# Voronoi diagrams and Delaunay triangulations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Voronoi diagrams and Delaunay triangulations Lecture 9, CS march 2004 Antoine Vigneron National University of Singapore NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.1/42

2 Outline today: geometry, no algorithm two problems proximity, mesh generation surprisingly, they are closely related geometric notions Voronoi diagram, Delaunay triangulation planar graph duality references D. Mount Lecture 16 (except algorithm) and 17 textbook chapter 7 (online!) and 9 demo (J. Snoeyink) at: NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.2/42

3 Voronoi diagrams NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.3/42

4 A Voronoi diagram NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.4/42

5 Proximity what is it? a dataset S of n points (called sites) in IR 2 let S = {s 1, s 2... s n } query point q, find closest site to q (=proximity queries) q closest site S NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.5/42

6 Problem how to address proximity? draw a diagram example with S = 2: Closer to s 1 Closer to s 2 s 1 s 2 Bisector of s 1 s 2 this is the Voronoi diagram of S = {s 1, s 2 } NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.6/42

7 Example with S = 3 Voronoi diagram of S = {s 1, s 2, s 3 } V(s 1 ) s 1 v s 2 V(s 2 ) V(s 3 ) s 3 v: a Voronoi vertex. Center of the circumcircle of triangle s 1 s 2 s 3 V(s i ): Voronoi cell of s i NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.7/42

8 Example V(s i ) s i V(s i ) = {x IR 2 j i, s i x < s j x } NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.8/42

9 Half plane h(s i, s j ) we denote h(s i, s j ) = { x IR 2 } s i x < s j x (Closer to s i ) h(s i, s j ) s i s j Bisector of s 1 s 2 NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.9/42

10 Voronoi cell it follows that V(s i ) = j i h(s i, s j ) V(s i ) s i NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.10/42

11 Properties the Voronoi diagram of S is not a planar straight line graph reason: it has infinite edges to fix this problem, we can restrict our attention to the portion of the Voronoi diagram that is within a large bounding box all the cells are convex, hence connected so the Voronoi diagram has n faces, one for each site it has O(n) edges and vertices non trivial; we can use the fact that vertices have degree at least 3+double counting+euler s relation NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.11/42

12 Algorithmic consequences V(s i ) is an intersection of n half planes so it can be computed in O(n log n) time (cf linear programming) we can compute the Voronoi diagram of S in O(n 2 log n) time we associate it with a point location data structure so we can answer proximity queries in: O(n 2 log n) preprocessing time expected O(n) space usage expected O(log n) query time next lecture: preprocessing time down to expected Θ(n log n) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.12/42

13 Voronoi cell 2 x V(s i ) the disk through s i centered at x contains no other site than s i V(s i ) s i x NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.13/42

14 Voronoi edges a Voronoi edge is an edge of the Voronoi diagram a point x on a Voronoi edge is equidistant to two nearest sites s i and s j hence the circle centered at x through s i and s j contains no site in its interior s i x s j a voronoi edge NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.14/42

15 General position assumption general position assumption: no four sites are cocircular a degenerate case: 4 sites lie on the same circle NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.15/42

16 Voronoi vertices a Voronoi vertex v is equidistant to three nearest sites s i, s j and s k hence the circle centered at v through s i, s j and s k contains no site in its interior s i v s j by our general position assumption, each Voronoi vertex has degree 3 (= adjacent to three edges) s k NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.16/42

17 Voronoi cells if V(s i ) is bounded, then it is a convex polygon V(s i ) is unbounded iff s i is a vertex of CH(S) CH(S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.17/42

18 Consequence knowing the Voronoi diagram, we can compute the convex hull in O(n) time so computing a Voronoi diagram takes Ω(n log n) time next lecture: an optimal O(n log n) time randomized algorithm there is also a deterministic O(n log n) time algorithm plane sweep algorithm see Lecture 16 of D. Mount or Lecture 7 in textbook you do not need to read it for CS4235 NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.18/42

19 Further remarks sites need not be points: one can define in the same way the Voronoi diagram of a set of line segments, or any shape we can also use different distance functions the Voronoi diagram can also be defined in IR d for any d but it has size Θ (n d/2 ) so it is only useful when d is small (say, at most 4) active research line: approximate Voronoi diagrams with smaller size NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.19/42

20 Dual of a planar graph NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.20/42

21 Example G NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.21/42

22 Example G G NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.22/42

23 Definition let G be a graph dual graph G each face f of G corresponds to a vertex f of G (f, g ) is an edge of G iff f and g are adjacent in G property: the dual of a planar graph is planar too What is the dual of a Voronoi diagram? NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.23/42

24 Delaunay triangulation NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.24/42

25 Triangulation of a set of points we are given a set S of n points in IR 2 we want to find a planar map with set of vertices S, where CH(S) is partitioned into triangles this is called a triangulation of S NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.25/42

26 The Delaunay triangulation DT (S) the Delaunay triangulation of the same set looks nicer has many interesting properties NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.26/42

27 Definition DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.27/42

28 Definition let S be a set of n points in IR 2 S is in general position: no 4 points are cocircular the Delaunay triangulation DT (S) of S is the embedding of the dual graph of the Voronoi diagram of S where the vertices are the sites: i, V(s i ) = s i the edges of DT (S) are straight line segments NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.28/42

29 Remarks DT (S): is it well defined? we need to prove that edges do not intersect (= it is a PSLG) left as an exercise faces are triangles the number of edges in a face of DT (S) is the degree of the corresponding Voronoi vertex general position assumption implies that Voronoi vertices have degree 3 NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.29/42

30 Convex hull the convex hull of S is the complement of the unbounded face of DT (S) CH(S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.30/42

31 Circumcircle property the circumcircle of any triangle in DT (S) is empty (contains no site in its interior) s 1 s 2 v s 3 DT (S) proof: let s 1 s 2 s 3 be a triangle in DT (S), let v be the corresponding Voronoi vertex. Property of Voronoi vertices: the circle centered at v through s 1 s 2 s 3 is empty NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.31/42

32 Empty circle property s i s j is an edge of DT (S) iff there is an empty circle through s i and s j s i s j DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.32/42

33 Proof (Empty circle property) s i s j DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.33/42

34 Closest pair property the closest two sites s i s j are connected by an edge of DT (S) s i sj DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.34/42

35 Proof (closest pair property) empty s i empty sj NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.35/42

36 Euclidean minimum spanning tree Euclidean graph set of vertices= S for all i j there is an edge between s i and s j with weight s i s j Euclidean Minimum Spanning Tree: minimum spanning tree of the euclidean graph Property: the EMST is a subgraph of DT (S) Corollary: it can be computed in O(n log n) time see D. Mount s notes pages NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.36/42

37 Angle sequence let T be a triangulation of S angle sequence Θ(T ): sequence of all the angles of the triangle of T in non decreasing order example π/4 π/2 π/3 π/3 π/4 π/3 Θ(T ) = (π/4, π/4, π/3, π/3, π/3, π/2) comparison: let T and T be two triangulations of S we compare Θ(T ) and Θ(T ) using lexicographic order example: (1, 1, 3, 4, 5) < (1, 2, 4, 4, 4) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.37/42

38 Optimality of DT (S) Theorem: the angle sequence of DT (S) is maximal among all triangulations of S in other words: the Delaunay triangulation maximizes the minimum angle intuition: avoids skinny triangles DT (P ) skinny triangle NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.38/42

39 Proof idea: flip edges to ensure the circumcircle property it decreases the angle sequence edge flip NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.39/42

40 Degenerate cases several possible Delaunay triangulations equally good in terms of angle sequence example: two possibilities NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.40/42

41 Applications generating good meshes jrs/mesh/ skinny triangles are bad in numerical analysis statistics: natural neighbor interpolation textbook example: height interpolation shape reconstruction... NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.41/42

42 Conclusion next lecture: an O(n log n) time RIC of the Delaunay triangulation from the Delaunay triangulation, we can obtain the Voronoi diagram in O(n) time (how?) the converse is also true, so these two problems are equivalent in an algorithmic point of view there are O(n log n) time deterministic algorithms divide and conquer plane sweep (D. Mount lecture 16) reduction to 3D convex hull (D. Mount lecture 28) not presented in CS 4235 in practice, RIC is used NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.42/42

### Divide-and-Conquer. Three main steps : 1. divide; 2. conquer; 3. merge.

Divide-and-Conquer Three main steps : 1. divide; 2. conquer; 3. merge. 1 Let I denote the (sub)problem instance and S be its solution. The divide-and-conquer strategy can be described as follows. Procedure

### Voronoi with Obstacles, Delaunay Constrained Triangulation and Delaunay 3D

Voronoi with Obstacles, Delaunay Constrained Triangulation and Delaunay 3D Marc Comino Trinidad December 13, 2014 Table of contents Shortest-path Voronoi Diagram Visibility Shortest-path Voronoi Diagram

### Voronoi Diagrams. (Slides mostly by Allen Miu)

Voronoi Diagrams (Slides mostly by Allen Miu) Post Office: What is the area of service? p i : site points q : free point e : Voronoi edge v : Voronoi vertex q p i v e Definition of Voronoi Diagram Let

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

### Algorithmica 9 1989 Springer-Verlag New York Inc.

Algorithmica (1989) 4:97-108 Algorithmica 9 1989 Springer-Verlag New York Inc. Constrained Delaunay Triangulations 1 L. Paul Chew 2 Abstract. Given a set of n vertices in the plane together with a set

### Computational Geometry. Lecture 1: Introduction and Convex Hulls

Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,

### Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

### Fast approximation of the maximum area convex. subset for star-shaped polygons

Fast approximation of the maximum area convex subset for star-shaped polygons D. Coeurjolly 1 and J.-M. Chassery 2 1 Laboratoire LIRIS, CNRS FRE 2672 Université Claude Bernard Lyon 1, 43, Bd du 11 novembre

### Approximation of an Open Polygonal Curve with a Minimum Number of Circular Arcs and Biarcs

Approximation of an Open Polygonal Curve with a Minimum Number of Circular Arcs and Biarcs R. L. Scot Drysdale a Günter Rote b,1 Astrid Sturm b,1 a Department of Computer Science, Dartmouth College b Institut

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### 1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

### Shortest Inspection-Path. Queries in Simple Polygons

Shortest Inspection-Path Queries in Simple Polygons Christian Knauer, Günter Rote B 05-05 April 2005 Shortest Inspection-Path Queries in Simple Polygons Christian Knauer, Günter Rote Institut für Informatik,

### Geometry: Euclidean. Through a given external point there is at most one line parallel to a

Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### Math 3372-College Geometry

Math 3372-College Geometry Yi Wang, Ph.D., Assistant Professor Department of Mathematics Fairmont State University Fairmont, West Virginia Fall, 2004 Fairmont, West Virginia Copyright 2004, Yi Wang Contents

### Shortest Path Algorithms

Shortest Path Algorithms Jaehyun Park CS 97SI Stanford University June 29, 2015 Outline Cross Product Convex Hull Problem Sweep Line Algorithm Intersecting Half-planes Notes on Binary/Ternary Search Cross

### Solving Geometric Problems with the Rotating Calipers *

Solving Geometric Problems with the Rotating Calipers * Godfried Toussaint School of Computer Science McGill University Montreal, Quebec, Canada ABSTRACT Shamos [1] recently showed that the diameter of

### Delaunay Refinement Algorithms for Triangular Mesh Generation

Delaunay Refinement Algorithms for Triangular Mesh Generation Jonathan Richard Shewchuk jrs@cs.berkeley.edu May 21, 2001 Department of Electrical Engineering and Computer Science University of California

### x 2 f 2 e 1 e 4 e 3 e 2 q f 4 x 4 f 3 x 3

Karl-Franzens-Universitat Graz & Technische Universitat Graz SPEZIALFORSCHUNGSBEREICH F 003 OPTIMIERUNG und KONTROLLE Projektbereich DISKRETE OPTIMIERUNG O. Aichholzer F. Aurenhammer R. Hainz New Results

### 3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES

3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES Brian Peasley * I propose an algorithm to construct a 3D point cloud from a sequence of stereo image pairs that show a full 360 degree view of an object.

### Lesson 2: Circles, Chords, Diameters, and Their Relationships

Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct

### Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

### Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the

### ABC is the triangle with vertices at points A, B and C

Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

### Final Review Geometry A Fall Semester

Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over

### Persistent Data Structures and Planar Point Location

Persistent Data Structures and Planar Point Location Inge Li Gørtz Persistent Data Structures Ephemeral Partial persistence Full persistence Confluent persistence V1 V1 V1 V1 V2 q ue V2 V2 V5 V2 V4 V4

### Disjoint Compatible Geometric Matchings

Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,

### Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

### Tetrahedron Meshes. Linh Ha

Tetrahedron Meshes Linh Ha Overview Fundamental Tetrahedron meshing Meshing polyhedra Tetrahedral shape Delaunay refinements Removing silver Variational Tetrahedral Meshing Meshing Polyhedra Is that easy?

### Lesson 5-3: Concurrent Lines, Medians and Altitudes

Playing with bisectors Yesterday we learned some properties of perpendicular bisectors of the sides of triangles, and of triangle angle bisectors. Today we are going to use those skills to construct special

### DEGREE CONSTRAINED TRIANGULATION. Roshan Gyawali

DEGREE CONSTRAINED TRIANGULATION by Roshan Gyawali Bachelor in Computer Engineering Institue of Engineering, Pulchowk Campus, Tribhuvan University, Kathmandu 2008 A thesis submitted in partial fulfillment

### Incenter Circumcenter

TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

### Finding and counting given length cycles

Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

### Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and

### 39 Symmetry of Plane Figures

39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

### Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

### Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key

Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,

### When is a graph planar?

When is a graph planar? Theorem(Euler, 1758) If a plane multigraph G with k components has n vertices, e edges, and f faces, then n e + f = 1 + k. Corollary If G is a simple, planar graph with n(g) 3,

### Terminology: When one line intersects each of two given lines, we call that line a transversal.

Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal.

### STABILITY AWARE DELAUNAY REFINEMENT. Bishal Acharya. Bachelor of Computer Engineering, Kathmandu Engineering College. Tribhuvan University, Nepal

STABILITY AWARE DELAUNAY REFINEMENT By Bishal Acharya Bachelor of Computer Engineering, Kathmandu Engineering College Tribhuvan University, Nepal 2006 A thesis submitted in partial fulfillment of the requirements

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### A NOTE ON EDGE GUARDS IN ART GALLERIES

A NOTE ON EDGE GUARDS IN ART GALLERIES R. Nandakumar (nandacumar@gmail.com) Abstract: We study the Art Gallery Problem with Edge Guards. We give an algorithm to arrange edge guards to guard only the inward

### INTERSECTION OF LINE-SEGMENTS

INTERSECTION OF LINE-SEGMENTS Vera Sacristán Discrete and Algorithmic Geometry Facultat de Matemàtiques i Estadística Universitat Politècnica de Catalunya Problem Input: n line-segments in the plane,

### 28 Closest-Point Problems -------------------------------------------------------------------

28 Closest-Point Problems ------------------------------------------------------------------- Geometric problems involving points on the plane usually involve implicit or explicit treatment of distances

### Chapter 6 Notes: Circles

Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment

### The minimum number of distinct areas of triangles determined by a set of n points in the plane

The minimum number of distinct areas of triangles determined by a set of n points in the plane Rom Pinchasi Israel Institute of Technology, Technion 1 August 6, 007 Abstract We prove a conjecture of Erdős,

### AN ALGORITHM FOR CENTRELINE EXTRACTION USING NATURAL NEIGHBOUR INTERPOLATION

AN ALGORITHM FOR CENTRELINE EXTRACTION USING NATURAL NEIGHBOUR INTERPOLATION Darka Mioc, François Anton and Girija Dharmaraj Department of Geomatics Engineering University of Calgary 2500 University Drive

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

### 4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

### 1 The Line vs Point Test

6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity

### Natural Neighbour Interpolation

Natural Neighbour Interpolation DThe Natural Neighbour method is a geometric estimation technique that uses natural neighbourhood regions generated around each point in the data set. The method is particularly

### Strictly Localized Construction of Planar Bounded-Degree Spanners of Unit Disk Graphs

Strictly Localized Construction of Planar Bounded-Degree Spanners of Unit Disk Graphs Iyad A. Kanj Ljubomir Perković Ge Xia Abstract We describe a new distributed and strictly-localized approach for constructing

### Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### From Perfect Matchings to the Four Colour Theorem

From Perfect Matchings to the Four Colour Theorem Aguilar (Cinvestav), Flores (UNAM), Pérez (HP), Santos (Cantabria), Zaragoza (UAM-A) Universidad Autónoma Metropolitana Unidad Azcapotzalco Departamento

### Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### The Essentials of CAGD

The Essentials of CAGD Chapter 2: Lines and Planes Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentials-cagd c 2000 Farin & Hansford

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Shortcut sets for plane Euclidean networks (Extended abstract) 1

Shortcut sets for plane Euclidean networks (Extended abstract) 1 J. Cáceres a D. Garijo b A. González b A. Márquez b M. L. Puertas a P. Ribeiro c a Departamento de Matemáticas, Universidad de Almería,

### Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

### INTERSECTION OF LINE-SEGMENTS

INTERSECTION OF LINE-SEGMENTS Vera Sacristán Computational Geometry Facultat d Informàtica de Barcelona Universitat Politècnica de Catalunya Problem Input: n line-segments in the plane, s i = (p i, q

### half-line the set of all points on a line on a given side of a given point of the line

Geometry Week 3 Sec 2.1 to 2.4 Definition: section 2.1 half-line the set of all points on a line on a given side of a given point of the line notation: is the half-line that contains all points on the

### Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

### Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

### (67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

### Geometry Module 4 Unit 2 Practice Exam

Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

### Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### The Geometer s Sketchpad: Non-Euclidean Geometry & The Poincaré Disk

The Geometer s Sketchpad: Non-Euclidean Geometry & The Poincaré Disk Nicholas Jackiw njackiw@kcptech.com KCP Technologies, Inc. ICTMT11 2013 Bari Overview. The study of hyperbolic geometry and non-euclidean

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### Medial Axis Construction and Applications in 3D Wireless Sensor Networks

Medial Axis Construction and Applications in 3D Wireless Sensor Networks Su Xia, Ning Ding, Miao Jin, Hongyi Wu, and Yang Yang Presenter: Hongyi Wu University of Louisiana at Lafayette Outline Introduction

### Geometry and Topology from Point Cloud Data

Geometry and Topology from Point Cloud Data Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 1 / 51

### Mean Value Coordinates

Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### TIgeometry.com. Geometry. Angle Bisectors in a Triangle

Angle Bisectors in a Triangle ID: 8892 Time required 40 minutes Topic: Triangles and Their Centers Use inductive reasoning to postulate a relationship between an angle bisector and the arms of the angle.

### Unknown Angle Problems with Inscribed Angles in Circles

: Unknown Angle Problems with Inscribed Angles in Circles Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central

### ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015

ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving

### 55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

### High degree graphs contain large-star factors

High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

### Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume *

Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume * Xiaosong Yang 1, Pheng Ann Heng 2, Zesheng Tang 3 1 Department of Computer Science and Technology, Tsinghua University, Beijing

### BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle

Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and

### Well-Separated Pair Decomposition for the Unit-disk Graph Metric and its Applications

Well-Separated Pair Decomposition for the Unit-disk Graph Metric and its Applications Jie Gao Department of Computer Science Stanford University Joint work with Li Zhang Systems Research Center Hewlett-Packard

### Chapter 5: Relationships within Triangles

Name: Chapter 5: Relationships within Triangles Guided Notes Geometry Fall Semester CH. 5 Guided Notes, page 2 5.1 Midsegment Theorem and Coordinate Proof Term Definition Example midsegment of a triangle

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### CHAPTER 1 CEVA S THEOREM AND MENELAUS S THEOREM

HTR 1 V S THOR N NLUS S THOR The purpose of this chapter is to develop a few results that may be used in later chapters. We will begin with a simple but useful theorem concerning the area ratio of two

### Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry

Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel

### CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the