Constrained Delaunay Triangulations (CDT)
|
|
- Sibyl Flynn
- 5 years ago
- Views:
Transcription
1 Constrained Delaunay Triangulations (CDT) Recall: Definition 1 (Constrained triangulation). A triangulation with prespecified edges or breaklines between nodes. Motivation: Geological faults in oil recovery Rivers, roads, lake boundaries in GIT Repr. non-convex boundaries and holes Linear features in CAD models Meshing for FEM (boundaries, interior & exterior) 1
2 2
3 Algorithm III from Preliminaries Given a predefined constraint E i,j. Suppose that the endpoints p i and p j are in. 1. For all T i in, If IntT i E ij, remove T i from We get one or more regions R i on each side of E i,j inside simple closed polygond. (More than one if E i,j intersects nodes.) 2. Triangulate each region R i with Algorithm I (Protruding point removal). (a) p i E i,j p j (b) p i E i,j p j May include holes and arbitrary exterior boundary 3
4 Delaunay Triangulation of a PSLG We will generalize the theory of Delaunay triangulation from P to G where G GP,E c is a PSLG (Planar Straight Line Graph). E c are constrained edges. The endpoints of E c are in P. Recall fom conventional Delaunay triangulations: MaxMin angle criterion Lexicographical ordering of indicator vectors Definition of local- and global optimum The circle criterion (will be redefined) The dual (Voronoi diagram) will not be considered here! (involved and not so useful) DEFINISJON (Visibility). p i and p j are visible to each other if p i p j does not intersect the interior of any edge in E c. 4
5 DEFINISJON (CDT). A CDT G of a PSLG GP,E c is a triangulation containing the edges E c such that Ct of any triangle t in G contains no point of P in its interior which is visible from all the three nodes of t. DEFINISJON: Modified circle criterion (relaxed) DEFINISJON: Edges in G that are not in E c are called Delaunay edges; triangles in G are called Delaunay triangles. (a) (b) (c) (d) a) GP,E c b) DTP c) CDTG 5
6 if E c G P and G P, P; a conventional Delaunay triangulation. 6
7 Generalization of the theory to CDTs (Brief, consult cited papers for details) (a) (b) p 2 θ 1 e e' p 1 θ 2 p 3 θ 2 e' p 4 In (b), e is constrained LEMMA. The modified circle criterion and the MaxMin angle criterion are equivalent for strictly convex quadrilaterals. Recall: indicator vector, lexicographical measure,... LEMMA(*). The indicator vector (of a contrained triangulation) becomes lexicographically larger each time an edge of a strictly convex quadrilateral is swapped according to the Delaunay swapping criteria. Basis for Lawson s LOP Algorithm 7
8 LOP applied to a G, whereg is a PSLG: 1. Start with an arbitrary G 2. Repeat swapping of edges according to modified circle criterion Does the LOP converge when applied to G? What does it converge to? Do we reach a global optimum? Lemma (*) and the fact that the number of possible triangulations of G is finite guarantees that the algorithm terminates after a finite number of edge swaps. DEFINITION. Locally optimal edge: 1. When the decision is not to swap it in the LOP. 2. Edges in E c and boundary edges are locally optimal by default. DEFINITION. Locally optimal triangulation; accordingly. 8
9 TEOREM. All interior edges of a triangulation G (G GP,E c ) are locally optimal (if and only if) the modified circle criterion holds for all triangles. Proof. See Exercise in lecture notes. So, LOP yields a CDT in accordance with DEFINITION (CDT). THEOREM. A triangulation G is a CDT in accordance withdefinition (CDT) (if and only if) its indicator vector is lexicographically maximum. Proof. See Exercise with guidelines in lecture notes. (Difficult since a dual construction (Voronoi) cannot be used.) 9
10 Uniqueness of a CDT can be deduced from the proof (under the usual assumption that no four points of P are cocircular). Unique characterization of Delaunay edge and Delaunay triangle in A CDT (Recall, edges in E c arenot Delaunay edges): THEOREM (Delaunay edge). Edge e ij between points p i and p j of P is Delaunay (if and only if) p i and p j are visible to each other and there exists a circle passing through p i and p j that does not contain any points of P in its interior visible from both p i and p j. Proof. See Exercise in lecture notes. 10
11 THEOREM (Delaunay triangle). Triangle t with nodes p i, p j and p k is Delaunay (if and only if) Ct contains no point of P in its interior which is visible from both p i, p j and p k. Proof. See Exercise in lecture notes. 11
12 Algorithms for Constrained Delaunay Triangulation Overview: Similar schemes as for conventional Delaunay, but 1. predefined constrained edges, and 2. modified circle criterion. Only incremental algorithms are considered here: basic operations: inserting a constrained edge into a CDT, and inserting a node into a CDT. 12
13 Given a PSLG GP,E c where endpoints of E c are in P. Algorithm (Compute G): 1. Compute P, (conventional Delaunay) 2. for each e in E c insert e into P,E c and update to a CDT. E c E c e 3. (add additional points into G) Recall from conventional Delaunay: two schemes for inserting a point p: remove triangles in (star shaped) influence region R p and retriangulate split t which contains the insertion point into three new triangles and apply recswap procedure three times. 13
14 Inserting an Edge into a CDT Insert constrained edge e c between p a and p b in P, E c P,E c e c. (a) p a e c pb e c L Q, (b) p a e c pb e c R Q, a) R e c b) Q e c,l and Q e c,r Influence region R e c of e c in G are triangles intersected by e c Note that no node is inserted or moved only R e c is affected! (Recall Theorem) Influence polygons, Q e c,l and Q e c,r on each side of e c. 14
15 Obtain P, E c e c by triangulating Q e c,l and Q e c,r on each side of the constrained edge e c separately using modified circle criterion: Step-by-step approach with base line and growing circles: Retriangulate one Q: 1. Let e c be the first baseline 2. Start with a growing circle with r and grow it into Q. 3. Make a triangle t with the first point p reached that is not separated from e c with a constraint. 4. Chose a new base line e c (an edge of t) and use the same circle. 5. GOTO 3 15
16 p' e b t p Y The growing The point p is uniquely defined as the point in P that, i) makes the largest angle at p spanned by the baseline e b, and ii) p is visible from the endpoints of e b. Note: No non-procedural approach as for point insertion. 16
17 e c L Q, (a) e c e c R Q, (b) (c) (d) Growing circles 17
18 Edge Insertion and Swapping Recall: T 2V I V B 2 I E 3V I 2V B 3 II E I 3V I V B 3 III T E V 1 IV, that is, a new constraint edge between two existing nodes does not change the cardinality of edges or triangles. This suggests that a new constrained edge can be swapped in place. 18
19 1. Swapping procedure for inserting a constraint e c into P,E c (operating inside R e c only). 2. LOP applied to edges inside each of Q e c,l and Q e c,r to obtain P,E c e c The swapping procedure: Principle: Swap edges away from the constrained edge e c such that eventually e c is included as an edge in the triangulation. Let p a, u 1,,u n,p b define the closed influence polygon Q ec,l to the left of the directed line from p a to p b. Note how the vertices of the polygon are enumerated when it is multiply connected; see figure. Consider a point u m each time where the angle m and swap away edges radiating from u m and intersecting e c. 1. is it always possible to find a point u m where m? 2. is there always a swappable edge at u m? 19
20 (a) e c L Q, u 2 u 4 u α 1 α 6 4 α 5 u 5 u 6 α 1 α 2 α 3 p a u 3 e c pb e c R Q, u 5 (b) α 1 u 1 u 3 u 4 α 3 α 4 α 5 p a u 2 α 2 e c pb (c) u 1 p a α 1 e c pb 20
21 LEMMA. A closed and simply connected polygon P has at least three interior angles smaller than. Proof. Let 1,, N, N 3, be the interior angles of P and suppose that k of them are smaller than. From elementary geometry we know that the sum of the interior angles is N 2 and thus, N N 2 i1 i i i i We have i N k, which inserted above gives, i k 2 i. i The right hand side is non-negative, so we have k 2. But since k 0 the right hand side is in fact positive and this gives k 3. i. Simply connected influence polygon: Lemma; at least one vertex different from p a and p b where m. multiply connected: see exercise. 21
22 w 0 = u m 1 α m u m w = r u + 1 m+1 e c w 1 w 2... w r Illustration for Lemma. Suppose that there are r edges radiating from u m and intersecting e c ; see figure. The endpoints of the r edges on the opposite side of e c from u m are denoted w 1,,w r, numbered counterclockwise. Conventions w 0 u m1 and w r1 u m1. LEMMA. There is at least one edge radiating from u m and intersecting e c, where m, that is a diagonal in a convex quadrilateral (and thus swappable). Proof. The closed polygon defined by the sequence w 0, w 1,,w r, w r1 (not including u m ) has at least three interior angles smaller than by Lemma. Thus, there is at least one point w s,1 s r such that the angle w s1 w s w s1 is smaller than. Then the quadrilateral with u m,w s as a diagonal must be convex since w s1,u m,w s1 is also smaller than. 22
23 Thus, all edges u m,w s,1s r, where m,can be swapped away from u m such that there are no edges left radiating from u m and intersecting e c. Result: u m is eliminated from the influence polygon. Algorithm (eliminate u m, with m ): 1. while r 1 2. Let u m, w s be a diagonal in a convex quadrilateral u m,w s1,w s,w s1. 3. Swap u m, w s to w s1,w s1 4. r r 1 Eventually, r 1 and and only one edge u m,w 1 radiates from u m and intersects e c. u m, w 1 is a diagonal in the quadrilateral u m, u m1,w 1, u m1 that is convex by Lemma. When u m,w 1 is swapped to u m1,u m1 in the r th cycle of the algorithm, u m is isolated from e c and eliminated from Q; seeu 1 in figure. 23
24 Repeat Algorithm on Q u m p a,u 1,u m1,u m1,u n,p b etc. and eventually on Q p a,u 1,p b. Algorithm (Insert constrained edge e c 1. while n 1 2. Find a point u m,1m n where m 3. Apply Algorithm to u m 4. n n 1 5. Q ec,l p a, u 1,,u m1,u m1,,u n,p b When n 1, Q p a,u 1,p b and the interior angle 1 at u 1 is smaller than by Lemma; see figure. When u 1,w 1 is swapped to p a,p b it takes the role as the constrained edge e c that has p a and p b as endpoints. 24
25 Finally, apply LOP to edges inside influence polygons Q e c,l and Q e c,r. LOP terminates as CDT with e c as a constraint! Existence follows from Lemma, extended to multiply connected polygons, and Lemma. 25
26 Inserting a Point into a CDT (a) (b) p e c e c Influence polygon Q p in a). Let P p,e c be the CDT obtained by inserting a point p into a CDT P,E c. Exact limitation of R p : LEMMA. A triangle t in P,E c will be modified when inserting a point p to obtain P p,e c if and only if the circumcircle of t contains p in its interior and p is visible from all the three nodes of t. Proof. The proof follows directly from Theorem (Delaunay triangle). 26
27 How can P p,e c be obtained? Alt. 1: THEOREM. All new triangles of P p,e c have p as a common node. Proof. See Exercise in lecture notes. P p,e c can be obtained from P,E c by removing all triangles of R p and connecting p to all points of Q p ;see Figure (b). Alt 2, swapping procedure: As for conventional Delaunay with this modification only: recswapdelaunay must not swap constrained edges. 27
Triangulations and Applications
Øyvind Hjelle and Morten Dæhlen Triangulations and Applications Figures algorithms and some equations as an aid to oral exam. November 27, 2007 Springer Berlin Heidelberg NewYork Hong Kong London Milan
More informationComputational Geometry
Lecture 12: Lecture 12: Motivation: Terrains by interpolation To build a model of the terrain surface, we can start with a number of sample points where we know the height. Lecture 12: Motivation: Terrains
More informationDelaunay Triangulations. Presented by Glenn Eguchi Computational Geometry October 11, 2001
Delaunay Triangulations Presented by Glenn Eguchi 6.838 Computational Geometry October 11, 2001 Motivation: Terrains Set of data points A R 2 Height ƒ(p) defined at each point p in A How can we most naturally
More informationDelaunay Triangulations
Delaunay Triangulations (slides mostly by Glenn Eguchi) Motivation: Terrains Set of data points A R 2 Height ƒ(p) defined at each point p in A How can we most naturally approximate height of points not
More informationCS 532: 3D Computer Vision 14 th Set of Notes
1 CS 532: 3D Computer Vision 14 th Set of Notes Instructor: Philippos Mordohai Webpage: www.cs.stevens.edu/~mordohai E-mail: Philippos.Mordohai@stevens.edu Office: Lieb 215 Lecture Outline Triangulating
More informationChapter 8. Voronoi Diagrams. 8.1 Post Oce Problem
Chapter 8 Voronoi Diagrams 8.1 Post Oce Problem Suppose there are n post oces p 1,... p n in a city. Someone who is located at a position q within the city would like to know which post oce is closest
More informationPreferred directions for resolving the non-uniqueness of Delaunay triangulations
Preferred directions for resolving the non-uniqueness of Delaunay triangulations Christopher Dyken and Michael S. Floater Abstract: This note proposes a simple rule to determine a unique triangulation
More informationDifferential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder]
Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder] Preliminaries Recall: Given a smooth function f:r R, the function
More informationVoronoi Diagrams and Delaunay Triangulations. O Rourke, Chapter 5
Voronoi Diagrams and Delaunay Triangulations O Rourke, Chapter 5 Outline Preliminaries Properties and Applications Computing the Delaunay Triangulation Preliminaries Given a function f: R 2 R, the tangent
More informationVoronoi Diagrams in the Plane. Chapter 5 of O Rourke text Chapter 7 and 9 of course text
Voronoi Diagrams in the Plane Chapter 5 of O Rourke text Chapter 7 and 9 of course text Voronoi Diagrams As important as convex hulls Captures the neighborhood (proximity) information of geometric objects
More informationThe Farthest Point Delaunay Triangulation Minimizes Angles
The Farthest Point Delaunay Triangulation Minimizes Angles David Eppstein Department of Information and Computer Science UC Irvine, CA 92717 November 20, 1990 Abstract We show that the planar dual to the
More informationSurface Mesh Generation
Surface Mesh Generation J.-F. Remacle Université catholique de Louvain September 22, 2011 0 3D Model For the description of the mesh generation process, let us consider the CAD model of a propeller presented
More informationArt Gallery, Triangulation, and Voronoi Regions
Art Gallery, Triangulation, and Voronoi Regions CS535 Fall 2016 Daniel G. Aliaga Department of Computer Science Purdue University [some slides based on Profs. Shmuel Wimer and Andy Mirzaian Topics Triangulation
More informationVoronoi Diagrams. A Voronoi diagram records everything one would ever want to know about proximity to a set of points
Voronoi Diagrams Voronoi Diagrams A Voronoi diagram records everything one would ever want to know about proximity to a set of points Who is closest to whom? Who is furthest? We will start with a series
More informationComputational Geometry: Lecture 5
Computational Geometry: Lecture 5 Don Sheehy January 29, 2010 1 Degeneracy In many of the algorithms that we have discussed so far, we have run into problems when that input is somehow troublesome. For
More informationVoronoi Diagram. Xiao-Ming Fu
Voronoi Diagram Xiao-Ming Fu Outlines Introduction Post Office Problem Voronoi Diagram Duality: Delaunay triangulation Centroidal Voronoi tessellations (CVT) Definition Applications Algorithms Outlines
More informationComputational Geometry Lecture Delaunay Triangulation
Computational Geometry Lecture Delaunay Triangulation INSTITUTE FOR THEORETICAL INFORMATICS FACULTY OF INFORMATICS 7.12.2015 1 Modelling a Terrain Sample points p = (x p, y p, z p ) Projection π(p) = (p
More informationPractical Linear Algebra: A Geometry Toolbox
Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 18: Putting Lines Together: Polylines and Polygons Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book
More information1/25 Warm Up Find the value of the indicated measure
1/25 Warm Up Find the value of the indicated measure. 1. 2. 3. 4. Lesson 7.1(2 Days) Angles of Polygons Essential Question: What is the sum of the measures of the interior angles of a polygon? What you
More informationPS Computational Geometry Homework Assignment Sheet I (Due 16-March-2018)
Homework Assignment Sheet I (Due 16-March-2018) Assignment 1 Let f, g : N R with f(n) := 8n + 4 and g(n) := 1 5 n log 2 n. Prove explicitly that f O(g) and f o(g). Assignment 2 How can you generalize the
More informationAdvanced Algorithms Computational Geometry Prof. Karen Daniels. Fall, 2012
UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Fall, 2012 Voronoi Diagrams & Delaunay Triangulations O Rourke: Chapter 5 de Berg et al.: Chapters 7,
More informationLecture 16: Voronoi Diagrams and Fortune s Algorithm
contains q changes as a result of the ith insertion. Let P i denote this probability (where the probability is taken over random insertion orders, irrespective of the choice of q). Since q could fall through
More informationLesson 7.1. Angles of Polygons
Lesson 7.1 Angles of Polygons Essential Question: How can I find the sum of the measures of the interior angles of a polygon? Polygon A plane figure made of three or more segments (sides). Each side intersects
More informationChapter 10 Polygons and Area
Geometry Concepts Chapter 10 Polygons and Area Name polygons according to sides and angles Find measures of interior angles Find measures of exterior angles Estimate and find areas of polygons Estimate
More informationCOMPUTING CONSTRAINED DELAUNAY
COMPUTING CONSTRAINED DELAUNAY TRIANGULATIONS IN THE PLANE By Samuel Peterson, University of Minnesota Undergraduate The Goal The Problem The Algorithms The Implementation Applications Acknowledgments
More informationCAD & Computational Geometry Course plan
Course plan Introduction Segment-Segment intersections Polygon Triangulation Intro to Voronoï Diagrams & Geometric Search Sweeping algorithm for Voronoï Diagrams 1 Voronoi Diagrams Voronoi Diagrams or
More informationAPPROXIMATING THE MAXMIN AND MINMAX AREA TRIANGULATIONS USING ANGULAR CONSTRAINTS. J. Mark Keil, Tzvetalin S. Vassilev
Serdica J. Computing 4 00, 3 334 APPROXIMATING THE MAXMIN AND MINMAX AREA TRIANGULATIONS USING ANGULAR CONSTRAINTS J. Mark Keil, Tzvetalin S. Vassilev Abstract. We consider sets of points in the two-dimensional
More informationVoronoi diagrams Delaunay Triangulations. Pierre Alliez Inria
Voronoi diagrams Delaunay Triangulations Pierre Alliez Inria Voronoi Diagram Voronoi Diagram Voronoi Diagram The collection of the non-empty Voronoi regions and their faces, together with their incidence
More information3. Voronoi Diagrams. 3.1 Definitions & Basic Properties. Examples :
3. Voronoi Diagrams Examples : 1. Fire Observation Towers Imagine a vast forest containing a number of fire observation towers. Each ranger is responsible for extinguishing any fire closer to her tower
More informationConstruction Of The Constrained Delaunay Triangulation Of A Polygonal Domain
Construction Of The Constrained Delaunay Triangulation Of A Polygonal Domain Reinhard Klein 1 A fast and easy to implement divide-and-conquer algorithm is presented for the construction of the Constrained
More informationPolygon decomposition. Motivation: Art gallery problem
CG Lecture 3 Polygon decomposition 1. Polygon triangulation Triangulation theory Monotone polygon triangulation 2. Polygon decomposition into monotone pieces 3. Trapezoidal decomposition 4. Convex decomposition
More informationLesson Plan #39. 2) Students will be able to find the sum of the measures of the exterior angles of a triangle.
Lesson Plan #39 Class: Geometry Date: Tuesday December 11 th, 2018 Topic: Sum of the measures of the interior angles of a polygon Objectives: Aim: What is the sum of the measures of the interior angles
More informationWeek 8 Voronoi Diagrams
1 Week 8 Voronoi Diagrams 2 Voronoi Diagram Very important problem in Comp. Geo. Discussed back in 1850 by Dirichlet Published in a paper by Voronoi in 1908 3 Voronoi Diagram Fire observation towers: an
More informationGeneralized barycentric coordinates
Generalized barycentric coordinates Michael S. Floater August 20, 2012 In this lecture, we review the definitions and properties of barycentric coordinates on triangles, and study generalizations to convex,
More informationWeek 2 Polygon Triangulation
Week 2 Polygon Triangulation What is a polygon? Last week A polygonal chain is a connected series of line segments A closed polygonal chain is a polygonal chain, such that there is also a line segment
More informationPebble Sets in Convex Polygons
2 1 Pebble Sets in Convex Polygons Kevin Iga, Randall Maddox June 15, 2005 Abstract 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
More informationMa/CS 6b Class 26: Art Galleries and Politicians
Ma/CS 6b Class 26: Art Galleries and Politicians By Adam Sheffer The Art Gallery Problem Problem. We wish to place security cameras at a gallery, such that they cover it completely. Every camera can cover
More informationLectures 19: The Gauss-Bonnet Theorem I. Table of contents
Math 348 Fall 07 Lectures 9: The Gauss-Bonnet Theorem I Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In
More informationPackage RTriangle. January 31, 2018
Package RTriangle January 31, 2018 Copyright 1993, 1995, 1997, 1998, 2002, 2005 Jonathan Richard Shewchuk; 2011-2018 License CC BY-NC-SA 4.0 Title Triangle - A 2D Quality Mesh Generator and Delaunay Triangulator
More informationAn angle that has a measure less than a right angle.
Unit 1 Study Strategies: Two-Dimensional Figures Lesson Vocab Word Definition Example Formed by two rays or line segments that have the same 1 Angle endpoint. The shared endpoint is called the vertex.
More informationPolygon Triangulation
Polygon Triangulation Definition Simple Polygons 1. A polygon is the region of a plane bounded by a finite collection of line segments forming a simple closed curve. 2. Simple closed curve means a certain
More informationCS6100: Topics in Design and Analysis of Algorithms
CS6100: Topics in Design and Analysis of Algorithms Guarding and Triangulating Polygons John Augustine CS6100 (Even 2012): Guarding and Triangulating Polygons The Art Gallery Problem A simple polygon is
More informationOptimal Angular Resolution for Face-Symmetric Drawings
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 15, no. 4, pp. 551 564 (2011) Optimal Angular Resolution for Face-Symmetric Drawings David Eppstein 1 Kevin A. Wortman 2 1 Department
More informationEXTREME POINTS AND AFFINE EQUIVALENCE
EXTREME POINTS AND AFFINE EQUIVALENCE The purpose of this note is to use the notions of extreme points and affine transformations which are studied in the file affine-convex.pdf to prove that certain standard
More informationElementary Planar Geometry
Elementary Planar Geometry What is a geometric solid? It is the part of space occupied by a physical object. A geometric solid is separated from the surrounding space by a surface. A part of the surface
More informationComputational Geometry
Motivation Motivation Polygons and visibility Visibility in polygons Triangulation Proof of the Art gallery theorem Two points in a simple polygon can see each other if their connecting line segment is
More informationLecture 3: Art Gallery Problems and Polygon Triangulation
EECS 396/496: Computational Geometry Fall 2017 Lecture 3: Art Gallery Problems and Polygon Triangulation Lecturer: Huck Bennett In this lecture, we study the problem of guarding an art gallery (specified
More informationPractical Linear Algebra: A Geometry Toolbox
Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 17: Breaking It Up: Triangles Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/pla
More informationQUADRATIC TIME COMPUTABLE INSTANCES OF MAXMIN AND MINMAX AREA TRIANGULATIONS OF CONVEX POLYGONS. Tigran Mirzoev, Tzvetalin S.
Serdica J. Computing 4 (2010), 335 348 QUADRATIC TIME COMPUTABLE INSTANCES OF MAXMIN AND MINMAX AREA TRIANGULATIONS OF CONVEX POLYGONS Tigran Mirzoev, Tzvetalin S. Vassilev Abstract. We consider the problems
More informationGeneralized Delaunay Triangulation for Planar Graphs*
Discrete Comput Geom 1:201-217 (1986) 1986 Spr~nger-Vedag New York Inc, y Generalized Delaunay Triangulation for Planar Graphs* D. T. Lee and A. K. Lin Department of Electrical Engineering and Computer
More information6-1 Properties and Attributes of Polygons
6-1 Properties and Attributes of Polygons Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up 1. A? is a three-sided polygon. triangle 2. A? is a four-sided polygon. quadrilateral Evaluate each expression
More informationSimplicial Complexes: Second Lecture
Simplicial Complexes: Second Lecture 4 Nov, 2010 1 Overview Today we have two main goals: Prove that every continuous map between triangulable spaces can be approximated by a simplicial map. To do this,
More informationCMSC 425: Lecture 9 Geometric Data Structures for Games: Geometric Graphs Thursday, Feb 21, 2013
CMSC 425: Lecture 9 Geometric Data Structures for Games: Geometric Graphs Thursday, Feb 21, 2013 Reading: Today s materials is presented in part in Computational Geometry: Algorithms and Applications (3rd
More informationBasic Euclidean Geometry
hapter 1 asic Euclidean Geometry This chapter is not intended to be a complete survey of basic Euclidean Geometry, but rather a review for those who have previously taken a geometry course For a definitive
More informationConstrained Boundary Recovery for Three Dimensional Delaunay Triangulations
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2004; 00:01 [Version: 2000/01/19 v2.0] Constrained Boundary Recovery for Three Dimensional Delaunay Triangulations
More informationComputational Geometry Lecture Duality of Points and Lines
Computational Geometry Lecture Duality of Points and Lines INSTITUTE FOR THEORETICAL INFORMATICS FACULTY OF INFORMATICS 11.1.2016 Duality Transforms We have seen duality for planar graphs and duality of
More informationLecture 11 Combinatorial Planning: In the Plane
CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University Lecture 11 Combinatorial Planning: In the Plane Instructor: Jingjin Yu Outline Convex shapes, revisited Combinatorial planning
More informationGeometry/Trigonometry Unit 5: Polygon Notes Period:
Geometry/Trigonometry Unit 5: Polygon Notes Name: Date: Period: # (1) Page 270 271 #8 14 Even, #15 20, #27-32 (2) Page 276 1 10, #11 25 Odd (3) Page 276 277 #12 30 Even (4) Page 283 #1-14 All (5) Page
More informationWarm-Up Exercises. 1. If the measures of two angles of a triangle are 19º and 80º, find the measure of the third angle. ANSWER 81º
Warm-Up Exercises 1. If the measures of two angles of a triangle are 19º and 80º, find the measure of the third angle. 81º 2. Solve (x 2)180 = 1980. 13 Warm-Up Exercises 3. Find the value of x. 126 EXAMPLE
More informationAMS 345/CSE 355 Computational Geometry
AMS 345/CSE 355 Computational Geometry Lecture: Polygons, Guarding Joe Mitchell Do You Like Puzzles? Come to the new Stony Brook Puzzle Society Meets: Friday 1:05-2:30 pm at CSE 2120 Organizer: Pramod
More informationVoronoi Diagrams and Delaunay Triangulation slides by Andy Mirzaian (a subset of the original slides are used here)
Voronoi Diagrams and Delaunay Triangulation slides by Andy Mirzaian (a subset of the original slides are used here) Voronoi Diagram & Delaunay Triangualtion Algorithms Divide-&-Conquer Plane Sweep Lifting
More informationComputational Geometry
Computational Geometry 600.658 Convexity A set S is convex if for any two points p, q S the line segment pq S. S p S q Not convex Convex? Convexity A set S is convex if it is the intersection of (possibly
More informationScientific Computing WS 2018/2019. Lecture 12. Jürgen Fuhrmann Lecture 12 Slide 1
Scientific Computing WS 2018/2019 Lecture 12 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 12 Slide 1 Recap For more discussion of mesh generation, see J.R. Shewchuk: Lecture Notes on Delaunay
More informationPolygon Partitioning. Lecture03
1 Polygon Partitioning Lecture03 2 History of Triangulation Algorithms 3 Outline Monotone polygon Triangulation of monotone polygon Trapezoidal decomposition Decomposition in monotone mountain Convex decomposition
More informationCS133 Computational Geometry
CS133 Computational Geometry Voronoi Diagram Delaunay Triangulation 5/17/2018 1 Nearest Neighbor Problem Given a set of points P and a query point q, find the closest point p P to q p, r P, dist p, q dist(r,
More informationpine cone Ratio = 13:8 or 8:5
Chapter 10: Introducing Geometry 10.1 Basic Ideas of Geometry Geometry is everywhere o Road signs o Carpentry o Architecture o Interior design o Advertising o Art o Science Understanding and appreciating
More informationarxiv: v1 [cs.cg] 7 Oct 2017
A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations Anna Lubiw 1, Zuzana Masárová 2, and Uli Wagner 2 arxiv:1710.02741v1 [cs.cg] 7 Oct 2017 1 School of Computer Science, University
More informationQuadtrees and Meshing
Computational Geometry Lecture INSTITUT FÜR THEORETISCHE INFORMATIK FAKULTÄT FÜR INFORMATIK Tamara Mchedlidze Darren Strash 14.12.2015 Motivation: Meshing PC Board Layouts To simulate the heat produced
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.854J / 18.415J Advanced Algorithms Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.415/6.854 Advanced
More informationOther Voronoi/Delaunay Structures
Other Voronoi/Delaunay Structures Overview Alpha hulls (a subset of Delaunay graph) Extension of Voronoi Diagrams Convex Hull What is it good for? The bounding region of a point set Not so good for describing
More informationThe National Strategies Secondary Mathematics exemplification: Y8, 9
Mathematics exemplification: Y8, 9 183 As outcomes, Year 8 pupils should, for example: Understand a proof that the sum of the angles of a triangle is 180 and of a quadrilateral is 360, and that the exterior
More informationMISCELLANEOUS SHAPES
MISCELLANEOUS SHAPES 4.1. INTRODUCTION Five generic shapes of polygons have been usefully distinguished in the literature: convex, orthogonal, star, spiral, and monotone. 1 Convex polygons obviously do
More informationLecture Notes on Delaunay Mesh Generation
Lecture Notes on Delaunay Mesh Generation Jonathan Richard Shewchuk September 20, 1999 Department of Electrical Engineering and Computer Science University of California at Berkeley Berkeley, CA 94720
More informationGeometry Chapter 8 Test Review
Geometry Chapter 8 Test Review Short Answer 1. Find the sum of the measures of the interior angles of the indicated convex polygon. Decagon 2. Find the sum of the measures of the interior angles of the
More informationLecture # 7 The Combinatorial Approach to Knot Theory Mike McCa rey. In this section we will define graphs in the combinatorial sense.
1 Graphs Lecture # 7 The Combinatorial Approach to Knot Theory Mike McCa rey In this section we will define graphs in the combinatorial sense. 1.1 Definitions and Facts Definition A graph is a finite set
More informationInner Approximation of Polygons and Polyhedra by Unions of Boxes
Inner Approximation of Polygons and Polyhedra by Unions of Boxes Christian Spielberger Martin Held Technical Report 2006-02 April 2006 Department of Computer Sciences Jakob-Haringer-Straße 2 5020 Salzburg
More informationPacking Two Disks into a Polygonal Environment
Packing Two Disks into a Polygonal Environment Prosenjit Bose, School of Computer Science, Carleton University. E-mail: jit@cs.carleton.ca Pat Morin, School of Computer Science, Carleton University. E-mail:
More informationHigh-Dimensional Computational Geometry. Jingbo Shang University of Illinois at Urbana-Champaign Mar 5, 2018
High-Dimensional Computational Geometry Jingbo Shang University of Illinois at Urbana-Champaign Mar 5, 2018 Outline 3-D vector geometry High-D hyperplane intersections Convex hull & its extension to 3
More informationContours & Implicit Modelling 1
Contouring & Implicit Modelling Visualisation Lecture 8 Institute for Perception, Action & Behaviour School of Informatics Contours & Implicit Modelling 1 Brief Recap Contouring Implicit Functions lecture
More information2D Geometry. Pierre Alliez Inria Sophia Antipolis
2D Geometry Pierre Alliez Inria Sophia Antipolis Outline Sample problems Polygons Graphs Convex hull Voronoi diagram Delaunay triangulation Sample Problems Line Segment Intersection Theorem: Segments (p
More information7. The Gauss-Bonnet theorem
7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationSaccheri Quadrilaterals
Saccheri Quadrilaterals efinition: Let be any line segment, and erect two perpendiculars at the endpoints and. ark off points and on these perpendiculars so that and lie on the same side of the line, and
More informationAngles of Polygons. Essential Question What is the sum of the measures of the interior angles of a polygon?
7.1 Angles of Polygons Essential Question What is the sum of the measures of the interior angles of a polygon? The Sum of the Angle Measures of a Polygon Work with a partner. Use dynamic geometry software.
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More informationOn Minimum Weight Pseudo-Triangulations
On Minimum Weight Pseudo-Triangulations Oswin Aichholzer Franz Aurenhammer Thomas Hackl Bettina Speckmann Abstract In this note we discuss some structural properties of minimum weight pseudo-triangulations.
More informationProjects & Polygon Triangulation
Computational Geometry Lecture Projects & INSTITUTE FOR THEORETICAL INFORMATICS FACULTY OF INFORMATICS Tamara Mchedlidze Darren Strash 02.11.2015 1 Projects 2 Project Specifications Groups: 2 or 3 students,
More informationOutline of the presentation
Surface Reconstruction Petra Surynková Charles University in Prague Faculty of Mathematics and Physics petra.surynkova@mff.cuni.cz Outline of the presentation My work up to now Surfaces of Building Practice
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationClipping a Mesh Against a Plane
Clipping a Mesh Against a Plane David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To
More informationLattice Polygon s and Pick s Theorem From Dana Paquin and Tom Davis 1 Warm-Up to Ponder
Lattice Polygon s and Pick s Theorem From Dana Paquin and Tom Davis http://www.geometer.org/mathcircles/pick.pdf 1 Warm-Up to Ponder 1. Is it possible to draw an equilateral triangle on graph paper so
More informationConvex Optimization MLSS 2015
Convex Optimization MLSS 2015 Constantine Caramanis The University of Texas at Austin The Optimization Problem minimize : f (x) subject to : x X. The Optimization Problem minimize : f (x) subject to :
More informationCS 532: 3D Computer Vision 11 th Set of Notes
1 CS 532: 3D Computer Vision 11 th Set of Notes Instructor: Philippos Mordohai Webpage: www.cs.stevens.edu/~mordohai E-mail: Philippos.Mordohai@stevens.edu Office: Lieb 215 Lecture Outline Line Intersection
More information6th Bay Area Mathematical Olympiad
6th Bay Area Mathematical Olympiad February 4, 004 Problems and Solutions 1 A tiling of the plane with polygons consists of placing the polygons in the plane so that interiors of polygons do not overlap,
More informationVolume Illumination and Segmentation
Volume Illumination and Segmentation Computer Animation and Visualisation Lecture 13 Institute for Perception, Action & Behaviour School of Informatics Overview Volume illumination Segmentation Volume
More informationGeometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review
Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon -
More informationSimultaneously flippable edges in triangulations
Simultaneously flippable edges in triangulations Diane L. Souvaine 1, Csaba D. Tóth 2, and Andrew Winslow 1 1 Tufts University, Medford MA 02155, USA, {dls,awinslow}@cs.tufts.edu 2 University of Calgary,
More information274 Curves on Surfaces, Lecture 5
274 Curves on Surfaces, Lecture 5 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 5 Ideal polygons Previously we discussed three models of the hyperbolic plane: the Poincaré disk, the upper half-plane,
More informationLecture Notes 2: The Simplex Algorithm
Algorithmic Methods 25/10/2010 Lecture Notes 2: The Simplex Algorithm Professor: Yossi Azar Scribe:Kiril Solovey 1 Introduction In this lecture we will present the Simplex algorithm, finish some unresolved
More informationThe Art Gallery Problem: An Overview and Extension to Chromatic Coloring and Mobile Guards
The Art Gallery Problem: An Overview and Extension to Chromatic Coloring and Mobile Guards Nicole Chesnokov May 16, 2018 Contents 1 Introduction 2 2 The Art Gallery Problem 3 2.1 Proof..................................
More information[Me] Meisters, G. H., Polygons have ears, American Mathematical Monthly, June/July 1975, pp
4. Applications Meisters [Me] Two-Ears Theorem was motivated by the problem of triangulating a simple polygon. In fact Meisters suggests a greedy, but concise algorithm to achieve this goal, i.e., find
More information