In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we s...

The scattering support is an estimate of the support of a source or scatterer, based on a limited set of far field measurements. In this paper, we suppose that the far field is measured at all wavenumbers, but only at a few, say N, angles θi ∈ . From these measurements, we produce a -convex polygon (a convex polygon with normals in the θi directions). We show that this polygon must be contained...

An analytical solution is presented that reconstructs residual stress field from limited and incomplete data.  The inverse problem of reconstructing residual stresses is solved using an appropriate form of the airy stress function.  This function is chosen to satisfy the stress equilibrium equations together with the boundary conditions for a domain within a convex polygon.  The analytical solu...

We extend a dynamic-programming algorithm of Keil and Snoeyink for finding a minimum convex decomposition of a simple polygon to the case when both convex polygons and pseudo-triangles are allowed. Our algorithm determines a minimum pseudo-convex decomposition of a simple polygon in O(n) time where n is the number of the vertices of the polygon. In this way we obtain a well-structured decomposi...

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n ≥ 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic progra...

Behavior of convex solution polygons to a general crystalline motion is investigated. A polygon is called admissible if the set of its normal angles equals that of the Wulff shape. We prove that if the initial polygon is not an admissible polygon, then all edges disappear simultaneously, or edge disappearing occurs at most finitely many instants and eventually a convex solution polygon becomes ...

A convex polygon that is nearly-similar to a model polygon P has sides parallel and in the same order to the corresponding sides of P. The lengths of the sides are unrestricted and may be zero. Given a set of target convex polygons in the plane with a total of n vertices, and a xed model convex stabbing polygon P, the minimum-perimeter polygon nearly-similar to P that stabs the targets can be f...

In many industrial and non-industrial applications, it is necessary to identify the largest inscribed rectangle in a certain shape. The problem is studied for convex and non-convex polygons. Another criterion is the direction of the rectangle: axis aligned or general. In this paper a heuristic algorithm is presented for finding the largest axis aligned inscribed rectangle in a general polygon. ...

The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P , we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P . We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the...