Degree Constrained Triangulation of Annular Regions and Point Sites
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Constrained a Fast Algorithm for Generating Delaunay
004s7949/93 56.00 + 0.00 if) 1993 Pcrgamon Press Ltd A FAST ALGORITHM FOR GENERATING CONSTRAINED DELAUNAY TRIANGULATIONS S. W. SLOAN Department of Civil Engineering and Surveying, University of Newcastle, Shortland, NSW 2308, Australia {Received 4 March 1992) Abstract-A fast algorithm for generating constrained two-dimensional Delaunay triangulations is described. The scheme permits certain edges to be specified in the final t~an~ation, such as those that correspond to domain boundaries or natural interfaces, and is suitable for mesh generation and contour plotting applications. Detailed timing statistics indicate that its CPU time requhement is roughly proportional to the number of points in the data set. Subject to the conditions imposed by the edge constraints, the Delaunay scheme automatically avoids the formation of long thin triangles and thus gives high quality grids. A major advantage of the method is that it does not require extra points to be added to the data set in order to ensure that the specified edges are present. I~RODU~ON to be degenerate since two valid Delaunay triangu- lations are possible. Although it leads to a loss of Triangulation schemes are used in a variety of uniqueness, degeneracy is seldom a cause for concern scientific applications including contour plotting, since a triangulation may always be generated by volume estimation, and mesh generation for finite making an arbitrary, but consistent, choice between element analysis. Some of the most successful tech- two alternative patterns. niques are undoubtedly those that are based on the One major advantage of the Delaunay triangu- Delaunay triangulation. lation, as opposed to a triangulation constructed To describe the construction of a Delaunay triangu- heu~stically, is that it automati~ily avoids the lation, and hence explain some of its properties, it is creation of long thin triangles, with small included convenient to consider the corresponding Voronoi angles, wherever this is possible. -
Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Illinois Digital Environment for Access to Learning and Scholarship Repository TRIANGULATION OF SIMPLE 3D SHAPES WITH WELL-CENTERED TETRAHEDRA EVAN VANDERZEE, ANIL N. HIRANI, AND DAMRONG GUOY Abstract. A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fields. We show how to triangulate simple domains using completely well-centered tetrahedra. The domains we consider here are space, infinite slab, infinite rectangular prism, cube, and regular tetrahedron. We also demonstrate single tetrahedra with various combinations of the properties of dihedral acuteness, 2-well-centeredness, and 3-well-centeredness. 1. Introduction 3 In this paper we demonstrate well-centered triangulation of simple domains in R . A well- centered simplex is one for which the circumcenter lies in the interior of the simplex [8]. This definition is further refined to that of a k-well-centered simplex which is one whose k-dimensional faces have the well-centeredness property. An n-dimensional simplex which is k-well-centered for all 1 ≤ k ≤ n is called completely well-centered [15]. These properties extend to simplicial complexes, i.e. to meshes. Thus a mesh can be completely well-centered or k-well-centered if all its simplices have that property. For triangles, being well-centered is the same as being acute-angled. But a tetrahedron can be dihedral acute without being 3-well-centered as we show by example in Sect. -
Exploring Topics of the Art Gallery Problem
The College of Wooster Open Works Senior Independent Study Theses 2019 Exploring Topics of the Art Gallery Problem Megan Vuich The College of Wooster, [email protected] Follow this and additional works at: https://openworks.wooster.edu/independentstudy Recommended Citation Vuich, Megan, "Exploring Topics of the Art Gallery Problem" (2019). Senior Independent Study Theses. Paper 8534. This Senior Independent Study Thesis Exemplar is brought to you by Open Works, a service of The College of Wooster Libraries. It has been accepted for inclusion in Senior Independent Study Theses by an authorized administrator of Open Works. For more information, please contact [email protected]. © Copyright 2019 Megan Vuich Exploring Topics of the Art Gallery Problem Independent Study Thesis Presented in Partial Fulfillment of the Requirements for the Degree Bachelor of Arts in the Department of Mathematics and Computer Science at The College of Wooster by Megan Vuich The College of Wooster 2019 Advised by: Dr. Robert Kelvey Abstract Created in the 1970’s, the Art Gallery Problem seeks to answer the question of how many security guards are necessary to fully survey the floor plan of any building. These floor plans are modeled by polygons, with guards represented by points inside these shapes. Shortly after the creation of the problem, it was theorized that for guards whose positions were limited to the polygon’s j n k vertices, 3 guards are sufficient to watch any type of polygon, where n is the number of the polygon’s vertices. Two proofs accompanied this theorem, drawing from concepts of computational geometry and graph theory. -
AN EXPOSITION of MONSKY's THEOREM Contents 1. Introduction
AN EXPOSITION OF MONSKY'S THEOREM WILLIAM SABLAN Abstract. Since the 1970s, the problem of dividing a polygon into triangles of equal area has been a surprisingly difficult yet rich field of study. This paper gives an exposition of some of the combinatorial and number theoretic ideas used in this field. Specifically, this paper will examine how these methods are used to prove Monsky's theorem which states only an even number of triangles of equal area can divide a square. Contents 1. Introduction 1 2. The p-adic Valuation and Absolute Value 2 3. Sperner's Lemma 3 4. Proof of Monsky's Theorem 4 Acknowledgments 7 References 7 1. Introduction If one tried to divide a square into triangles of equal area, one would see in Figure 1 that an even number of such triangles would work, but could an odd number work? Fred Richman [3], a professor at New Mexico State, considered using this question in an exam for graduate students in 1965. After being unable to find any reference or answer to this question, he decided to ask it in the American Mathematical Monthly. n Figure 1. Dissection of squares into an even number of triangles After 5 years, Paul Monsky [4] finally found an answer when he proved that only an even number of triangles of equal area could divide a square, which later came to be known as Monsky's theorem. Formally, the theorem states: Theorem 1.1 (Monsky [4]). Let S be a square in the plane. If a triangulation of S into m triangles of equal area is given, then m is even. -
CSE 5319-001 (Computational Geometry) SYLLABUS
CSE 5319-001 (Computational Geometry) SYLLABUS Spring 2012: TR 11:00-12:20, ERB 129 Instructor: Bob Weems, Associate Professor, http://ranger.uta.edu/~weems Office: 627 ERB, 817/272-2337 ([email protected]) Hours: TR 12:30-1:50 PM and by appointment (please email by 8:30 AM) Prerequisite: Advanced Algorithms (CSE 5311) Objective: Ability to apply and expand geometric techniques in computing. Outcomes: 1. Exposure to algorithms and data structures for geometric problems. 2. Exposure to techniques for addressing degenerate cases. 3. Exposure to randomization as a tool for developing geometric algorithms. 4. Experience using CGAL with C++/STL. Textbooks: M. de Berg et.al., Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag, 2000. https://libproxy.uta.edu/login?url=http://www.springerlink.com/content/k18243 S.L. Devadoss and J. O’Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. References: Adobe Systems Inc., PostScript Language Tutorial and Cookbook, Addison-Wesley, 1985. (http://Www-cdf.fnal.gov/offline/PostScript/BLUEBOOK.PDF) B. Casselman, Mathematical Illustrations: A Manual of Geometry and PostScript, Springer-Verlag, 2005. (http://www.math.ubc.ca/~cass/graphics/manual) CGAL User and Reference Manual (http://www.cgal.org/Manual) T. Cormen, et.al., Introduction to Algorithms, 3rd ed., MIT Press, 2009. E.D. Demaine and J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007. (occasionally) J. O’Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987. (http://maven.smith.edu/~orourke/books/ArtGalleryTheorems/art.html, occasionally) J. O’Rourke, Computational Geometry in C, 2nd ed., Cambridge Univ. -
Optimal Higher Order Delaunay Triangulations of Polygons*
Optimal Higher Order Delaunay Triangulations of Polygons Rodrigo I. Silveira and Marc van Kreveld Department of Information and Computing Sciences Utrecht University, 3508 TB Utrecht, The Netherlands {rodrigo,marc}@cs.uu.nl Abstract. This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of qual- ity measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O(n3) running time required for normal triangulations to O(k2n log k + kn log n) expected time, where n is the number of vertices of the polygon. An extension to polygons with points inside is also presented, allowing to compute an optimal triangulation of a polygon with h ≥ 1 components inside in O(kn log n)+O(k)h+2n expected time. Furthermore, through experimental results we show that, in practice, it can be used to triangulate point sets optimally for small values of k. This represents the first practical result on optimization of higher order Delaunay triangulations for k>1. 1 Introduction One of the best studied topics in computational geometry is the triangulation. When the input is a point set P , it is defined as a subdivision of the plane whose bounded faces are triangles and whose vertices are the points of P .When the input is a polygon, the goal is to decompose it into triangles by drawing diagonals. Triangulations have applications in a large number of fields, including com- puter graphics, multivariate analysis, mesh generation, and terrain modeling. Since for a given point set or polygon, many triangulations exist, it is possible to try to find one that is the best according to some criterion that measures some property of the triangulation. -
Polygon Decomposition Motivation: Art Gallery Problem Art Gallery Problem
CG Lecture 3 Motivation: Art gallery problem Definition: two points q and r in a Polygon decomposition simple polygon P can see each other if the open segment qr R lies entirely within P. 1. Polygon triangulation p q • Triangulation theory A point p guards a region • Monotone polygon triangulation R ⊆ P if p sees all q∈R 2. Polygon decomposition into monotone r pieces Problem: Given a polygon P, what 3. Trapezoidal decomposition is the minimum number of guards 4. Convex decomposition required to guard P, and what are their locations? 5. Other results 1 2 Simple observations Art gallery problem: upper bound • Convex polygon: all points • Theorem: Every simple planar are visible from all other polygon with n vertices has a points only one guard in triangulation of size n-2 (proof any location is necessary! later). convex • Star-shaped polygon: all • n-2 guards suffice for an n-gon: points are visible from any • Subdivide the polygon into n–2 point in the kernel only triangles (triangulation). one guard located in its • Place one guard in each triangle. kernel is necessary. star-shaped 3 4 Art gallery problem: lower bound Simple polygon triangulation • There exists a Input: a polygon P polygon with n described by an ordered vertices, for which sequence of vertices ⎣n/3⎦ guards are <v0, …vn–1>. necessary. Output: a partition of P Can we improve the upper into n–2 non-overlapping • Therefore, ⎣n/3⎦ bound? triangles and the guards are needed in adjacencies between Yes! In fact, at most ⎣n/3⎦ the worst case. -
Triangulations and Simplex Tilings of Polyhedra
Triangulations and Simplex Tilings of Polyhedra by Braxton Carrigan A dissertation submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 4, 2012 Keywords: triangulation, tiling, tetrahedralization Copyright 2012 by Braxton Carrigan Approved by Andras Bezdek, Chair, Professor of Mathematics Krystyna Kuperberg, Professor of Mathematics Wlodzimierz Kuperberg, Professor of Mathematics Chris Rodger, Don Logan Endowed Chair of Mathematics Abstract This dissertation summarizes my research in the area of Discrete Geometry. The par- ticular problems of Discrete Geometry discussed in this dissertation are concerned with partitioning three dimensional polyhedra into tetrahedra. The most widely used partition of a polyhedra is triangulation, where a polyhedron is broken into a set of convex polyhedra all with four vertices, called tetrahedra, joined together in a face-to-face manner. If one does not require that the tetrahedra to meet along common faces, then we say that the partition is a tiling. Many of the algorithmic implementations in the field of Computational Geometry are dependent on the results of triangulation. For example computing the volume of a polyhedron is done by adding volumes of tetrahedra of a triangulation. In Chapter 2 we will provide a brief history of triangulation and present a number of known non-triangulable polyhedra. In this dissertation we will particularly address non-triangulable polyhedra. Our research was motivated by a recent paper of J. Rambau [20], who showed that a nonconvex twisted prisms cannot be triangulated. As in algebra when proving a number is not divisible by 2012 one may show it is not divisible by 2, we will revisit Rambau's results and show a new shorter proof that the twisted prism is non-triangulable by proving it is non-tilable. -
Two Algorithms for Constructing a Delaunay Triangulation 1
International Journal of Computer and Information Sciences, Vol. 9, No. 3, 1980 Two Algorithms for Constructing a Delaunay Triangulation 1 D. T. Lee 2 and B. J. Schachter 3 Received July 1978; revised February 1980 This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set of Npoints. The first algorithm uses a divide-and-conquer approach. It runs in O(Nlog N) time, which is asymptotically optimal. The second algorithm is iterative and requires O(N 2) time in the worst case. However, its average case performance is comparable to that of the first algorithm. KEY WORDS: Delaunay triangulation; triangulation; divide-and-con- quer; Voronoi tessellation; computational geometry; analysis of algorithms. 1. INTRODUCTION In this paper we consider the problem of triangulating a set of points in the plane. Let V be a set of N ~> 3 distinct points in the Euclidean plane. We assume that these points are not all colinear. Let E be the set of (n) straight- line segments (edges) between vertices in V. Two edges el, e~ ~ E, el ~ e~, will be said to properly intersect if they intersect at a point other than their endpoints. A triangulation of V is a planar straight-line graph G(V, E') for which E' is a maximal subset of E such that no two edges of E' properly intersect.~16~ 1 This work was supported in part by the National Science Foundation under grant MCS-76-17321 and the Joint Services Electronics Program under contract DAAB-07- 72-0259. -
Incremental Construction of Constrained Delaunay Triangulations Jonathan Richard Shewchuk Computer Science Division University of California Berkeley, California, USA
Incremental Construction of Constrained Delaunay Triangulations Jonathan Richard Shewchuk Computer Science Division University of California Berkeley, California, USA 1 7 3 10 0 9 4 6 8 5 2 The Delaunay Triangulation An edge is locally Delaunay if the two triangles sharing it have no vertex in each others’ circumcircles. A Delaunay triangulation is a triangulation of a point set in which every edge is locally Delaunay. Constraining Edges Sometimes we need to force a triangulation to contain specified edges. Nonconvex shapes; internal boundaries Discontinuities in interpolated functions 2 Ways to Recover Segments Conforming Delaunay triangulations Edges are all locally Delaunay. Worst−case input needsΩ (n²) to O(n2.5 ) extra vertices. Constrained Delaunay triangulations (CDTs) Edges are locally Delaunay or are domain boundaries. Goal segments Input: planar straight line graph (PSLG) Output: constrained Delaunay triangulation (CDT) Every edge is locally Delaunay except segments. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Randomized Incremental CDT Construction Start with Delaunay triangulation of vertices. Do segment location. Insert segment. Topics Inserting a segment in expected linear time. -
An Introductory Study on Art Gallery Theorems and Problems
AN INTRODUCTORY STUDY ON ART GALLERY THEOREMS AND PROBLEMS JEFFRY CHHIBBER M Dept of Mathematics, Noorul Islam Centre of Higher Education, Kanyakumari, Tamil Nadu, India. E-mail: [email protected] Abstract - In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Visibility graphs may also be used to calculate the placement of radio antennas, or as a tool used within architecture and urban planningthrough visibility graph analysis. This is a brief survey on the visibility graphs application in Art Gallery Problems and Theorems. Keywords- Art galley theorems, orthogonal polygon, triangulation, Visibility graphs I. INTRODUCTION point y outside of P if the segment xy is nowhere interior to P; xy may intersect ∂P, the boundary of P. In a visibility graph, each node in the graph Star polygon: A polygon visible from a single interior represents a point location, and each edge represents point. Diagonal: A segment inside a polygon whose a visible connectionbetween them. That is, if the line endpoints are vertices, and which otherwise does not segment connecting two locations does not pass touch ∂P. Floodlight: A light that illuminates from the through any obstacle, an edge is drawn between them apex of a cone with aperture α. Vertex floodlight: in the graph. Lozano-Perez & Wesley (1979) attribute One whose apex is at a vertex (at most one per the visibility graph method for Euclidean shortest vertex). paths to research in 1969 by Nils Nilsson on motion planning for Shakey the robot, and also cite a 1973 The problem: description of this method by Russian mathematicians What is the art gallery problem? M. -
Generalized Barycentric Coordinate Finite Element Methods on Polytope Meshes
Generalized Barycentric Coordinate Finite Element Methods on Polytope Meshes Andrew Gillette Department of Mathematics University of Arizona Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 1 / 41 What are a priori FEM error estimates? n n Poisson’s equation in R : Given a domain D ⊂ R and f : D! R, find u such that strong form −∆u = f u 2 H2(D) Z Z weak form ru · rφ = f φ 8φ 2 H1(D) D D Z Z 1 discrete form ruh · rφh = f φh 8φh 2 Vh finite dim. ⊂ H (D) D D Typical finite element method: ! Mesh D by polytopes fPg with vertices fvi g; define h := max diam(P). ! Fix basis functions λi with local piecewise support, e.g. barycentric functions. P ! Define uh such that it uses the λi to approximate u, e.g. uh := i u(vi )λi A linear system for uh can then be derived, admitting an a priori error estimate: p p+1 jju − uhjjH1(P) ≤ Ch jujHp+1(P); 8u 2 H (P); | {z } | {z } approximation error optimal error bound provided that the λi span all degree p polynomials on each polytope P. Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 2 / 41 The generalized barycentric coordinate approach Let P be a convex polytope with vertex set V . We say that λv : P ! R are generalized barycentric coordinates (GBCs) on P X if they satisfy λv ≥ 0 on P and L = L(vv)λv; 8 L : P ! R linear. v2V Familiar properties are implied by this definition: X X λv ≡ 1 vλv(x) = x λvi (vj ) = δij v2V v2V | {z } | {z } | {z } interpolation partition of unity linear precision traditional FEM family of GBC reference elements Unit Affine Map T Bilinear Map Diameter Reference Physical T Ω Ω Element Element Andrew Gillette - U.