Steiner polygons in the Steiner problem

A polygon, whose vertices are points in a given setA ofn points, is defined to be a Steiner polygon ofA if all Steiner minimal trees forA lie in it. Cockayne first found that a Steiner polygon can be obtained by repeatedly deleting triangles from the boundary of the convex hull ofA. We generalize this concept and give a method to construct Steiner polygons by repeatedly deletingk-gons,k n. We also prove the uniqueness of Steiner polygons obtained by our method.     

Pebble Sets in Convex Polygons

Lukács and András posed the problem of showing the existence of a set of n−2 points in the interior of a convex n-gon so that the interior of every triangle determined by three vertices of the polygon contains a unique point of S. Such sets have been called pebble sets by De Loera, Peterson, and Su. We seek to characterize all such sets for any given convex polygon in the plane. We first consider a certain class of pebble sets, called peripheral because they consist of points that lie close to the boundary of the polygon. We characterize all peripheral pebble sets, and show that for regular polygons, these are the only ones. Though we demonstrate examples of polygons where there are other pebble sets, we nevertheless provide a characterization of the kinds of points that can be involved in non-peripheral pebble sets. We furthermore describe algorithms to find such points.     

Covering and guarding polygons using L k -sets

The Art Gallery Problem is the problem of finding a minimum number of points (called guards) in a given polygon such that every point in the polygon is visible to at least one of the guards. Chvátal [5] was the first to show that, in the worst case, [n/3] such points will suffice for any polygon of n sides. O'Rourke [15] later showed that only [n/4] guards were needed if line segments, rather than points, were allowed as guards. In this paper, we unify these results, and extend them to many other classes of guards, while using a generalization of visibility known as link-visibility. In particular, we present the following theorems:

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

Isoperimetric Polygons of Maximum Width

The value is shown to be an upper bound on the width of any n-sided polygon with unit perimeter. This bound is reached when n is not a power of 2, and the corresponding optimal solutions are the regular polygons when n is odd and clipped regular Reuleaux polygons when n is even but not a power of 2. Using a global optimization algorithm, we show that the optimal width for the quadrilateral is with a precision of 10⁻⁴. We propose two mathematical programs to determine the maximum width when n=2 ^s with s≥3 and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n=8, 16, and 32. Work of the first author was supported by NSERC grant 239436-01, AFOSR FA9550-07-1-0302, and ExxonMobil. Work of the second author was supported by NSERC grant 239436-01.     

Enumeration of Almost-Convex Polygons on the Square Lattice

We classify self-avoiding polygons on the square lattice according to a concavity measure, m, where 2m is the difference between the number of steps in the polygon and the perimeter of the minimal rectangle bounding the polygon. We generate series expansions for the perimeter generating functions S_m(x) for polygons of concavity m. We analyze the series S_m(x) for m = 0 to 3. If N_m,n denotes the number of polygons of perimeter 2n and concavity m, with m = o(n^1/2), we prove that N_m,n ? 2^2n?m?7n^m+1/m!, and that the radius of convergence of the series counting all polygons with m = o(n) is equal to 1/4. Our numerical data leads us to conjecture that in fact for m = o(n^1/2), a result confirmed for m = 0 and 1.     

Probability Theory of Random Polygons from the Quaternionic Viewpoint

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Hausmann and Knutson, using the Hopf map on quaternions from the complex Stiefel manifold of 2‐frames in n‐space to the space of closed n‐gons in 3‐space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons that comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges. © 2014 Wiley Periodicals, Inc.     

Rigid pentagons in hypercubes

Two sets of vertices of a hypercubes in ⁿ and ^m are said to be equivalent if there exists a distance preserving linear transformation of one hypercube into the other taking one set to the other. A set of vertices of a hypercube is said to be weakly rigid if up to equivalence it is a unique realization of its distance pattern and it is called rigid if the same holds for any multiple of its distance pattern. A method of describing all rigid and weakly rigid sets of vertices of hypercube of a given size is developed. It is also shown that distance pattern of any rigid set is on the face of convex cone of all distance patterns of sets of vertices in hypercubes.Rigid pentagons (i.e. rigid sets of size 5 in hypercubes) are described. It is shown that there are exactly seven distinct types of rigid pentagons and one type of rigid quadrangle. It is also shown that there is a unique weakly rigid pentagon which is not rigid. An application to the study of all rigid pentagons and quadrangles inL ¹ having integral distance pattern is also given.This work was done during a visit of both the authors to Mehta Research Institute, Allahabad, India.     

Sets determined by finitely many X-rays

A measurable set in _n which is uniquely determined among all measurable sets (modulo null sets) by its X-rays in a finite set L of directions, or more generally by its X-rays parallel to a finite set L of subspaces, is called L-unique, or simply unique. Some subclasses of the L-unique sets are known. The L-additive sets are those measurable sets E which can be written E {x _n : _i f _i (x) * 0}. Here, denotes equality modulo null sets, * is either or >, and the terms in the sum are the values of ridge functions f _i orthogonal to subspaces S _i in L. If n=2, the L-inscribable convex sets are those whose interiors are the union of interiors of inscribed convex polygons, all of whose sides are parallel to the lines in L. Relations between these classes are investigated. It is shown that in _n each L-inscribable convex set is L-additive, but L-additive convex sets need not be L-inscribable. It is also shown that every ellipsoid in _n is unique for any set of three directions. Finally, some results are proved concerning the structure of convex sets in _n , unique with respect to certain families of coordinate subspaces.     

Decomposing a simple polygon into pseudo-triangles and convex polygons

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudo-triangles and convex polygons. We call the resulting decomposition PT-convex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably.The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles are allowed. Our algorithm determines such a decomposition in O(n³) time and space, where n is the number of the vertices of the polygon.     

Multiple-guard kernels of simple polygons

We study a generalization of the kernel of a polygon. A polygonP isk guardable if there arek points inP such that, for all pointsp inP, there is at least one of thek points that seesp. We call thek points ak-guard set ofP and thek-kernel of a polygonP is the union of allk-guard sets ofP. The usual definition of the kernel of a polygon is now the one-kernel in this notation.We show that the two-kernel of a simple polygonP withn edges has O(n⁴) components and that there are polygons whose two-kernel has (n) components. Moreover, we show that the components of the two-kernel of a simple polygon can be paired in a natural manner which implies that the two-kernel of a simple polygon has either one component or an even number of components. Finally, we consider the question of whether there is a non-star-shaped simple polygonP such that 2-kernel(P) = P. We show that when the two-kernel has only one component, it contains a hole; hence, the two-kernel of a simple polygonP is neverP itself. For everyk 1, there are, however, polygonsP _k with holes such thatk-kernel(P_k) = P_k.This work was supported under grant No. Ot 64/8-1, Diskrete Probleme from the the Deutsche Forschungsgemeinschaft, grants from the Natural Sciences and Engineering Council of Canada, from the Information Technology Research Centre of Ontario, and from the Research Grants Council of Hong Kong.     

Properties of tangential and cyclic polygons: an application of circulant matrices

In this paper, the properties of tangential and cyclic polygons proposed by Lopez-Real are proved rigorously using the theory of circulant matrices. In particular, the concepts of slippable tangential polygons and conformable cyclic polygons are defined. It is shown that an n-sided tangential (or cyclic) polygon P _n with n even is slippable (or conformable) and the sum of a set of non-adjacent sides (or interior angles) of P _n satisfies certain equalities. On the other hand, for a tangential (or cyclic) polygon P _n with n odd, it is rigid and the sum of a set of non-adjacent sides (or interior angles) of P _n satisfies certain inequalities. These inequalities give a definite answer to the question raised by Lopez-Real concerning the alternating sum of interior angles of a cyclic polygon.     

Algorithms for minimum length partitions of polygons

We show that it is possible to find a diagonal partition of anyn-vertex simple polygon into smaller polygons, each of at mostm edges, minimizing the total length of the partitioning diagonals, in timeO(n ³ m ²). We derive the same asymptotic upper time-bound for minimum length diagonal partitions of simple polygons into exactlym-gons provided that the input polygon can be partitioned intom-gons. Also, in the latter case, if the input polygon is convex, we can reduce the upper time-bound toO(n ³ logm).     

Finding the Shortest Watchman Route in a Simple Polygon

We present the first polynomial time algorithm that finds the shortest route in a simple polygon such that all points of the polygon are visible from the route. This route is called the shortest watchman route, and we do not assume any restrictions on the route or on the simple polygon. Our algorithm runs in worst case O(n ⁶ ) time, but it is adaptive, making it run faster on polygons with a simple structure. Received December 12, 1997, and in revised form September 30, 1998.     

Unions of orthogonally convex or orthogonally starshaped polygons

LetT be a simply connected orthogonal polygon having the property that for every three points ofT, at least two of these points see each other via staircases inT. ThenT is a union of three orthogonally convex polygons. The number three is best possible.ForT a simply connected orthogonal polygon,T is a union of two orthogonally convex polygons if and only if for every sequencev ₁,...,v _n,v _n+1 =v ₁ inT, n odd, at least one consecutive pairv _i ,v _i+1 sees each other via staircase paths inT, 1 i n. An analogous result says thatT is a union of two orthogonal polygons which are starshaped via staircase paths if and only if for every odd sequence inT, at least one consecutive pair sees a common point via staircases inT.Supported in part by NSF grants DMS-8908717 and DMS-9207019.     

Completely classifying all vertex-transitive and edge-transitive polyhedra

Recently A. Dress completed the classification of the regular polyhedra in E ³ by adding one class to the enumeration given by Grünbaum on this subject. This classification is the only systematic study of a collection of polyhedra possessing special symmetries which uses the generalized definition of a polygon allowing for skew polygons as well as planar polygons in E ³. This study gives necessary conditions for polyhedra to be vertex-transitive and edge-transitive. These conditions are restrictive enough to make the task of completely enumerating such polyhedra realizable and efficient. Examples of this process are given, and an explanation of the basic process is discussed. These new polyhedra are appearing more frequently in applications of geometry, and this examination is a beginning of the classifications of polyhedra having special symmetries even though there are many other such classes which lack this scrutiny.     

THE MINIMUM AREA OF A SIMPLE POLYGON WITH GIVEN SIDE LENGTHS

Answering a question of H. Harborth, for any given a ₁,...,a _n > 0, satisfying we determine the infimum of the areas of the simple n-gons in the Euclidean plane, having sides of length a ₁,...,a _n (in some order). The infimum is attained (in limit) if the polygon degenerates into a certain kind of triangle, plus some parts of zero area. We show the same result for simple polygons on the sphere (of not too great length), and for simple polygons in the hyperbolic plane. Replacing simple n-gons by convex ones, we answer the analogous questions. The infimum is attained also here for degeneration into a certain kind of triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Intersection of Maximal Orthogonally k-Starshaped Polygons

Let F be a simply connected orthogonal polygon in R ² and let P denote the intersection of all maximal orthogonally k-starshaped polygons in F for any fixed integer k,k2. If P and for every x,y P which are joined in F by a staircase path having two segments there is a similar staircase path from x to y in P, then there exists a maximal orthogonally k-starshaped polygon Q in F such that the staircase k-kernel of Q is a subset of the staircase k-kernel of P. In particular, F is either an orthogonally k-starshaped simply connected polygon in F or empty.     

D.c. sets,d.c. functions and nonlinear equations

Ad.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in ⁿ into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.The revision of this paper was produced during the author's stay supported by a Sophia lecturing-research grant at Sophia University (Tokyo, Japan).