CS 177 Homework 1. Julian Panetta. October 22, We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F:
|
|
- Olivia Joseph
- 5 years ago
- Views:
Transcription
1 CS 177 Homework 1 Julian Panetta October, Euler Characteristic 1.1 Polyhedral Formula We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F: V E + F = 1 First, let s show that this relationship holds for any individual polygon. It is clearly true for all triangles (n = ) since they have vertices, edges, and 1 face. Notice that any polygon with n > vertices can be turned into a polygon with n 1 vertices by removing a vertex and the two edges incident on that vertex then inserting a single new edge to close the polygon. This operation does not alter the Euler characteristic, since V reduces by 1, but E also reduces by 1 to counter its effect. So all polygons with n vertices have the same Euler characteristic as a polygon with n 1 vertices. Hence, if all polygons with n 1 vertices have an Euler characteristic of 1, all polygons with n vertices will also have an Euler characteristic of 1. Considering our base case that showed the Euler characteristic is 1 for all polygons with n = vertices, we see by induction that any polygon with n vertices has an Euler characteristic of 1 n. Thus the relationship is satisfied for all polygons. Now, we want to extend this to show that the relationship is satisfied for all polygonal disks. Because all polygons satisfy the relationship, clearly we have a base case that all polygonal disks made up of one polygon satisfy the relationship. Consider a polygonal disk made up of n > polygons. This disk can be turned into a polygonal disk (with the same silhouette!) by finding two adjacent polygons within the disk (these are guaranteed to exist since n > 1), and merging them by removing all edges they share in common. Say they share s edges. Then E is reduced by s, lowering the Euler characteristic. However, we obviously must remove a vertex when we remove both edges incident on it, so if s > 1, we remove s 1 vertices as we remove the s shared edges. Also, in merging the two faces into one, we reduce the face count, F by exactly one. Thus, the net effect on the Euler characteristic is V E + F = (s 1) ( s) 1 = s+1+s 1 = 0. Thus all polygonal disks of n polygons have the same Euler characteristic as some polygonal disk of n 1 polygons. That means if all polygonal disks made up of n 1 polygons have satisfy the relationship, all polygonal disks of n polygons must also satisfy the relationship. Because we already saw the base case where a polygonal disk of 1 polygon satisfies the relationship, by induction we see that all polygonal disks satisfy V E + F = 1. Furthermore, we can use this fact to prove something about the Euler characteristic of a polyhedron. Notice that if we remove a single face and no attached vertices or edges from any polyhedron, we get a polygonal disk (i.e., the resultant mesh with a boundary can be deformed into a disk without cutting or gluing). In performing this operation, we reduced the Euler characteristic by 1. Therefore the Euler characteristic of any polyhedron is exactly one more than the Euler characteristic of any polygonal disk, and we must have: 1
2 V E + F = 1. Euler-Poincaré Formula There is nothing to prove, but we are presented the Euler characteristic, V E + F = g. Tessellation.1 Regular Valence Consider a (connected, orientable) simplicial surface, K, with all vertices of regular valence. K is, by definition, made up only of triangular faces. We want to show that K s Euler characteristic is 0, which implies that g = 1. Let s begin by counting the edges, E, and faces, F in terms of V, the number of vertices in K. Count the edges by looping over each vertex and adding up the 6 edges incident on that vertex. After this procedure terminates, we have touched each edge twice (once for each endpoint), and we have therefore double-counted. It is easy to see that E = 6 V = V. Count the faces by again looping over each vertex and adding up the 6 triangles incident on that vertex. After this procedure terminates, we have touched each face three times (once for each of its three vertices), and we have therefore triple-counted. It is evident, then, that F = 6 V = V. Now we can compute the Euler characteristic: X = V E + F = V V + V = 0 = g = g = 1 Therefore, the only (connected, orientable) simplicial surface for which every vertex has regular valence is a torus (g = 1).. Minimally Irregular Valence We ve just seen a case with a torus that has all regular vertices, so clearly m(k) = 0 when g = 1. Now, let s consider the case where g = 0. By Euler-Poincaré: V E + F = g = To get an expression only in terms of vertex and face counts, let s count the edges in terms of faces. If we loop over all the faces and count three edges per face, we will end up counting each edge twice because each edge is shared by two faces. Therefore the number of edges is E = F, and V F + F = V F = = V = F + 4 Let I be the number of irregular vertices. Then V I is the number of regular vertices. Let s count the number of faces in terms of vertices. Each regular vertex touches 6 triangles, and each irregular vertex must touch at least triangles by the assumption that all vertices have at least valence. If we count faces by adding up all the triangles incident on all vertices, we will count each triangle times (once for each vertex belonging to the triangle), so:
3 This means that: 6( V I) + I F = V I + I = V I V = F + 4 V I + 4 = I 4 In other words, a simplicial surface of genus g = 0 must have at least 4 irregular vertices. To show that this bound is tight, consider a tetrahedron; this is a simplicial surface with 4 irregular vertices. Therefore m(k) = 4 when g = 0. Finally, let s consider the case where g. We already know from.1 there must be at least one irregular vertex because we showed that the only case where all vertices are regular is when g = 1. However, we can show that having just a single vertex of irregular valence is sufficient to bring down the Euler characteristic to any negative value resulting from g. Let X be the valence of our single irregular vertex. There are V 1 regular vertices if only one is irregular, so we can count the faces to be: F = 6( V 1) + X = V + X So, since we already found E = F, the Euler characteristic of the simplicial surface is: V E + F = V F = V V + X = V V x 6 = 1 X 6 Our only restriction is that X, so we see that regardless of how low the RHS of the Euler-Poincaré formula goes as g is increased, we can always increase X, the valence of the single irregular vertex, to lower the LHS so equality is maintained. Thus, when g, m(k) = 1. Collecting everything we ve shown, we see: 4, g = 0 m(k) = 0, g = 1 1, g Discrete Gaussian Curvature.1 Area of a Spherical Triangle Consider the spherical triangle and its diangles shown in figure 1. Because each diangle covers ( α i ) π = α i π of the unit sphere s surface, the surface area of each diangle is: ( A i = (4πr αi ) ( αi ) ) = (4 π) = 4α i π π We can find the spherical triangle s area by combining A 1, A, and A. Note that all three diangles overlap exactly at the shaded spherical triangle and at the identical spherical triangle on the opposite side of the sphere to cover them each three times. Furthermore, the diangles collectively cover all the remaining sphere surface exactly once. If we take A s to be the area of the sphere, and A = A to be the area of the spherical triangle on the opposite side, we find:
4 .1 Area of a Spherical Triangle Show that the area of a spherical triangle on the unit sphere with interior angles α 1, α, α is (Hint: consider the areas A = α 1 + α + α π. A 1, A, A of the three shaded regions (called diangles ) pictured below.) 1 Figure 1: The diangles of spherical triangle with angles α 1 α α. A 1 corresponds to α 1, A to α, and A to α. Note that there is an identical version of the spherical triangle in the back of the sphere (it has the same angles as the original triangle.) A 1 + A + A = A + A + (A s A A ) = A + A + 4π r = 4A + 4π A = 1 4 (A 1 + A + A ) π = 1 4 (4α 1 + 4α + 4α ) π = α 1 + α + α π. Area of a Spherical Polygon Using the standard ear trimming algorithm (or another more efficient algorithm), one can triangulate any n-gon, P, into n polygons. This algorithm, which forms and removes a triangle by taking two outside edges and creating one internal edge, will clearly work on the surface of a sphere. Because a triangulation of a polygon will have the same area as the original polygon, we can measure the area of any spherical polygon by adding up the area of the individual triangular components of the triangularization, T which we know how to measure from the last problem: A(P ) = A(t) = t,1 + α t, + α t, π) t T t T(α ( ) = α t,1 + α t, + α t, (n )π = ( n)π + α α T t T 4
5 Where α T means all angles in the triangulation. The important thing to note here is that the all the angles in the triangularization of the n-gon can be partitioned into n nonempty partitions whose members add up to a polygon angle β i i {1,..., n}. This partition is constructed by grouping all the interior angles at polygon vertex i of all the triangles incident on i. As a result of this fact: And so we have found: α = α T i=1 β i A(P ) = ( n)π + i=1 β i. Angle Defect Figure : Here we see a polygonal surface and two orthogonal planes that are spanned by face f s normal and one of each of the two neighboring face normals. The face normals are shown in green lines, the two angles we want to relate are shown in blue, and the perpendicular relationship between the edges of face f and the planes is also noted. If we can find the angles of the polygon defined by the intersection of the face normals with the unit sphere, we can use the result of. to find the area of this polygon, the Gaussian curvature. For consistency with., let s call these angles β f, where f is the face number of the face whose normal defines the polygon vertex at this angle. Then, if we number the faces so that f F v = {1,..., n}, we can use our formula from. verbatim to express the Gaussian curvature: K = ( n)π + β i (1) i=1 5
6 We want to express this curvature in terms of f (v), the interior angle of face f at vertex v. In figure, this is shown as f. Also, the β f shown in this figure is in fact the angle at vertex f of the spherical polygon. This is clear since the edges of the spherical polygon are the great circle arcs defined by the intersection with the unit sphere of the planes spanned by two adjacent face normals, and β f is the angle between two such adjacent planes. To find β f in terms of f (v), we note that the edges of face f are actually normal to the planes. This is because the normal to the edges of face f can be thought range from face f s normal to the adjacent face s normal. In other words, the normal of the edge lies in a sector on the plane spanned by the two adjacent face normals, so the edge itself must be perpendicular to that plane. This is illustrated by the blue lines on figure. From that figure, we see that β f and f (v) therefore belong to a quadrilateral whose other two angles are both π. So that means: β f + f (v) + π + π = π = β f = π f (v) Plugging this angle into (1) we see: K = ( n)π + (π i=1 K = π i(v)) = ( n)π + nπ f (v) = d(v) f (v) And we have shown that the Gaussian curvature is equal to the angle defect, d(v)..4 Discrete Gauss-Bonnet Theorem First, we count the number of edges in terms of the number of faces. If we loop over all the faces and count three edges per face, we will end up counting each edge twice because each edge is shared by two faces. Therefore the number of edges is E = F. That means we can write the Euler characteristic as: And so: X = V E + F = V F + F = V F πx = π V π F Now looking at the sum over all vertices of the angle defect, we see: d(v ) = π f (v) = π V f (v) This double summation adds up all the angles incident on every vertex, so it will add up each angle in the surface exactly once. Since each face s angles adds up to π, the sum of all angles in the simplicial surface is π F. Thus: 6
7 d(v ) = π V π F = πx 7
GAUSS-BONNET FOR DISCRETE SURFACES
GAUSS-BONNET FOR DISCRETE SURFACES SOHINI UPADHYAY Abstract. Gauss-Bonnet is a deep result in differential geometry that illustrates a fundamental relationship between the curvature of a surface and its
More informationConvex Hulls (3D) O Rourke, Chapter 4
Convex Hulls (3D) O Rourke, Chapter 4 Outline Polyhedra Polytopes Euler Characteristic (Oriented) Mesh Representation Polyhedra Definition: A polyhedron is a solid region in 3D space whose boundary is
More informationWeek 7 Convex Hulls in 3D
1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons
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 information6.2 Classification of Closed Surfaces
Table 6.1: A polygon diagram 6.1.2 Second Proof: Compactifying Teichmuller Space 6.2 Classification of Closed Surfaces We saw that each surface has a triangulation. Compact surfaces have finite triangulations.
More informationSurfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
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 informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 8 Solutions
Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 8 Solutions Exercises from Chapter 2: 5.5, 5.10, 5.13, 5.14 Exercises from Chapter 3: 1.2, 1.3, 1.5 Exercise 5.5. Give an example
More informationMatching and Planarity
Matching and Planarity Po-Shen Loh June 010 1 Warm-up 1. (Bondy 1.5.9.) There are n points in the plane such that every pair of points has distance 1. Show that there are at most n (unordered) pairs of
More informationPick up some wrapping paper.
Pick up some wrapping paper. What is the area of the following Christmas Tree? There is a nice theorem that allows one to compute the area of any simply-connected (i.e. no holes) grid polygon quickly.
More informationEuler s Theorem. Brett Chenoweth. February 26, 2013
Euler s Theorem Brett Chenoweth February 26, 2013 1 Introduction This summer I have spent six weeks of my holidays working on a research project funded by the AMSI. The title of my project was Euler s
More informationCurvature Berkeley Math Circle January 08, 2013
Curvature Berkeley Math Circle January 08, 2013 Linda Green linda@marinmathcircle.org Parts of this handout are taken from Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill
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 informationTangencies between disjoint regions in the plane
June 16, 20 Problem Definition Two nonoverlapping Jordan regions in the plane are said to touch each other or to be tangent to each other if their boundaries have precisely one point in common and their
More informationShape Modeling and Geometry Processing
252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry
More informationCS195H Homework 5. Due:March 12th, 2015
CS195H Homework 5 Due:March 12th, 2015 As usual, please work in pairs. Math Stuff For us, a surface is a finite collection of triangles (or other polygons, but let s stick with triangles for now) with
More informationTWO CONTRIBUTIONS OF EULER
TWO CONTRIBUTIONS OF EULER SIEMION FAJTLOWICZ. MATH 4315 Eulerian Tours. Although some mathematical problems which now can be thought of as graph-theoretical, go back to the times of Euclid, the invention
More informationGeometry of Flat Surfaces
Geometry of Flat Surfaces Marcelo iana IMPA - Rio de Janeiro Xi an Jiaotong University 2005 Geometry of Flat Surfaces p.1/43 Some (non-flat) surfaces Sphere (g = 0) Torus (g = 1) Bitorus (g = 2) Geometry
More informationAnswer Key: Three-Dimensional Cross Sections
Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection
More informationTutorial 3 Comparing Biological Shapes Patrice Koehl and Joel Hass
Tutorial 3 Comparing Biological Shapes Patrice Koehl and Joel Hass University of California, Davis, USA http://www.cs.ucdavis.edu/~koehl/ims2017/ What is a shape? A shape is a 2-manifold with a Riemannian
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 informationMesh data structures. Abstract. 1 Introduction. 2 Method. 2.1 Implement the half edge mesh LIU TNM
Mesh data structures Fredrik Salomonsson fresa517@student.liu.se LIU TNM079 2010-05-28 Abstract When dealing with meshes containing large amount of triangles or quadrilaterals one need to have some kind
More informationDISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU /858B Fall 2017
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B Fall 2017 LECTURE 10: DISCRETE CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B
More informationLectures in Discrete Differential Geometry 3 Discrete Surfaces
Lectures in Discrete Differential Geometry 3 Discrete Surfaces Etienne Vouga March 19, 2014 1 Triangle Meshes We will now study discrete surfaces and build up a parallel theory of curvature that mimics
More information1 Descartes' Analogy
Ellipse Line Paraboloid Parabola Descartes' Analogy www.magicmathworks.org/geomlab Hyperbola Hyperboloid Hyperbolic paraboloid Sphere Cone Cylinder Circle The total angle defect for a polygon is 360 ;
More informationOne simple example is that of a cube. Each face is a square (=regular quadrilateral) and each vertex is connected to exactly three squares.
Berkeley Math Circle Intermediate I, 1/23, 1/20, 2/6 Presenter: Elysée Wilson-Egolf Topic: Polygons, Polyhedra, Polytope Series Part 1 Polygon Angle Formula Let s start simple. How do we find the sum of
More informationThe following is a summary, hand-waving certain things which actually should be proven.
1 Basics of Planar Graphs The following is a summary, hand-waving certain things which actually should be proven. 1.1 Plane Graphs A plane graph is a graph embedded in the plane such that no pair of lines
More information751 Problem Set I JWR. Due Sep 28, 2004
751 Problem Set I JWR Due Sep 28, 2004 Exercise 1. For any space X define an equivalence relation by x y iff here is a path γ : I X with γ(0) = x and γ(1) = y. The equivalence classes are called the path
More informationINTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES
INTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES RACHEL CARANDANG Abstract. This paper provides an overview of the homology groups of a 2- dimensional complex. It then demonstrates a proof of the Invariance
More informationCoxeter Decompositions of Hyperbolic Polygons
Europ. J. Combinatorics (1998) 19, 801 817 Article No. ej980238 Coxeter Decompositions of Hyperbolic Polygons A. A. FELIKSON Let P be a polygon on hyperbolic plane H 2. A Coxeter decomposition of a polygon
More informationThe Graphs of Triangulations of Polygons
The Graphs of Triangulations of Polygons Matthew O Meara Research Experience for Undergraduates Summer 006 Basic Considerations Let Γ(n) be the graph with vertices being the labeled planar triangulation
More informationThe radius for a regular polygon is the same as the radius of the circumscribed circle.
Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
More informationDefinitions. Topology/Geometry of Geodesics. Joseph D. Clinton. SNEC June Magnus J. Wenninger
Topology/Geometry of Geodesics Joseph D. Clinton SNEC-04 28-29 June 2003 Magnus J. Wenninger Introduction Definitions Topology Goldberg s polyhedra Classes of Geodesic polyhedra Triangular tessellations
More informationIntroduction to Rational Billiards II. Talk by John Smillie. August 21, 2007
Introduction to Rational Billiards II Talk by John Smillie August 21, 2007 Translation surfaces and their singularities Last time we described the Zemlyakov-Katok construction for billiards on a triangular
More informationWhat would you see if you live on a flat torus? What is the relationship between it and a room with 2 mirrors?
DAY I Activity I: What is the sum of the angles of a triangle? How can you show it? How about a quadrilateral (a shape with 4 sides)? A pentagon (a shape with 5 sides)? Can you find the sum of their angles
More informationNotes on Spherical Geometry
Notes on Spherical Geometry Abhijit Champanerkar College of Staten Island & The Graduate Center, CUNY Spring 2018 1. Vectors and planes in R 3 To review vector, dot and cross products, lines and planes
More informationDefinition 1 (Hand-shake model). A hand shake model is an incidence geometry for which every line has exactly two points.
Math 3181 Dr. Franz Rothe Name: All3181\3181_spr13t1.tex 1 Solution of Test I Definition 1 (Hand-shake model). A hand shake model is an incidence geometry for which every line has exactly two points. Definition
More informationCLASSIFICATION OF SURFACES
CLASSIFICATION OF SURFACES JUSTIN HUANG Abstract. We will classify compact, connected surfaces into three classes: the sphere, the connected sum of tori, and the connected sum of projective planes. Contents
More informationTriangulations of hyperbolic 3-manifolds admitting strict angle structures
Triangulations of hyperbolic 3-manifolds admitting strict angle structures Craig D. Hodgson, J. Hyam Rubinstein and Henry Segerman segerman@unimelb.edu.au University of Melbourne January 4 th 2012 Ideal
More informationTHE DNA INEQUALITY POWER ROUND
THE DNA INEQUALITY POWER ROUND Instructions Write/draw all solutions neatly, with at most one question per page, clearly numbered. Turn in the solutions in numerical order, with your team name at the upper
More information1 Discrete Connections
CS 177: Discrete Differential Geometry Homework 4: Vector Field Design (due: Tuesday Nov 30th, 11:59pm) In the last homework you saw how to decompose an existing vector field using DEC. In this homework
More informationarxiv: v1 [math.co] 15 Apr 2018
REGULAR POLYGON SURFACES IAN M. ALEVY arxiv:1804.05452v1 [math.co] 15 Apr 2018 Abstract. A regular polygon surface M is a surface graph (Σ, Γ) together with a continuous map ψ from Σ into Euclidean 3-space
More informationColoring. Radhika Gupta. Problem 1. What is the chromatic number of the arc graph of a polygonal disc of N sides?
Coloring Radhika Gupta 1 Coloring of A N Let A N be the arc graph of a polygonal disc with N sides, N > 4 Problem 1 What is the chromatic number of the arc graph of a polygonal disc of N sides? Or we would
More informationChapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings
Chapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 12: Planar Graphs Math 184A / Fall 2017 1 / 45 12.1 12.2. Planar graphs Definition
More informationPlanar graphs. Math Prof. Kindred - Lecture 16 Page 1
Planar graphs Typically a drawing of a graph is simply a notational shorthand or a more visual way to capture the structure of the graph. Now we focus on the drawings themselves. Definition A drawing of
More informationThe angle measure at for example the vertex A is denoted by m A, or m BAC.
MT 200 ourse notes on Geometry 5 2. Triangles and congruence of triangles 2.1. asic measurements. Three distinct lines, a, b and c, no two of which are parallel, form a triangle. That is, they divide the
More informationGrade VIII. Mathematics Geometry Notes. #GrowWithGreen
Grade VIII Mathematics Geometry Notes #GrowWithGreen Polygons can be classified according to their number of sides (or vertices). The sum of all the interior angles of an n -sided polygon is given by,
More informationProceedings - AutoCarto Columbus, Ohio, USA - September 16-18, Alan Saalfeld
Voronoi Methods for Spatial Selection Alan Saalfeld ABSTRACT: We define measures of "being evenly distributed" for any finite set of points on a sphere and show how to choose point subsets of arbitrary
More informationEuler Characteristic
Euler Characteristic Rebecca Robinson May 15, 2007 Euler Characteristic Rebecca Robinson 1 PLANAR GRAPHS 1 Planar graphs v = 5, e = 4, f = 1 v e + f = 2 v = 6, e = 7, f = 3 v = 4, e = 6, f = 4 v e + f
More information1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D.
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationAppendix E. Plane Geometry
Appendix E Plane Geometry A. Circle A circle is defined as a closed plane curve every point of which is equidistant from a fixed point within the curve. Figure E-1. Circle components. 1. Pi In mathematics,
More informationPlanar Graphs. 1 Graphs and maps. 1.1 Planarity and duality
Planar Graphs In the first half of this book, we consider mostly planar graphs and their geometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter
More informationMath 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi
Math 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi 1. Euler Characteristic of Surfaces Leonhard Euler noticed that the number v of vertices, the number e of edges and
More informationTHE HALF-EDGE DATA STRUCTURE MODELING AND ANIMATION
THE HALF-EDGE DATA STRUCTURE MODELING AND ANIMATION Dan Englesson danen344@student.liu.se Sunday 12th April, 2011 Abstract In this lab assignment which was done in the course TNM079, Modeling and animation,
More informationHustle Geometry SOLUTIONS MAΘ National Convention 2018 Answers:
Hustle Geometry SOLUTIONS MAΘ National Convention 08 Answers:. 50.. 4. 8 4. 880 5. 6. 6 7 7. 800π 8. 6 9. 8 0. 58. 5.. 69 4. 0 5. 57 6. 66 7. 46 8. 6 9. 0.. 75. 00. 80 4. 8 5 5. 7 8 6+6 + or. Hustle Geometry
More informationAcknowledgement: Scott, Foresman. Geometry. SIMILAR TRIANGLES. 1. Definition: A ratio represents the comparison of two quantities.
1 cknowledgement: Scott, Foresman. Geometry. SIMILR TRINGLS 1. efinition: ratio represents the comparison of two quantities. In figure, ratio of blue squares to white squares is 3 : 5 2. efinition: proportion
More information1 Appendix to notes 2, on Hyperbolic geometry:
1230, notes 3 1 Appendix to notes 2, on Hyperbolic geometry: The axioms of hyperbolic geometry are axioms 1-4 of Euclid, plus an alternative to axiom 5: Axiom 5-h: Given a line l and a point p not on l,
More informationWe have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.
Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the
More information5. THE ISOPERIMETRIC PROBLEM
Math 501 - Differential Geometry Herman Gluck March 1, 2012 5. THE ISOPERIMETRIC PROBLEM Theorem. Let C be a simple closed curve in the plane with length L and bounding a region of area A. Then L 2 4 A,
More informationEXTERNAL VISIBILITY. 1. Definitions and notation. The boundary and interior of
PACIFIC JOURNAL OF MATHEMATICS Vol. 64, No. 2, 1976 EXTERNAL VISIBILITY EDWIN BUCHMAN AND F. A. VALENTINE It is possible to see any eleven vertices of an opaque solid regular icosahedron from some appropriate
More informationGeometry 10 and 11 Notes
Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into
More informationImpulse Gauss Curvatures 2002 SSHE-MA Conference. Howard Iseri Mansfield University
Impulse Gauss Curvatures 2002 SSHE-MA Conference Howard Iseri Mansfield University Abstract: In Riemannian (differential) geometry, the differences between Euclidean geometry, elliptic geometry, and hyperbolic
More information: Mesh Processing. Chapter 8
600.657: Mesh Processing Chapter 8 Handling Mesh Degeneracies [Botsch et al., Polygon Mesh Processing] Class of Approaches Geometric: Preserve the mesh where it s good. Volumetric: Can guarantee no self-intersection.
More informationSpherical Geometry MATH430
Spherical Geometry MATH430 Fall 014 In these notes we summarize some results about the geometry of the sphere that complement should the textbook. Most notions we had on the plane (points, lines, angles,
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 informationPlanarity. 1 Introduction. 2 Topological Results
Planarity 1 Introduction A notion of drawing a graph in the plane has led to some of the most deep results in graph theory. Vaguely speaking by a drawing or embedding of a graph G in the plane we mean
More informationBands: A Physical Data Structure to Represent Both Orientable and Non-Orientable 2-Manifold Meshes
Bands: A Physical Data Structure to Represent Both Orientable and Non-Orientable 2-Manifold Meshes Abstract This paper presents a physical data structure to represent both orientable and non-orientable
More informationGeometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts
Geometry Definitions and Theorems Chapter 9 Definitions and Important Terms & Facts A circle is the set of points in a plane at a given distance from a given point in that plane. The given point is the
More informationFIGURES FOR SOLUTIONS TO SELECTED EXERCISES. V : Introduction to non Euclidean geometry
FIGURES FOR SOLUTIONS TO SELECTED EXERCISES V : Introduction to non Euclidean geometry V.1 : Facts from spherical geometry V.1.1. The objective is to show that the minor arc m does not contain a pair of
More informationChapter 10 Similarity
Chapter 10 Similarity Def: The ratio of the number a to the number b is the number. A proportion is an equality between ratios. a, b, c, and d are called the first, second, third, and fourth terms. The
More informationLecture 4: Trees and Art Galleries
Math 7H: Honors Seminar Professor: Padraic Bartlett Lecture 4: Trees and Art Galleries Week 2 UCSB 2014 1 Prelude: Graph Theory In this talk, we re going to refer frequently to things called graphs and
More informationPolygon Angle-Sum Theorem:
Name Must pass Mastery Check by: [PACKET 5.1: POLYGON -SUM THEOREM] 1 Write your questions here! We all know that the sum of the angles of a triangle equal 180. What about a quadrilateral? A pentagon?
More informationSimple Graph. General Graph
Graph Theory A graph is a collection of points (also called vertices) and lines (also called edges), with each edge ending at a vertex In general, it is allowed for more than one edge to have the same
More informationHyperbolic Geometry on the Figure-Eight Knot Complement
Hyperbolic Geometry on the Figure-Eight Knot Complement Alex Gutierrez Arizona State University December 10, 2012 Hyperbolic Space Hyperbolic Space Hyperbolic space H n is the unique complete simply-connected
More informationGeometric structures on manifolds
CHAPTER 3 Geometric structures on manifolds In this chapter, we give our first examples of hyperbolic manifolds, combining ideas from the previous two chapters. 3.1. Geometric structures 3.1.1. Introductory
More informationScheduling, Map Coloring, and Graph Coloring
Scheduling, Map Coloring, and Graph Coloring Scheduling via Graph Coloring: Final Exam Example Suppose want to schedule some ;inal exams for CS courses with following course numbers: 1007, 3137, 3157,
More informationGraph theory. Po-Shen Loh. June We begin by collecting some basic facts which can be proved via bare-hands techniques.
Graph theory Po-Shen Loh June 013 1 Basic results We begin by collecting some basic facts which can be proved via bare-hands techniques. 1. The sum of all of the degrees is equal to twice the number of
More informationTheta Circles & Polygons 2015 Answer Key 11. C 2. E 13. D 4. B 15. B 6. C 17. A 18. A 9. D 10. D 12. C 14. A 16. D
Theta Circles & Polygons 2015 Answer Key 1. C 2. E 3. D 4. B 5. B 6. C 7. A 8. A 9. D 10. D 11. C 12. C 13. D 14. A 15. B 16. D 17. A 18. A 19. A 20. B 21. B 22. C 23. A 24. C 25. C 26. A 27. C 28. A 29.
More informationSMMG September 16 th, 2006 featuring Dr. Jessica Purcell Geometry out of the Paper: An Introduction to Manifolds
1. Explore a Cylinder SMMG September 16 th, 2006 featuring Dr. Jessica Purcell Geometry out of the Paper: An Introduction to Manifolds Take a strip of paper. Bring the two ends of the strip together to
More informationThe geometry and combinatorics of closed geodesics on hyperbolic surfaces
The geometry and combinatorics of closed geodesics on hyperbolic surfaces CUNY Graduate Center September 8th, 2015 Motivating Question: How are the algebraic/combinatoric properties of closed geodesics
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 informationJordan Curves. A curve is a subset of IR 2 of the form
Jordan Curves A curve is a subset of IR 2 of the form α = {γ(x) : x [0, 1]}, where γ : [0, 1] IR 2 is a continuous mapping from the closed interval [0, 1] to the plane. γ(0) and γ(1) are called the endpoints
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationConvex Hulls in Three Dimensions. Polyhedra
Convex Hulls in Three Dimensions Polyhedra Polyhedron 1.A polyhedron is the generalization of a 2- D polygon to 3-D A finite number of flat polygonal faces The boundary or surface of a polyhedron - Zero-dimensional
More informationDepartment: Course: Chapter 1
Department: Course: 2016-2017 Term, Phrase, or Expression Simple Definition Chapter 1 Comprehension Support Point Line plane collinear coplanar A location in space. It does not have a size or shape The
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 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 informationMathematics Scope & Sequence Geometry
Mathematics Scope & Sequence Geometry Readiness Standard(s) First Six Weeks (29 ) Coordinate Geometry G.7.B use slopes and equations of lines to investigate geometric relationships, including parallel
More informationThe Cyclic Cycle Complex of a Surface
The Cyclic Cycle Complex of a Surface Allen Hatcher A recent paper [BBM] by Bestvina, Bux, and Margalit contains a construction of a cell complex that gives a combinatorial model for the collection of
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 informationThe Farey Tessellation
The Farey Tessellation Seminar: Geometric Structures on manifolds Mareike Pfeil supervised by Dr. Gye-Seon Lee 15.12.2015 Introduction In this paper, we are going to introduce the Farey tessellation. Since
More informationEULER S FORMULA AND THE FIVE COLOR THEOREM
EULER S FORMULA AND THE FIVE COLOR THEOREM MIN JAE SONG Abstract. In this paper, we will define the necessary concepts to formulate map coloring problems. Then, we will prove Euler s formula and apply
More informationMath 443/543 Graph Theory Notes 5: Planar graphs and coloring
Math 443/543 Graph Theory Notes 5: Planar graphs and coloring David Glickenstein October 10, 2014 1 Planar graphs The Three Houses and Three Utilities Problem: Given three houses and three utilities, can
More informationA simple problem that has a solution that is far deeper than expected!
The Water, Gas, Electricity Problem A simple problem that has a solution that is far deeper than expected! Consider the diagram below of three houses and three utilities: water, gas, and electricity. Each
More informationFace Width and Graph Embeddings of face-width 2 and 3
Face Width and Graph Embeddings of face-width 2 and 3 Instructor: Robin Thomas Scribe: Amanda Pascoe 3/12/07 and 3/14/07 1 Representativity Recall the following: Definition 2. Let Σ be a surface, G a graph,
More informationExample: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices:
11.1: Space Figures and Cross Sections Polyhedron: solid that is bounded by polygons Faces: polygons that enclose a polyhedron Edge: line segment that faces meet and form Vertex: point or corner where
More informationEuler Characteristic
Euler Characteristic Sudesh Kalyanswamy 1 Introduction Euler characteristic is a very important topological property which started out as nothing more than a simple formula involving polyhedra. Euler observed
More information7th Bay Area Mathematical Olympiad
7th Bay Area Mathematical Olympiad February 22, 2005 Problems and Solutions 1 An integer is called formidable if it can be written as a sum of distinct powers of 4, and successful if it can be written
More informationDiscrete Mathematics I So Practice Sheet Solutions 1
Discrete Mathematics I So 2016 Tibor Szabó Shagnik Das Practice Sheet Solutions 1 Provided below are possible solutions to the questions from the practice sheet issued towards the end of the course. Exercise
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 information