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.     

Improved heuristics for the minimum weight triangulation problem

Given a planar point setS, a triangulation ofS is a maximal set of non-intersecting line segments connecting the points. The minimum weight triangulation problem is to find a triangulation ofS such that the sum of the lengths of the line segments in it is the smallest. No polynomial time algorithm is known to produce the optimal or even a constant approximation of the optimal solution, and it is also unknown whether the problem is NP-hard. In this paper, we propose two improved heuristics, which triangulate a set ofn points in a plane inO(n ³) time and never do worse than the minimum spanning tree triangulation algorithm given by Lingas and the greedy spanning tree triangulation algorithm given by Heath and Pemmaraju. These two algorithms both produce an optimal triangulation if the points are the vertices of a convex polygon, and also do the same in some special cases.     

Optimizing a constrained convex polygonal annulus

In this paper we give solutions to several constrained polygon annulus placement problems for offset and scaled polygons, providing new efficient primitive operations for computational metrology and dimensional tolerancing. Given a convex polygon P and a planar point set S, the goal is to find the thinnest annulus region of P containing S. Depending on the application, there are several ways this problem can be constrained. In the variants that we address the size of the polygon defining the inner (respectively, outer) boundary of the annulus is fixed, and the annulus is minimized by minimizing (respectively, maximizing) the outer (respectively, inner) boundary. We also provide solutions to a related known problem: finding the smallest homothetic copy of a polygon containing a set of points. For all of these problems, we solve for the cases where smallest and largest are defined by either the offsetting or scaling of a polygon. We also provide some experimental results from implementations of several competing approaches to a primitive operation important to all the above variants: finding the intersection of n copies of a convex polygon.     

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.     

Packing and covering the plane with translates of a convex polygon

A covering of the Euclidean plane by a polygon P is a system of translated copies of P whose union is the plane, and a packing of P in the plane is a system of translated copies of P whose interiors are disjoint. A lattice covering is a covering in which the translates are defined by the points of a lattice, and a lattice packing is defined similarly. We show that, given a convex polygon P with n vertices, the densest lattice packing of P in the plane can be found in O(n) time. We also show that the sparsest lattice covering of the plane by a centrally symmetric convex polygon can be solved in O(n) time. Our approach utilizes results from classical geometry that reduce these packing and covering problems to the problems of finding certain extremal enclosed figures within the polygon.     

A four-vertex theorem for polygons and its generalization to polytopes

A vertex v of a convex polygon P is called minimal (respectively maximal) if the circle going through v and its neighbouring vertices encloses the interior of P (respectively has no vertex of P in its interior) The main result of this paper is a generalization to the convex polytopes of R ^d of the following theorem: Every convex polygon has at least two minimal and two maximal vertices The proof uses a duality theory which translates some spherical properties of a convex polytope of R ^d into combinatorial properties of a convex polyhedron of R ^d+1.     

A linear-time algorithm for computing the voronoi diagram of a convex polygon

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in (n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.This research began while the first and fourth authors were visiting the Mathematical Sciences Research Institute in Berkeley, California. Work by the fourth author was supported in part by NSF Grant No. 8120790.     

The orthogonal convex skull problem

We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n ²) algorithm is presented, which is a substantial improvement over theO(n ⁷) time algorithm for the general problem.The work of the first author was supported under a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692 and the work of the second author was partially supported by NSF Grants Nos. DCR-84-01898 and DCR-84-01633.     

Isodiametric Problems for Polygons

The maximal area of a polygon with n = 2m edges and unit diameter is not known when m ≥ 5, nor is the maximal perimeter of a convex polygon with n = 2^m edges and unit diameter known when m ≥ 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c₁/n³ in the area problem and c₂/n⁵ in the perimeter problem, for certain constants c₁ and c₂.     

Flipturning Polygons

   Abstract. A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most

Finding minimal nested polygons

An algorithm for finding a polygon with minimum number of edges nested in two simplen-sided polygons is presented. The algorithm solves the problem in at mostO(n logn) time, and improves the time complexity of two previousO(n ²) algorithms.The work was supported by NSERC grant OPG0041629.     

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.     

Visibility with One Reflection

We extend the concept of the polygon visible from a source point S in a simple polygon by considering visibility with two types of reflection, specular and diffuse. In specular reflection a light ray reflects from an edge of the polygon according to the rule: the angle of incidence equals the angle of reflection. In diffuse reflection a light ray reflects from an edge of the polygon in all inward directions. Several geometric and combinatorial properties of visibility polygons under these two types of reflection are described, when at most one reflection is permitted. We show that the visibility polygon Vs(S) under specular reflection may be nonsimple, while the visibility polygon Vd(S) under diffuse reflection is always simple. We present a Θ(n ² ) worst-case bound on the combinatorial complexity of both Vs(S) and Vd(S) and describe simple O(n ² log ² n) time algorithms for constructing the sets. Received September 27, 1995, and in revised form October 24, 1997.     

Efficient Algorithms for the Smallest Enclosing Ball Problem

Consider the problem of computing the smallest enclosing ball of a set of m balls in ⁿ. Existing algorithms are known to be inefficient when n > 30. In this paper we develop two algorithms that are particularly suitable for problems where n is large. The first algorithm is based on log-exponential aggregation of the maximum function and reduces the problem into an unconstrained convex program. The second algorithm is based on a second-order cone programming formulation, with special structures taken into consideration. Our computational experiments show that both methods are efficient for large problems, with the product mn on the order of 10⁷. Using the first algorithm, we are able to solve problems with n = 100 and m = 512,000 in about 1 hour.His work was supported by Australian Research Council.Research supported in part by the Singapore-MIT Alliance.     

Largest Placement of One Convex Polygon Inside Another

We show that the largest similar copy of a convex polygon P with m edges inside a convex polygon Q with n edges can be computed in O(mn ² log n) time. We also show that the combinatorial complexity of the space of all similar copies of P inside Q is O(mn ² ) , and that it can also be computed in O(mn ² log n) time. Received December 11, 1995, and in revised form March 3, 1997.     

Untangling a Polygon

   Abstract. The following problem was raised by M. Watanabe. Let P be a self-intersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n) ^2/3 ) vertices. On the other hand, every polygon P can be untangled in at most

Convex drawings of graphs with non-convex boundary constraints

In this paper, we study a new problem of convex drawing of planar graphs with non-convex boundary constraints, and call a drawing in which every inner-facial cycle is drawn as a convex polygon an inner-convex drawing. It is proved that every triconnected plane graph with the boundary fixed with a star-shaped polygon whose kernel has a positive area admits an inner-convex drawing. We also prove that every four-connected plane graph whose boundary is fixed with a crown-shaped polygon admits an inner-convex drawing. We present linear time algorithms to construct inner-convex drawings for both cases.     

Convexity preserving interpolation by algebraic curves and surfaces

The problem of interpolation by a convex curve to the vertices of a convex polygon is considered. A natural 1-parameter family ofC algebraic curves solving this problem is presented. This is extended to a solution, of a general Hermite-type problem, in, which the curve also interpolates to one or two prescribedtangents at any desired vertices of the polygon. The construction of these curves is a generalization of well known methods for generatingconic sections. Several properties of this family of algebraic curves are discussed. In addition, the method is generalized to convexC interpolation of strictly convex data sets inR ³ by algebraicsurfaces.     

About strongly polynomial time algorithms for quadratic optimization over submodular constraints

We present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O(NM log(N ²/M)) algorithm for the problemNetwork defined on a network onM arcs andN nodes; an O(n logn) algorithm for thetree problem onn variables; an O(n logn) algorithm for theNested problem, and a linear time algorithm for theGeneralized Upper Bound problem. These algorithms are the best known so far for these problems. The status of the general problem and open questions are presented as well.This research has been supported in part by ONR grant N00014-91-J-1241.Corresponding author.     

Optimally Computing the Shortest Weakly Visible Subedge of a Simple Polygon

Given ann-vertex simple polygonP, the problem of computing the shortest weakly visible subedge ofPis that of finding a shortest line segmentson the boundary ofPsuch thatPis weakly visible froms(ifsexists). In this paper, we present new geometric observations that are useful for solving this problem. Based on these geometric observations, we obtain optimal sequential and parallel algorithms for solving this problem. Our sequential algorithm runs inO(n) time, and our parallel algorithm runs inO(log n) time usingO(n/log n) processors in the CREW PRAM computational model. Using the previously best known sequential algorithms to solve this problem would takeO(n²) time. We also give geometric observations that lead to extremely simple and optimal algorithms for solving, both sequentially and in parallel, the case of this problem where the polygons are rectilinear.