Polygonal Problem Research Articles

Equidissections of Trapezoids

Suppose a plane polygon is dissected into triangles of equal areas. What numbers of triangles are possible? Given a polygon K, a dissection of K into m triangles of equal areas is called an m-equidissection. The spectrum of a polygon K, denoted S(K), is the set of integers m such that K has an m-equidissection. If S(K) consists of all multiples of a single integer m, we say that S(K) is principal and write S(K)= . Monsky [2] showed that if K is a square, then S(K)= . Kasimatis [1] showed that if K is a regular n-gon with n 2 5, then S(K) = . A discussion of this problem for other polygons is contained in Chapter 5 of [4]. What can we say about the spectrum of a trapezoid? We may confine our attention to trapezoids with vertices (0, 0), (1, 0), (0, 1), (a, 1), a > O, since any trapezoid is affinely equivalent to such a trapezoid. Denote this trapezoid by T(a). The following are shown in [4] (pp. 121-122):

Outside-in: an alternative approach to forest map digitizing

ABSTRACT Abstract. This paper examines the problem of polygon digitizing, and suggests an inversion of the traditional approach for polygons of the environmental type, where each individual polygon, rather than its boundaries, is of primary interest. Many workers in geology, forestry, soil science, etc. have found the tedious specification of boundary arcs and their topological linkage and editing to be a major block to productivity. An alternative digitizing approach is described using Thiessen/Voronoi polygons to generate topology automatically and to reduce map digitizing times significantly. The digitizing of representative points within each polygon is performed using the GIS. The stand-alone Voronoi polygon building and boundary extraction module then generates linked arcs that are returned to the GIS for further analysis. Examples of the productivity enhancement in a production environment are given, along with the algorithms used.

STOCK CUTTING OF COMPLICATED DESIGNS BY COMPUTING MINIMAL NESTED POLYGONS

Abstract This paper studies the following problem in stock cutting: when it is required to cut out complicated designs from parent material, it is cumbersome to cut out the exact design or shape, especially if the cutting process involves optimization. In such cases, it is desired that, as a first step, the machine cut out a relatively simpler approximation of the original design, in order to facilitate the optimization techniques that are then used to cut out the actual design. This paper studies this problem of approximating complicated designs or shapes. The problem is defined formally first and then it is shown that this problem is equivalent to the Minimal Nested Polygon problem in geometry. Some properties of the problem are then shown and it is demonstrated that the problem is related to the Minimal Turns Path problem in geometry. With these results, an efficient approximate algorithm is obtained for the original stock cutting problem. Numerical examples are provided to illustrate the working of the algorithm in different cases.

Optimal matching of general polygons based on the minimum zone error

This paper presents a parametric approach for the matching of polygonal profiles based on the tolerancing requirements defined by the ANSI standards. The polygonal matching problem is formulated as a minimax optimization model and a procedure for solving this optimization model is developed.

Formula omitted][formula omitted]-Algorithms for Minimum Link Path and Related Problems

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R to equal the minimum number of line segments or links that are needed to construct a path in R between the points. The minimum link path problem is to compute a path consisting of the minimum number of links between two points in R, when R is the inside of an n-sided simple polygon. The minimum nested polygon problem asks for a minimum link closed path (girth) when R is an annular region defined by a pair of nested simple polygons. Efficient sequential algorithms based on greedy methods have been described for both problems. However, neither problem was known to be N C . In this paper we present algorithms that require O(log n loglog n) time and O( n) space using O( n) processors for both problems. The approach used involves new results on the parallel ( N C 1) computation of the complete visibility polygon of a simple polygon from a set of points inside it, along with an algebraic technique based on fractional linear transforms that permits effective parallelization of the "greedy" computations. The complexity results of this paper are with respect to the CREW-PRAM model of computation. The time × processor product of these algorithms is within a small polylog factor of the best known sequential algorithms for the respective problems.

Optimal parallel hypercube algorithms for polygon problems

We present parallel techniques on hypercubes for solving optimally a class of polygon problems. We thus obtain optimal O(log n) time, n-processor hypercube algorithms for the problems of computing the portions of an n-vertex simple polygonal chain C that are visible from a given source point, computing the convex hull of C, testing an n-vertex simple polygon P for monotonicity, and other related problems as well. Previously it was not known how to achieve these complexity bounds on hypercubes, one of the main difficulties being that there is no known optimal sorting hypercube algorithm that achieves these bounds. In fact these are the first optimal geometric hypercube algorithms that do not assume that the input is given already sorted by x or y coordinates. The hypercube model we use is the standard one, with O(1) local memory per processor, and with one port communication. >

Intervertex distances in convex polygons

We summarize recent progress on intervertex distance problems for convex polygons, prove several new results for multiple intervertex distances and different intervertex distances, and raise open questions for further research.

DETERMINING THE SEPARATION OF SIMPLE POLYGONS

Given simple polygons P and Q, their separation, denoted by σ(P, Q), is defined to be the minimum distance between their boundaries. We present a parallel algorithm for finding a closest pair among all pairs (p, q), p ∈ P and q ∈ Q. The algorithm runs in O ( log n) time using O(n) processors on a CREW PRAM, where n = |P| + |Q|. This algorithm is time-optimal and improves by a factor of O ( log n) on the time complexity of previous parallel methods. The algorithm can be implemented serially in Θ (n) time, which gives the first optimal sequential algorithm for determining the separation of simple polygons. Our results are obtained by providing a unified treatment of the separation and the closest visible vertex problems for simple polygons.

Tight bounds for the number of edge guards for spiral polygons

In this paper we give tight lower and upper bounds for the number of edge guards required for covering spiral polygons. We have proved that [(n + 2)/5]edge guards are necessary and sufficient to cover a spiral polygon. It has been shown by Aggarwal [2] that [(n + 2)/5]diagonal guards are necessary and sufficient to cover a spiral polygon. Edge guards are more restrictive than diagonal guards. Hence the previous theorem can be got as a corollary using our theorem. The necessary condition of the edge guard problem for spiral polygons has not been investigated although the diagonal guard problem for the same has been solved [2]. The necessary proof of the edge guard problem follows from the necessary condition of the diagonal guard problem but we have given an alternate proof of necessity.

An optimal algorithm to solve the minimum weakly cooperative guards problem for 1-spiral polygons

An optimal algorithm to solve the minimum weakly cooperative guards problem for 1-spiral polygons

An efficient algorithm for finding the CSG representation of a simple polygon

Modeling two-dimensional and three-dimensional objects is an important theme in computer graphics. Two main types of models are used in both cases: boundary representations, which represent the surface of an object explicitly but represent its interior only implicitly, and constructive solid geometry representations, which model a complex object, surface and interior together, as a boolean combination of simpler objects. Because neither representation is good for all applications, conversion between the two is often necessary. We consider the problem of converting boundary representations of polyhedral objects into constructive solid geometry (CSG) representations. The CSG representations for a polyhedronP are based on the half-spaces supporting the faces ofP. For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane only once. Our main contribution is an efficient and practicalO(n logn) algorithm for doing this boundary-to-CSG conversion for a simple polygon ofn sides. We also prove that such nice formulae do not always exist for general polyhedra in three dimensions.

An addendum to parallel methods for visibility and shortest-path problems in simple polygons

An addendum to parallel methods for visibility and shortest-path problems in simple polygons

Location of the collapsed phase for two-dimensional, directed, interacting polymers

By mapping a model for a directed polymer onto a novel polygon problem, we give a physically appealing proof for the location of the boundary of the collapsed phase in phase space, applicable also when interactions with a surface are included.

Minimum dissection of a rectilinear polygon with arbitrary holes into rectangles

In this paper, the problem of dissecting a plane rectilinear polygon with arbitrary (possibly, degenerate) holes into a minimum number of rectangles is shown to be solvable inO(n3/2 logn) time. This fact disproves a famous assertion about the NP-hardness of the minimum rectangular dissection problem for rectilinear polygons with point holes.

Parallel methods for visibility and shortest-path problems in simple polygons

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest-path problems for simple polygons. Our algorithms all run inO(logn) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygonP, which we call thestratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility ofP from an edge, constructing the visibility graph of the vertices ofP (using an output-sensitive number of processors), constructing the shortest-path tree from a vertex ofP, and determining all-farthest neighbors for the vertices inP. The computational model we use is the CREW PRAM.

CONVEX POLYGON PROBLEMS ON MESHES WITH MULTIPLE BROADCASTING

We propose time-optimal algorithms for a number of convex polygon problems on meshes with multiple broadcasting. Specifically, we show that on a mesh with multiple broadcasting of size n × n, the task of deciding whether an n-gon is convex, deciding whether two convex n-gons edge-intersect, deciding whether one convex n-gon lies in the interior of another, as well as variants of the tasks of computing the intersection and union of two convex n-gons can be accomplished in Θ( log n) time. We also show that detecting whether two convex n-gons are separable takes O(1) time.

Optimal parallel algorithms for point-set and polygon problems

In this paper we give parallel algorithms for a number of problems defined on point sets and polygons. All our algorithms have optimalT(n) * P(n) products, whereT(n) is the time complexity andP(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. Our algorithms provide parallel analogues to well-known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently than point-set problems, and that nearest-neighbor problems can be solved without explicitly constructing a Voronoi diagram.

PARALLEL COMPUTATION OF INTERNAL AND EXTERNAL FARTHEST NEIGHBORS IN SIMPLE POLYGONS

We present efficient parallel algorithms for two problems in simple polygons: the all-farthest neighbors problem and the external all-farthest neighbors problem. The all-farthest neighbors problem is that of computing, for each vertex p of a simple polygon P, a point ψ(p) in P farthest from p when the distance between p and ψ(p) is measured by the shortest path between them constrained to lie inside P. The external all-farthest neighbors problem is that of computing, for each vertex p of a simple polygon P, a point ϕ(p) on (the boundary of) P farthest from p when the distance between p and ϕ(p) is measured by the shortest path between them constrained to lie outside (the interior of) P. Both our algorithms run in O( log 2 n) time on a CREW PRAM with O(n) processors. Our divide-and-conquer method for the external all-farthest neighbors problem, in fact, leads to a new O(n log n) time serial algorithm that matches the currently best serial algorithm for this problem, but is simpler.

The contour problem for restricted-orientation polygons

The problem of finding the contour of the union of a collection of polygons in which all vertices have integer coordinates and the slopes of the sides belong to a finite fixed collection of orientations is studied from two perspectives. The first is that of determining which algorithms for finding the contour of the union for rectilinear polygons and/or rectangles can be generalized to handle polygons with sides of two or more additional directions while still retaining efficiency. The second is that of determining which algorithms for computing the union of arbitrary polygons in general position can be revised to handle degeneracies and possibly to take advantage of the restricted number of orientations to improve robustness, efficiency, or both. Three distinct rectilinear algorithms are detailed, and general preprocessing and postprocessing procedures which allow all three rectilinear algorithms to operate on polygons of additional orientations are presented. Two general algorithms for computing the contour of union based on line segment intersection algorithms and plane sweeping are also presented and analyzed. >

The intersection searching problem for c-oriented polygons

We have shown that a c-oriented polygon intersection query can be answered in O(log n+t) time using O(n log n) space, where n is the number of c-oriented polygons, each with a bounded number of edges, and t is the number of reported polygons.