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.     

Tilings of Polygons with Similar Triangles, II

Let A be a polygon, and let s (A) denote the number of distinct nonsimilar triangles Δ such that A can be dissected into finitely many triangles similar to Δ . If A can be decomposed into finitely many similar symmetric trapezoids, then s(A)=∞ . This implies that if A is a regular polygon, then s(A)=∞ . In the other direction, we show that if s(A)=∞ , then A can be decomposed into finitely many symmetric trapezoids with the same angles. We introduce the following classification of tilings: a tiling is regular if Δ has two angles, α and β , such that at each vertex of the tiling the number of angles α is the same as that of β . Otherwise the tiling is irregular. We prove that for every polygon A the number of triangles that tile A irregularly is at most c ⋅ n ⁶ , where n is the number of vertices of A. If A has a regular tiling, then A can be decomposed into finitely many symmetric trapezoids with the same angles. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p411.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received February 17, 1997, and in revised form June 16, 1997.     

Deforming star-shaped polygons in the plane

Given a star-shaped polygon in the euclidean plane and a vertex v of the polygon, the author characterizes all those points w in the plane such that when the vertex v moves to w along a straight line path, while all other vertices of the polygon are fixed, the polygon remains star-shaped. An example is given to show that this characterization fails for the star-shaped polyhedral spheres in the 3-dimensional euclidean space.     

Triangle Contact Representations and Duality

A?contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A?primal?Cdual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal?Cdual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.     

A Pedestrian Approach to Ray Shooting: Shoot a Ray, Take a Walk

We propose a very simple ray-shooting algorithm, whose only data structure is a triangulation. The query algorithm, after locating the triangle containing the origin of the ray, walks along the ray, advancing from one triangle to a neighboring one until the polygon boundary is reached. The key result of the paper is a Steiner triangulation of a simple polygon with the property that a ray can intersect at most O(log n) triangles before reaching the polygon boundary. We are able to compute such a triangulation in linear sequential time, or in O(log n) parallel time using O(n/log n) processors. This gives a simple, yet optimal, ray-shooting algorithm for a simple polygon. Using a well-known technique, we can extend our triangulation procedure to a multiconnected polygon with k components and n vertices, so that a ray intersects at most O(√k log n) triangles.     

Allocating Vertex π-Guards in Simple Polygons via Pseudo-Triangulations

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: (1) Any simple polygon with n vertices can be monitored by at most \lfloor n/2 \rfloor general vertex π-guards. This bound is tight up to an additive constant of 1. (2) Any simple polygon with n vertices, k of which are convex, can be monitored by at most \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.     

On fat partitioning, fat covering and the union size of polygons

The complexity of the contour of the union of simple polygons with n vertices in total can be O(n²) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O(n log log n), which is a generalization of a similar result for fat triangles (Matou ek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles.     

Triangulations (tilings) and certain block triangular matrices

For a convex polygonP withn sides, a ‘partitioning’ ofP inton−2 nonoverlapping triangles each of whose vertices is a vertex ofP is called a triangulation or tiling, and each triangle is a tile. Each tile has a given cost associated with it which may differ one from another. This paper considers the problem of finding a tiling ofP such that the sum of the costs of the tiles used is a minimum, and explores the curiosity that (an abstract formulation of) it can be cast as a linear program. Further the special structure of the linear program permits a recursive O(n ³) algorithm. Research and reproduction of this report were partially supported by the National Science Foundation Grants MCS-8119774, MCS-7926009 and ECS-8012974; Department of Energy Contract DE-AM03-76SF00326, PA# DE-AT03-76ER72018; Office of Naval Research Contract N00014-75-C-0267. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and donot necessarily reflect the views of the above sponsors.     

The Minimum Number of Interior H-Points of Convex H-Dodecagons

Mathematical Notes - An H-polygon is a simple polygon whose vertices are H-points, which are points of the set of vertices of a tiling of ℝ2 by regular hexagons of unit edge. Let G(v) denote...     

Primitive and mensurable hex-triangles

When the corners of a planar polygon P are restricted to lie in the set H of vertices of a tiling of the plane by hexagons of unit area, then very often the area of P is given by the Pick-type formula A(P)=b/4+i/2+c/12-1, where b and i count the number of points of H on the boundary P and in the interior of P respectively, and c is the boundary characteristic. We now characterize all primitive triangles for which this formula holds, and consider the magnitude of the error when it fails.     

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.     

Configurations of inscribed equilateral triangles

This paper explores equilateral triangles XYZ with vertices on sidelines of a given triangle ABC such that one side of XYZ is parallel to the corresponding side of ABC. There are six such triangles. They have many interesting properties which we investigate using trilinear coordinates. Our results improve and add to the earlier results of Blas Herrera Gómez about these configurations. We obtain new characterizations of several central points of the triangles and identify interesting pairs of triangles that are homologic (or perspective) and orthologic. The recognition of the Darboux cubic of a triangle is also accomplished in these configurations. Triples of circles intersecting in a point and six points on a conic also appear.      

Realizable Quadruples for Hex-polygons

There are many interesting combinatorial results and problems dealing with lattice polygons, that is, polygons in ℝ² with vertices in the integral lattice ℤ². Geometrically, ℤ² is the set of corners of a tiling of ℝ² by unit squares. Denote by H the set of corners of a tiling of the plane by regular hexagons of unit area and call a polygon P a Hex-polygon or an H-polygon if all vertices of P are in H. Our purpose is to study several combinatorial properties of H-polygons that are analogous to properties of lattice polygons. In particular we aim to find some relationships between the numbers b and i of points from H on the boundary and in the interior of an H-polygon P with the numbers v and c of vertices and the so-called boundary characteristic of P. We also pose three open problems dealing with convex Hex-polygons.     

On Simple Polygonalizations with Optimal Area

We discuss the problem of finding a simple polygonalization for a given set of vertices P that has optimal area. We show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P and prove that it is NP-complete to find a minimum weight polygon or a maximum weight polygon for a given vertex set, resulting in a proof of NP-completeness for the corresponding area optimization problems. This answers a generalization of a question stated by Suri in 1989. Finally, we turn to higher dimensions, where we prove that, for 1 k d , 2 d , it is NP-hard to determine the smallest possible total volume of the k -dimensional faces of a d -dimensional simple nondegenerate polyhedron with a given vertex set, answering a generalization of a question stated by O'Rourke in 1980. Received June 26, 1997, and in revised form February 13, 1999, and May 19, 1999.     

Tilings of polygons with similar triangles

We prove that if a polygonP is decomposed into finitely many similar triangles then the tangents of the angles of these triangles are algebraic over the field generated by the coordinates of the vertices ofP. IfP is a rectangle then, apart from four sporadic cases, the triangles of the decomposition must be right triangles. Three of these sporadic triangles tile the square. In any other decomposition of the square into similar triangles, the decomposition consists of right triangles with an acute angle such that tan is a totally positive algebraic number. Most of the proofs are based on the following general theorem: if a convex polygonP is decomposed into finitely many triangles (not necessarily similar) then the coordinate system can be chosen in such a way that the coordinates of the vertices ofP belong to the field generated by the cotangents of the angles of the triangles in the decomposition.This work was completed while the author had a visiting position at the Mathematical Institute of the Hungarian Academy of Sciences.     

Illuminating a convex polygon with vertex lights

We prove that the interior of every convex polygon with n vertices (n ≥ 4) can be illuminated by four 45°-vertex lights. We restrict each vertex to anchoring at most one floodlight. This answers a question of O’Rourke, Shermer and Streinu [5].      

On the interior lattice points of convex lattice 11-gon

A lattice point in the plane is a point with integer coordinates. A lattice polygon K is a polygon whose vertices are lattice points. In this note we prove that any convex lattice 11-gon contains at least 15 interior lattice points.     

A note on tropical triangles in the plane

We define transversal tropical triangles （affine and projective） and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.     

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.