Geometric Modeling Mortenson Chapter 11. Complex Model Construction
|
|
|
- Toby Davis
- 5 years ago
- Views:
Transcription
1 Geometric Modeling Mortenson Chapter 11 Complex Model Construction
2 Topics Topology of Models Connectivity and other intrinsic properties Graph-Based Models Emphasize topological structure Boolean Models Set theory, set membership classification, Boolean operators Boolean Model Construction Constructive Solid Geometry Boundary Models (B-Rep)
3 Model Topology Euler s Formula for 3D Polyhedra: V E F 2 Poincare s Generalization to n-dimensional Space: N N N typo fixed 2 n Euler-Poincare Formula: (G = genus = number of handles ) V E F 2(1 G) 0
4 Model Topology (continued)
5 Model Topology (continued) 2 ways to join a pair of edges (match numbers) preserves orientation reverses orientation
![Model Topology (continued) Edge 0 of face 1 matched with edge 0 of face 2 [(1,0) (2,0)] [(1,1) (5,0)] [(1,2) (4,0)]](/docs-images/90/102181825/images/6-0.jpg "[(1,3) (3,0)] [(2,1) (3,3)] [(2,2) (6,2)] [(2,3) (5,1)] [(3,1) (4,3)] [(3,2) (6,3)] [(4,1) (5,3)] [(4,2) (6,0)] [(5,2)")
6 Model Topology (continued) Edge 0 of face 1 matched with edge 0 of face 2 [(1,0) (2,0)] [(1,1) (5,0)] [(1,2) (4,0)] [(1,3) (3,0)] [(2,1) (3,3)] [(2,2) (6,2)] [(2,3) (5,1)] [(3,1) (4,3)] [(3,2) (6,3)] [(4,1) (5,3)] [(4,2) (6,0)] [(5,2) (6,1)]
7 Model Topology (continued) Cylinder: orientable Torus: orientable Mobius strip: non-orientable, open surface Klein bottle: nonorientable and does not fit into 3D without self-intersections Orientability is intrinsically defined: left and right are never reversed. Non-orientable: right & left are not intrinsically defined.
8 Model Topology (continued) Sphere: orientable Torus: orientable Klein bottle: non-orientable Projective plane: non-orientable Transition Parity = 1 means match up normally. Transition Parity = -1 means match up in reverse.
9 Model Topology (continued) Curvature of piecewise flat surfaces K Curvature concentrated at vertices Sum up angle excesses of small paths around each vertex. Let: i V E i be excess of a path around vertex i. T i be total turning of a path around vertex i. 1 E i i V 1 (2 T i V i V T i where last part is for closed, piecewise flat surface V ) E 2 F 1 so = Euler characteristic, which is an intrinsic, topological invariant. K 2 V 2 i F 1 f i 2 ( V E F) f i = sum of interior angles of face i. Note this does not require knowledge of how edges are joined.
10 Model Topology (continued) Topology of Closed, Curved Surfaces Net = arbitrary collection of simple arcs (terminated at each end by a vertex) that divide the surface everywhere into topological disks. All valid nets on the same closed surface have the same Euler characteristic. 2 elementary net transformations Adding (or deleting) a face by modifying an edge Adding (or deleting) a vertex is invariant under these net transformations.
11 Model Topology (continued) Euler Operators Euler Object = connected network of faces, vertices, edges All valid nets on the same closed surface have the same Euler characteristic. Euler s formula for polyhedra requires: All faces are topological disks. Object s complement is connected. Each edge adjoins 2 faces with vertex at each end. At least 3 edges meet at each vertex.
12 Model Topology (continued) Spherical net example Nets are proper: collection of simple arcs (edges) terminated at each end by a vertex divide surface into topological disks Curving edges preserves validity of Euler s formula
13 Model Topology (continued) valid modifications to spherical nets
14 Model Topology (continued) valid modifications of (a) and (b) invalid modification of (c)
15 Model Topology (continued) C = number of polyhedral cells in 3D
16 Model Topology (continued) (a) Object with hole. External faces of hole are inadmissible. H = # holes in faces P = # holes entirely through object B = # separate objects formula modification (b) Edges added to correct inadmissibility. (c) Acceptable concavity. (d) Adding edges satisfies original Euler formula.
17 Model Topology (continued) V 9 classes of topological relationships between pairs of 3 types of elements
18 Graph-Based Models Geometric model emphasizing topological structure Data pointers link object s faces, edges, vertices Trade-off: redundancy yields search speed
19 Graph-Based Models (continued) For planar-faced polyhedra connectivity (adjacency) matrices can be used.
20 Graph-Based Models (continued)
21 Boolean Models
22 Boolean Models (continued) Set Membership Classification X Goal: define regularized set closure of interior no dangling edges or disconnected lower-dimensional parts Set membership classification differentiates between 3 subsets of any regularized set X: bx: boundary of X ix: interior of X bx cx: complement of X ix
23 Boolean Models (continued) Set Membership Classification Some similar geometric modeling problems: Point inclusion: point inside or outside a solid? Line/polygon clipping: line segment vs. polygon Polygon intersection: 2 polygons Solid interference: 2 solids
24 Boolean Models (continued) Set Membership Classification (a) 2 points same or different? (b) point vs. curve: 3 cases (c) point vs. curve or polygon (d) point vs. curved or polyhedral object
25 Boolean Models (continued) Set Membership Classification applicable to topological disc 9 classes of topological relationships between pairs of 3 types of elements
26 Boolean Models (continued) Set Membership Classification (a) point vs. sphere as parametric surface (assumes knowledge of closest point q) (b) point vs. parallelepiped defined as Boolean intersection of half-spaces
27 Boolean Models (continued)
28 Boolean Models (continued) edge of B intersects A in 4 ways 4 th way: intersection is a point 2 regularized polygons A and B
29 Boolean Models (continued) Point 2 is problematic with respect to intersection of A and B.
30 Boolean Models (continued) Outward pointing normals can aid intersection of 3D solids A and B.
31 Boolean Models (continued)
32 Boolean Models (continued) Degenerate intersection of 2 well-defined 2D objects.
33 Boolean Models (continued) Find intersection points. Segment intersected edges. For Union: - Find point on boundary of A outside B. - Trace around loop of edges. - Trace additional loops if needed.
34 Boolean Models (continued) Intersection Set-theoretic and regularized Boolean intersections.
35 Boolean Models (continued) Intersection Need to distinguish between segments 1 & 2 (see next slide). ba bb ia bb ba ib ia ib C ( ba bb) ( ia bb) ( ba ib) ( ia ib) Boundary points can become interior points. Interior points cannot become boundary points.
36 Boolean Models (continued) Intersection Simpler test would use consistent parameterization directions and tangent vector directions. Summarizing overall intersection approach C* bc* ic* Valid ( ba bb) ( ia bb) ( ba ib) ( ia ib) b
37 Boolean Models (continued) Union C ( ba Candidate components of a regularized Boolean union. bb) ( ia bb) ( ba ib) ( ia ib) ba ic* ia ib [ Valid ( ba bb)] bc* ba bb [( ba ib) ( bb ia) Valid ( ba bb)] i b bb ia ib
38 Boolean Models (continued) Difference (c) & (d) appear same! Candidate components of a regularized Boolean difference. C* ( ba bb ib) ( ia bb) Valid ( ba bb) ( ia bb ib)
39 Boolean Models (continued) A encloses B. Useful for modeling holes.
40 Boolean Models (continued)
41 Boolean Models (continued) (a) (c) produce standard results. (d) (f) produce invalid results. Regularizing (d) (f) yields null results.
42 Boolean Model Construction Boolean Model: combination of > 1 simpler solid objects. Boolean Model is procedural: shows how to combine parts. Assume regularization.
43 Boolean Model Construction (continued)
44 Boolean Model Construction (continued)
45 Boolean Model Construction (continued) (a) union of disjoint A and B (b) difference of disjoint A and B: A - B (c) union of A encompassing B (d) difference of A encompassing B: A - B (e) B - A (f) A B yields 2 objects (g) (h): union of A and B (h) makes concavity
46 Boolean Model Construction (continued) (a) 2 intersecting, closed, planar curves intersect an even number of times. (b) If curves A and B do not intersect & a point of B is inside curve A, then B is inside A. (c) Closed, planar curve intersects 3D solid an even number of times. (d) Plane P intersects bounding surface of S in 3 disjoint, closed loops.
47 Constructive Solid Geometry (CSG) CSG: Modeling methods defining complex solids as compositions of simpler solids. Root node represents final result. Internal nodes represent Boolean operations & their results. Leaves are primitive shapes.
48 Constructive Solid Geometry (continued)
49 Constructive Solid Geometry (continued)
50 Constructive Solid Geometry (continued)
51 Constructive Solid Geometry (continued) Refer to Figure on previous slide.
52 Constructive Solid Geometry (continued) B
53 Boundary Models Boundary Model: complete representation of a solid as an organized collection of surfaces. Boundary of a solid must be: - closed - orientable - non-self-intersecting - bounding - connected Region R n is finite, bounded portion of E n. R [ R i, Rb ]
54 Boundary Models (continued) Boundary Representation (B-Rep) B-Rep minimal face conditions: Number of faces is finite. Face is subset of solid s boundary. Union of faces defines boundary. Face is subset of more extensive surface (e.g. plane). Face has finite area. Face is dimensionally homogeneous (regularized).
55 Boundary Models (continued) Boundary Representation (B-Rep) Curved boundary faces require inside/outside convention.
56 Boundary Models (continued) Boundary Representation (B-Rep) Boundary representations are not unique.
57 Boundary Models (continued) Boundary Representation (B-Rep) Merging vertex 1 with vertex 2 makes object invalid.
58 Boundary Models (continued) Boundary Representation (B-Rep) Powerful b-rep systems view solid as union of general faces (e.g. parametric curves).
59 Boundary Models (continued) Boundary Representation (B-Rep) 0 value of parametric variable U 2-step A U B: -Locate u 1, u 2 -Identify active parametric regions. Include C.
60 Boundary Models (continued) Boundary Representation (B-Rep) Union of sphere with skew-truncated cylinder
Chapter 12 Solid Modeling. Disadvantages of wireframe representations
Chapter 12 Solid Modeling Wireframe, surface, solid modeling Solid modeling gives a complete and unambiguous definition of an object, describing not only the shape of the boundaries but also the object
Solid Modeling Lecture Series. Prof. Gary Wang Department of Mechanical and Manufacturing Engineering The University of Manitoba
Solid Modeling 25.353 Lecture Series Prof. Gary Wang Department of Mechanical and Manufacturing Engineering The University of Manitoba Information complete, unambiguous, accurate solid model Solid Modeling
Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry
Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry Representation Ab initio design Rendering Solid modelers Kinematic
Solid Modelling. Graphics Systems / Computer Graphics and Interfaces COLLEGE OF ENGINEERING UNIVERSITY OF PORTO
Solid Modelling Graphics Systems / Computer Graphics and Interfaces 1 Solid Modelling In 2D, one set 2D line segments or curves does not necessarily form a closed area. In 3D, a collection of surfaces
Solid Modeling. Ron Goldman Department of Computer Science Rice University
Solid Modeling Ron Goldman Department of Computer Science Rice University Solids Definition 1. A model which has a well defined inside and outside. 2. For each point, we can in principle determine whether
Lecture 17: Solid Modeling.... a cubit on the one side, and a cubit on the other side Exodus 26:13
Lecture 17: Solid Modeling... a cubit on the one side, and a cubit on the other side Exodus 26:13 Who is on the LORD's side? Exodus 32:26 1. Solid Representations A solid is a 3-dimensional shape with
Computer Aided Engineering Design Prof. Anupam Saxena Department of Mechanical Engineering Indian Institute of Technology, Kanpur.
(Refer Slide Time: 00:28) Computer Aided Engineering Design Prof. Anupam Saxena Department of Mechanical Engineering Indian Institute of Technology, Kanpur Lecture - 6 Hello, this is lecture number 6 of
Week 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
Boolean Component. Chapter 1. Boolean Operations
Chapter 1. Boolean Component Component: The Boolean Component (BOOL), in the bool directory, performs Boolean operations on the model topology of bodies, first finding the intersections between bodies,
4.2 Simplicial Homology Groups
4.2. SIMPLICIAL HOMOLOGY GROUPS 93 4.2 Simplicial Homology Groups 4.2.1 Simplicial Complexes Let p 0, p 1,... p k be k + 1 points in R n, with k n. We identify points in R n with the vectors that point
Solids as point set. Solid models. Solid representation schemes (cont d) Solid representation schemes. Solid representation schemes (cont d)
Solid models Solid models developed to address limitations of wireframe modeling. Attempt was to create systems which create only complete representations. Modelers would support direct creation of 3D
Manifolds. Chapter X. 44. Locally Euclidean Spaces
Chapter X Manifolds 44. Locally Euclidean Spaces 44 1. Definition of Locally Euclidean Space Let n be a non-negative integer. A topological space X is called a locally Euclidean space of dimension n if
Topic: Orientation, Surfaces, and Euler characteristic
Topic: Orientation, Surfaces, and Euler characteristic The material in these notes is motivated by Chapter 2 of Cromwell. A source I used for smooth manifolds is do Carmo s Riemannian Geometry. Ideas of
What 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
VALLIAMMAI ENGINEERING COLLEGE
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203 DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK M.E: CAD/CAM I SEMESTER ED5151 COMPUTER APPLICATIONS IN DESIGN Regulation 2017 Academic
Introduction to Solid Modeling
Introduction to Solid Modeling Hongxin Zhang and Jieqing Feng 2007-01-15 State Key Lab of CAD&CG Zhejiang University Contents Solid Representations: An Introduction Wireframe Models Boundary Representations
Geometric Modeling. Introduction
Geometric Modeling Introduction Geometric modeling is as important to CAD as governing equilibrium equations to classical engineering fields as mechanics and thermal fluids. intelligent decision on the
CLASSIFICATION OF SURFACES
CLASSIFICATION OF SURFACES YUJIE ZHANG Abstract. The sphere, Möbius strip, torus, real projective plane and Klein bottle are all important examples of surfaces (topological 2-manifolds). In fact, via the
CLASSIFICATION 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
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling Welcome to the lectures on computer graphics. We have
Invariant Measures of Convex Sets. Eitan Grinspun, Columbia University Peter Schröder, Caltech
Invariant Measures of Convex Sets Eitan Grinspun, Columbia University Peter Schröder, Caltech What will we measure? Subject to be measured, S an object living in n-dim space convex, compact subset of R
SOME 024: Computer Aided Design. E. Rozos
SOME 024: Computer Aided Design E. Rozos Introduction to CAD theory part 2 Lesson structure Why Solid modelling Solid modelling methods Representation based Manufacturing based Solid modelling storage
L1 - Introduction. Contents. Introduction of CAD/CAM system Components of CAD/CAM systems Basic concepts of graphics programming
L1 - Introduction Contents Introduction of CAD/CAM system Components of CAD/CAM systems Basic concepts of graphics programming 1 Definitions Computer-Aided Design (CAD) The technology concerned with the
EXPERIENCING GEOMETRY
EXPERIENCING GEOMETRY EUCLIDEAN AND NON-EUCLIDEAN WITH HISTORY THIRD EDITION David W. Henderson Daina Taimina Cornell University, Ithaca, New York PEARSON Prentice Hall Upper Saddle River, New Jersey 07458
Flavor of Computational Geometry. Convex Hull in 2D. Shireen Y. Elhabian Aly A. Farag University of Louisville
Flavor of Computational Geometry Convex Hull in 2D Shireen Y. Elhabian Aly A. Farag University of Louisville February 2010 Agenda Introduction Definitions of Convexity and Convex Hulls Naïve Algorithms
CSG obj. oper3. obj1 obj2 obj3. obj5. obj4
Solid Modeling Solid: Boundary + Interior Volume occupied by geometry Solid representation schemes Constructive Solid Geometry (CSG) Boundary representations (B-reps) Space-partition representations Operations
1. Introduction to Constructive Solid Geometry (CSG)
opyright@010, YZU Optimal Design Laboratory. All rights reserved. Last updated: Yeh-Liang Hsu (010-1-10). Note: This is the course material for ME550 Geometric modeling and computer graphics, Yuan Ze University.
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.
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
Lecture 7: Jan 31, Some definitions related to Simplical Complex. 7.2 Topological Equivalence and Homeomorphism
CS 6170 Computational Topology: Topological Data Analysis University of Utah Spring 2017 School of Computing Lecture 7: Jan 31, 2017 Lecturer: Prof. Bei Wang Scribe: Avani Sharma,
Geometric and Solid Modeling. Problems
Geometric and Solid Modeling Problems Define a Solid Define Representation Schemes Devise Data Structures Construct Solids Page 1 Mathematical Models Points Curves Surfaces Solids A shape is a set of Points
Lecture notes: Object modeling
Lecture notes: Object modeling One of the classic problems in computer vision is to construct a model of an object from an image of the object. An object model has the following general principles: Compact
Topology and Boundary Representation. The ACIS boundary representation (B-rep) of a model is a hierarchical decomposition of the model s topology:
Chapter 6. Model Topology Topology refers to the spatial relationships between the various entities in a model. Topology describes how geometric entities are connected (connectivity). On its own, topology
Polygon Meshes and Implicit Surfaces
CSCI 420 Computer Graphics Lecture 9 Polygon Meshes and Implicit Surfaces Polygon Meshes Implicit Surfaces Constructive Solid Geometry [Angel Ch. 10] Jernej Barbic University of Southern California 1 Modeling
Polygon Meshes and Implicit Surfaces
CSCI 420 Computer Graphics Lecture 9 and Constructive Solid Geometry [Angel Ch. 10] Jernej Barbic University of Southern California Modeling Complex Shapes An equation for a sphere is possible, but how
INTRODUCTION 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
Math 366 Lecture Notes Section 11.4 Geometry in Three Dimensions
Math 366 Lecture Notes Section 11.4 Geometry in Three Dimensions Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. A sphere is the set of all points at a
CS3621 Midterm Solution (Fall 2005) 150 points
CS362 Midterm Solution Fall 25. Geometric Transformation CS362 Midterm Solution (Fall 25) 5 points (a) [5 points] Find the 2D transformation matrix for the reflection about the y-axis transformation (i.e.,
Math 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
Shape 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
1 Euler characteristics
Tutorials: MA342: Tutorial Problems 2014-15 Tuesday, 1-2pm, Venue = AC214 Wednesday, 2-3pm, Venue = AC201 Tutor: Adib Makroon 1 Euler characteristics 1. Draw a graph on a sphere S 2 PROBLEMS in such a
Lofting 3D Shapes. Abstract
Lofting 3D Shapes Robby Prescott Department of Computer Science University of Wisconsin Eau Claire Eau Claire, Wisconsin 54701 robprescott715@gmail.com Chris Johnson Department of Computer Science University
Triangle meshes I. CS 4620 Lecture 2
Triangle meshes I CS 4620 Lecture 2 2014 Steve Marschner 1 spheres Andrzej Barabasz approximate sphere Rineau & Yvinec CGAL manual 2014 Steve Marschner 2 finite element analysis PATRIOT Engineering 2014
11/1/13. Polygon Meshes and Implicit Surfaces. Shape Representations. Polygon Models in OpenGL. Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 7 and Constructive Solid Geometry [Angel Ch. 12.1-12.3] Jernej Barbic University of Southern California Modeling Complex Shapes An equation for a sphere is possible,
Topology of Surfaces
EM225 Topology of Surfaces Geometry and Topology In Euclidean geometry, the allowed transformations are the so-called rigid motions which allow no distortion of the plane (or 3-space in 3 dimensional geometry).
Euler 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
Triangle meshes I. CS 4620 Lecture Steve Marschner. Cornell CS4620 Spring 2017
Triangle meshes I CS 4620 Lecture 2 2017 Steve Marschner 1 spheres Andrzej Barabasz approximate sphere Rineau & Yvinec CGAL manual 2017 Steve Marschner 2 finite element analysis PATRIOT Engineering 2017
Lecture 5 CLASSIFICATION OF SURFACES
Lecture 5 CLASSIFICATION OF SURFACES In this lecture, we present the topological classification of surfaces. This will be done by a combinatorial argument imitating Morse theory and will make use of the
Planar 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
TWO 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
3D Modeling: Solid Models
CS 430/536 Computer Graphics I 3D Modeling: Solid Models Week 9, Lecture 18 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science
Geometric structures on 2-orbifolds
Geometric structures on 2-orbifolds Section 1: Manifolds and differentiable structures S. Choi Department of Mathematical Science KAIST, Daejeon, South Korea 2010 Fall, Lectures at KAIST S. Choi (KAIST)
Surfaces Beyond Classification
Chapter XII Surfaces Beyond Classification In most of the textbooks which present topological classification of compact surfaces the classification is the top result. However the topology of 2- manifolds
Aspects of Geometry. Finite models of the projective plane and coordinates
Review Sheet There will be an exam on Thursday, February 14. The exam will cover topics up through material from projective geometry through Day 3 of the DIY Hyperbolic geometry packet. Below are some
GEOMETRY OF SURFACES. b3 course Nigel Hitchin
GEOMETRY OF SURFACES b3 course 2004 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on surfaces. Your mental image of a surface should be something like this: or this However we
Zipper Unfoldings of Polyhedral Complexes
Zipper Unfoldings of Polyhedral Complexes Erik D. Demaine Martin L. Demaine Anna Lubiw Arlo Shallit Jonah L. Shallit Abstract We explore which polyhedra and polyhedral complexes can be formed by folding
3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
Handles. The justification: For a 0 genus triangle mesh we can write the formula as follows:
Handles A handle in a 3d mesh is a through hole. The number of handles can be extracted of the genus of the 3d mesh. Genus is the number of times we can cut 2k edges without disconnecting the 3d mesh.
Triangle meshes I. CS 4620 Lecture Kavita Bala (with previous instructor Marschner) Cornell CS4620 Fall 2015 Lecture 2
Triangle meshes I CS 4620 Lecture 2 1 Shape http://fc00.deviantart.net/fs70/f/2014/220/5/3/audi_r8_render_by_smiska333-d7u9pjt.jpg spheres Andrzej Barabasz approximate sphere Rineau & Yvinec CGAL manual
Surfaces: 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
16.7 Surface Integrals
16 Vector Calculus 16.7 Surface Integrals Copyright Cengage Learning. All rights reserved. 1 Copyright Cengage Learning. All rights reserved. Surface Integrals The relationship between surface integrals
Introduction to Solid Modeling Parametric Modeling. Mechanical Engineering Dept.
Introduction to Solid Modeling Parametric Modeling 1 Why draw 3D Models? 3D models are easier to interpret. Simulation under real-life conditions. Less expensive than building a physical model. 3D models
4. Definition: topological space, open set, topology, trivial topology, discrete topology.
Topology Summary Note to the reader. If a statement is marked with [Not proved in the lecture], then the statement was stated but not proved in the lecture. Of course, you don t need to know the proof.
Introduction to 2D and 3D Computer Graphics. Realistic Rendering. -- Solids Modeling --
Introduction to 2D and 3D Computer Graphics Realistic Rendering -- Solids Modeling -- CS447/547 10-1 CS447/547 10-2 Solid objects can be defined......by sweeping an object along a trajectory through space...this
Lectures 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
Modeling 3D Objects: Part 2
Modeling 3D Objects: Part 2 Patches, NURBS, Solids Modeling, Spatial Subdivisioning, and Implicit Functions 3D Computer Graphics by Alan Watt Third Edition, Pearson Education Limited, 2000 General Modeling
Pick 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.
Polygons and Convexity
Geometry Week 4 Sec 2.5 to ch. 2 test Polygons and Convexity section 2.5 convex set has the property that any two of its points determine a segment contained in the set concave set a set that is not convex
A 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
GRAPH THEORY and APPLICATIONS. Planar Graphs
GRAPH THEORY and APPLICATIONS Planar Graphs Planar Graph A graph is planar if it can be drawn on a plane surface with no two edges intersecting. G is said to be embedded in the plane. We can extend the
Geometric Modeling Systems
Geometric Modeling Systems Wireframe Modeling use lines/curves and points for 2D or 3D largely replaced by surface and solid models Surface Modeling wireframe information plus surface definitions supports
3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
Isotopy classes of crossing arcs in hyperbolic alternating links
Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 1 altern / 21 Isotopy classes of crossing arcs in hyperbolic alternating links Anastasiia Tsvietkova Rutgers
Surface and Solid Geometry. 3D Polygons
Surface and Solid Geometry D olygons Once we know our plane equation: Ax + By + Cz + D = 0, we still need to manage the truncation which leads to the polygon itself Functionally, we will need to do this
Triangle meshes. Computer Graphics CSE 167 Lecture 8
Triangle meshes Computer Graphics CSE 167 Lecture 8 Examples Spheres Andrzej Barabasz Approximate sphere Rineau & Yvinec CGAL manual Based on slides courtesy of Steve Marschner 2 Examples Finite element
Lecture 4b. Surface. Lecture 3 1
Lecture 4b Surface Lecture 3 1 Surface More complete and less ambiguous representation than its wireframe representation Can be considered as extension to wireframe representation In finite element, surface
Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND
Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 2 WHAT YOU WILL LEARN Transformational geometry,
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
Development of Reverse Engineering System for Machine Engineering Using 3D Bit-map Data. Tatsuro Yashiki* and Tarou Takagi*
Development of Reverse Engineering System for Machine Engineering Using 3D Bit-map Data Tatsuro Yashiki* and Tarou Takagi* *Power & Industrial Systems R&D Laboratory, Hitachi, Ltd. Abstract In this paper,
Chapter 4-3D Modeling
Chapter 4-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 1 The 3D rendering
Course Number: Course Title: Geometry
Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line
Outcomes List for Math Multivariable Calculus (9 th edition of text) Spring
Outcomes List for Math 200-200935 Multivariable Calculus (9 th edition of text) Spring 2009-2010 The purpose of the Outcomes List is to give you a concrete summary of the material you should know, and
Curvature 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
Curves and Surfaces. Chapter 7. Curves. ACIS supports these general types of curves:
Chapter 7. Curves and Surfaces This chapter discusses the types of curves and surfaces supported in ACIS and the classes used to implement them. Curves ACIS supports these general types of curves: Analytic
Measuring Lengths The First Fundamental Form
Differential Geometry Lia Vas Measuring Lengths The First Fundamental Form Patching up the Coordinate Patches. Recall that a proper coordinate patch of a surface is given by parametric equations x = (x(u,
Geometrization and the Poincaré conjecture
Geometrization and the Poincaré conjecture Jan Metzger BRIGFOS, 2008 History of the Poincaré conjecture In 1904 Poincaré formulated his conjecture. It is a statement about three dimensional geometric objects,
13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY
13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 1 Nicholas M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright 2003 c Massachusetts Institute of Technology
THE LABELLED PEER CODE FOR KNOT AND LINK DIAGRAMS 26th February, 2015
THE LABELLED PEER CODE FOR KNOT AND LINK DIAGRAMS 26th February, 2015 A labelled peer code is a descriptive syntax for a diagram of a knot or link on a two dimensional sphere. The syntax is able to describe
Engineering Drawing II
Instructional Unit Basic Shading and Rendering -Basic Shading -Students will be able -Demonstrate the ability Class Discussions 3.1.12.B, -Basic Rendering to shade a 3D model to apply shading to a 3D 3.2.12.C,
A Genus Bound for Digital Image Boundaries
A Genus Bound for Digital Image Boundaries Lowell Abrams and Donniell E. Fishkind March 9, 2005 Abstract Shattuck and Leahy [4] conjectured and Abrams, Fishkind, and Priebe [1],[2] proved that the boundary
Fathi El-Yafi Project and Software Development Manager Engineering Simulation
An Introduction to Geometry Design Algorithms Fathi El-Yafi Project and Software Development Manager Engineering Simulation 1 Geometry: Overview Geometry Basics Definitions Data Semantic Topology Mathematics
Geometric 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
Two Connections between Combinatorial and Differential Geometry
Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut für Mathematik, Technische Universität Berlin Berlin Mathematical School DFG Research Group Polyhedral Surfaces
Math 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
Euler 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
Planar Graphs and Surfaces. Graphs 2 1/58
Planar Graphs and Surfaces Graphs 2 1/58 Last time we discussed the Four Color Theorem, which says that any map can be colored with at most 4 colors and not have two regions that share a border having
EULER 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
BAR-MAGNET POLYHEDRA AND NS-ORIENTATIONS OF MAPS
University of Ljubljana Institute of Mathematics, Physics and Mechanics Department of Mathematics Jadranska 19, 1111 Ljubljana, Slovenia Preprint series, Vol. 42 (2004), 940 BAR-MAGNET POLYHEDRA AND NS-ORIENTATIONS
Chapter 9 3D Modeling
Chapter 9 3D Modeling Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3D Modeling Snapshot Since Mid 1980 s become common place in industry Software Types Wireframe
) for all p. This means however, that the map ϕ 0 descends to the quotient
Solutions to sheet 6 Solution to exercise 1: (a) Let M be the Möbius strip obtained by a suitable identification of two opposite sides of the unit square [0, 1] 2. We can identify the boundary M with S
One 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