In this paper, we study a new problem of finding a convex drawing of graphs with a non-convex boundary. It is proved that every triconnected plane graph whose boundary is fixed with a star-shaped polygon admits a drawing in which every inner facial cycle is drawn as a convex polygon. Such a drawing, called an inner-convex drawing, can be obtained in linear time.

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 ...

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...

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...

The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP -hard, even for polygons without holes [3]. We propose a polynomial-time approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(logn), where n is the number of vertices in the input polygon. ...

A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edgelabeled graph, in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative ...

We prove that there is a motion from any convex polygon to any convex polygon with the same counterclockwise sequence of edge lengths, that preserves the lengths of the edges, and keeps the polygon convex at all times. Furthermore, the motion is “direct” (avoiding any intermediate canonical configuration like a subdivided triangle) in the sense that each angle changes monotonically throughout t...