving several new structural properties of the MAP, and combining them with a rotating technique of Toussaint [10]. We also discuss applications of our result to th...

Let P be a polygon inscribed in a circle. The shadow of P is a polygon P ′ whose vertices are at the midpoints of the arcs of consecutive points of P . The shadow sequence P , P , P , . . . is a sequence of inscribed polygons such that each P t is the shadow of P t−1 for all t ≥ 0. We show in this abstract that the shadow sequence converges to the regular polygon, and in such way that variance ...

We consider the geometric optimization problem of nding the maximum area axis-parallel rectangle (MAAPR) in an n-vertex general polygon. We characterize the MAAPR for general polygons by considering di erent cases based on the types of contacts between the rectangle and the polygon. We present a general framework for solving a key subcase of the MAAPR problem which dominates the running time fo...

A convex polygon is defined as a sequence (V0, . . . , Vn−1) of points on a plane such that the union of the edges [V0, V1], . . . , [Vn−2, Vn−1], [Vn−1, V0] coincides with the boundary of the convex hull of the set of vertices {V0, . . . , Vn−1}. It is proved that all sub-polygons of any convex polygon with distinct vertices are convex. It is also proved that, if all sub-(n − 1)-gons of an n-g...

In this paper we resolve the following problem: Given a simple polygon , what is the maximum-area polygon that is axially symmetric and is contained by ? We propose an algorithm for answering this question, analyze the algorithm’s complexity, and describe our implementation of it (for convex polygons). The algorithm is based on building and investigating a planar map, each cell of which corresp...

We give a new proof of an isoperimetric inequality for family closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from convex, spherical polygon, with each vertex polygon leading non-smooth point on surface. For example, lune revolution, two tips. Combined straightforward approximation argument, this was first proved ...

In this paper we present a polynomial time algorithm for computing a Hausdorff core of a polygon with a single reflex vertex. A Hausdorff core of a polygon P is a convex polygon Q contained inside P which minimizes the Hausdorff distance between P andQ. Our algorithm essentially consists of rotating a line about the reflex vertex; this line defines a convex polygon by cutting P . To determine t...

In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional (2D) space is proposed. First, the vertices of a tight polygon that contains the convex intersection of the ellipses are found in an efficient manner. To do so, the intersection points of the ellipses that fall on the boundary of the intersection region are d...