Free polygon enumeration and the area of an integral polygon

Free polygon enumeration and the area of an integral polygon

We introduce the notion of free polygons as combinatorial building blocks for convex integral polygons; that is, polygons with vertices having integer coordinates. In this context, an Euler-type formula is derived for the number of integer points in the interior of an integral polygon. This leads in turn to a formula for the area of an integral polygon P via the enumeration of free integral tri...

Finding the Maximum Area Centrosymmetric Polygon in a Convex Polygon

Maximizing the area of an axis-symmetric polygon inscribed by a convex polygon

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...

Maximizing the area of an axially symmetric polygon inscribed in a simple polygon

In this paper we solve the following optimization problem: Given a simple polygon P , what is the maximum-area polygon that is axially symmetric and is contained by P? We propose an algorithm for solving this problem, analyze its complexity, and describe our implementation of it (for the case of a convex polygon). The algorithm is based on building and investigating a planar map, each cell of w...

Geometry-Free Polygon Splitting

A polygon splitting algorithm is a combinatorial recipe. The description and the implementation of polygon splitting should not depend on the embedding geometry. Whether a polygon is being split in Euclidean, in spherical, in oriented projective, or in hyperbolic geometry should not be part of the description of the algorithm. The algorithm should be purely combinatorial, or geometry free. The ...