Implicit Convex Polygons

Convex polygons in the plane can be defined explicitly as an ordered list of vertices, or given implicitly, for example by a list of linear constraints. The latter representation has been considered in several fields such as facility location, robotics and computer graphics. In this paper, we investigate many fundamental geometric problems for implicitly represented polygons and give simple and fast algorithms that are easy to implement. We uncover an interesting partition of the problems into two classes: those that exhibit an (nlogn) lower bound on their complexity, and those that yield O(n) time algorithms via prune-and-search methods.     

关于平面19-点集的空凸分划问题

主要研究了平面上处于一般位置的19-点集,根据其凸包边数的不同,分别讨论了其所含空凸多边形的个数,得出G(19)≤5.在此基础上,对平面上处于一般位置的n-点集得出G(n)≤[11n/42],从而改进了G(n)的上界.     

Convex Polygons are Cover-Decomposable

We show that for any open convex polygon P, there is a constant k(P) such that any k(P)-fold covering of the plane with translates of P can be decomposed into two coverings.     

Asymptotic Enumeration of Convex Polygons

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice ^dtaking at least one step in every dimension. A polygon on ^dis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonsp_{n, d}of length 2nwithd(n)→∞ is asymptoticallywherer=r(n, d) is the unique solution ofr coth r=2n/d−1andb(r)=d(r coth r−r²/sinh² r). The convergence is uniform over alld?ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d⁻¹)^2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.     

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.     

Davenport's Theorem in the Theory of Irregularities of Point Distribution

We study distributions of N points in the unit square U ² with minimal order of L ₂-discrepancy ₂[ ] < C(log N)^1/2, where the constant C is independent of N. We present an approach that uses Walsh functions and can be generalized to higher dimensions. Bibliography: 19 titles.     

Circle Numbers of Regular Convex Polygons

The circle number function is extended here to regular convex polygons. To this end, the radius of the polygonal circle is defined as the Minkowski functional of the circumscribed polygonal disc, and the arc-length of the polygonal circle is measured in a generalized Minkowski space having the rotated polar body as the unit disc.     

Affine Properties of Convex Equal-Area Polygons

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.     

凸多边形上的Bezier曲面片

本文建立了正多边形上的重心坐标，并构造了多边形上的Ｂｅｚｉｅｒ曲面片、     

Tilings of Convex Polygons with Congruent Triangles

By the spectrum of a polygon A we mean the set of triples (??,??,??) such that A can be dissected into congruent triangles of angles ??,??,??. We propose a technique for finding the spectrum of every convex polygon. Our method is based on the following classification. A tiling is called regular if there are two angles of the triangles, ?? and ?? such that at every vertex of the tiling the number of triangles having angle ?? equals the number of triangles having angle ??. Otherwise the tiling is irregular. We list all pairs (A,T) such that A is a convex polygon and T is a triangle that tiles A regularly. The list of triangles tiling A irregularly is always finite, and can be obtained, at least in principle, by considering the system of equations satisfied by the angles, examining the conjugate tilings, and comparing the sides and the area of the triangles to those of A. Using this method we characterize the convex polygons with infinite spectrum, and determine the spectrum of the regular triangle, the square, all rectangles, and the regular N-gons with N large enough.     

Helly Type Problems forSome Special Convex Polygons

In the combinatorial geometry of convex sets the question of how efficiently a family ofconvex sets can be pierced by points has led to various problems which may be regarded asextensions of the Helly-type problems. A family of sets is said to be n-pierceable (abbreviatedas n) if there exists a set of n points such that each member of the family contains at leastone of them. A family of sets is said to be nk: if every subfamily of size k or less is n. Thefamous Helly theorem in combinatorial …     

Tiling Convex Polygons with Congruent Equilateral Triangles

We study the sets $\mathcal{T}_{v}=\{m \in\{1,2,\ldots\}: \mbox{there is a convex polygon in }\mathbb{R}^{2}\mbox{ that has }v\mbox{ vertices and can be tiled with $m$ congruent equilateral triangles}\}$ , v=3,4,5,6. $\mathcal{T}_{3}$ , $\mathcal{T}_{4}$ , and $\mathcal{T}_{6}$ can be quoted completely. The complement $\{1,2,\ldots\} \setminus\mathcal{T}_{5}$ of $\mathcal{T}_{5}$ turns out to be a subset of Euler’s numeri idonei. As a consequence, $\{1,2,\ldots\} \setminus\mathcal{T}_{5}$ can be characterized with up to two exceptions, and a complete characterization is given under the assumption of the Generalized Riemann Hypothesis.     

On Minimal Triangulations of Products of Convex Polygons

We give new lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving the lower bounds in Bowen et al. (Topology 44:321–339, 2005).     

On Polygons Excluding Point Sets

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that ${B\cup R}$ is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K = K(l), which is bounded from above by a polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k ≥ K. Some other related problems are also considered.     

Ray Shooting Amidst Convex Polygons in 2D

We consider the problem of ray shooting in a two-dimensional scene consisting ofmconvex polygons with a total ofnedges. We present a data structure that requiresO(mn log m) space and preprocessing time and that answers a ray shooting query inO(log² m log² n) time. If the polygons are pairwise disjoint, the space and preprocessing time can be improved toO((m²+n)log m) andO((m²+n log n)log m), respectively. Our algorithm also works for a collection of disjoint simple polygons. We also show that if we allow onlyO(n) space, a ray shooting query among a collection of disjoint simple polygons can be answered in timeO(m/[formula]¹⁺ log² n) time, for any >0.     

On Polygons Enclosing Point Sets II

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56 |Conv(B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p ₁ , ... , p _n and S a set of m points contained in the interior of P, m ≤ n − 1. Then there is a convex decomposition {P ₁ , ... , P _n } of P such that all points from S lie on the boundaries of P ₁ , ... , P _n , and each P _i contains a whole edge of P on its boundary. F. Hurtado partially supported by projects MEC MTM2006-01267 and DURSI 2005SGR00692. C. Merino supported by CONACYT of Mexico, Proyecto 43098. J. Urrutia supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876, and MCYT BFM2003-04062. I. Ventura partially supported by Project MCYT BFM2003-04062.     

On Irregularities of Distribution II

Let a=(a₁, a₂, a₃, ...) be an arbitrary infinite sequence inU=[0, 1). Let Van der Corput [5] conjectured that d(a, n) (n=1, 2, ...) isunbounded, and this was proved in 1945 by van Aardenne-Ehrenfest[1]. Later she refined this [2], obtaining for infinitely many n. Here and later c₁, c₂, ... denote positiveabsolute constants. In 1954, Roth [8] showed that the quantity is closely related to the discrepancy of a suitable point setin U².     

The Unit Distance Problem for Centrally Symmetric Convex Polygons

   Abstract. Let f(n) be the maximum number of unit distances determined by the vertices of a convex n -gon. Erdos and Moser conjectured that this function is linear. Supporting this conjecture we prove that f ^sym (n)