Geometric Modeling Summer Semester 2010 Introduction
|
|
- Darleen Dennis
- 5 years ago
- Views:
Transcription
1 Geometric Modeling Summer Semester 2010 Introduction Motivation Topics Basic Modeling Techniques
2 Today... Topics: Formalities & Organization Introduction: Geometric Modeling Mathematical Tools (1) Geometric Modeling Summer Semester 2010 Introduction 2 / 93
3 Today... Topics: Formalities & Organization Introduction: Geometric Modeling Motivation Overview: Topics Basic modeling techniques Mathematical Tools (1) Geometric Modeling Summer Semester 2010 Introduction 3 / 93
4 Motivation
5 Motivation This lecture covers two related areas: Classic geometric modeling Geometry processing Common techniques (math, models, terminology), but different goals Geometric Modeling Summer Semester 2010 Introduction 5 / 93
6 Geometric Modeling Geometric Modeling: You start with a blank screen, design a geometric model Typical techniques: Triangle meshes Constructive Solid Geometry (CSG) Spline curves & surfaces Subdivision surfaces Goal is interactive modeling Mathematical tools are designed with the user in mind Geometric Modeling Summer Semester 2010 Introduction 6 / 93
7 Geometry Processing Geometry Processing You already have a geometric model Typically from a 3D range scanner (read: not nice) You need to process & edit the geometry The original model has not been build with the user in mind (stupid range scanner) Typical techniques: Noise removal, filtering Surface reconstruction Registration Freeform deformation modeling Statistical analysis (features, symmetry, hole-filling etc...) Geometric Modeling Summer Semester 2010 Introduction 7 / 93
8 Our Perspective The perspective of this lecture: The basic mathematical tools for handling geometry are the same Different usage, adaptation, specific algorithms We will discuss The basic concepts and tools (mathematical foundation, representations, basic algorithms)...and applications in both areas. Geometric Modeling Summer Semester 2010 Introduction 8 / 93
9 Examples: Geometric Modeling
10 The Modern World... designed on a computer (the building) designed on a computer as well (the cars) fortunately, not (yet) designed on a computer (the trees) (c.f. Danny Hillis, Siggraph 2001 keynote) Geometric Modeling Summer Semester 2010 Introduction 10 / 93
11 Impact of Geometric Modeling We live in a world designed using CAD Almost any man-made structure is nowadays planed and designed using computers Architecture Commodities: Chairs, furniture, your microwave & toaster Your car (in case you have one, but probably the bike as well) spline curves have actually been invented in the automotive industry Typesetting <advertising> Our abilities in geometric modeling shapes the world we live in each day. </advertising> Geometric Modeling Summer Semester 2010 Introduction 11 / 93
12 Different Modeling Tasks CAD / CAM Precision Guarantees Handle geometric constraints exactly (e.g. exact circles) Modeling guided by rules and constraints Geometric Modeling Summer Semester 2010 Introduction 12 / 93
13 Different Modeling Tasks Photorealistic Rendering Has to look good Ad-hoc techniques are ok Using textures & shaders to fake details More complexity, but less rigorous Just two examples, lots of stuff in between... Geometric Modeling Summer Semester 2010 Introduction 13 / 93
14 Examples: Geometry Processing
15 Geometry Processing A rather new area Motivation: 3D scanning You (your company) can buy devices that scan real world 3D objects You get (typically) clouds of measurement points Many other sources of geometry as well: Science (CT, [F]MRI, ET, Cryo-EM,...) 3D movie making The design department of your company has dozens of TByte of polygon soup... Crawl the internet Need to process the geometry further Geometric Modeling Summer Semester 2010 Introduction 15 / 93
16 Photoshopping Geometry Geometry Processing: Cleanup: Remove inconsistencies Make watertight (well defined inside/outside, for 3D printers) Simplify keep only the main structure Remove noise, small holes, etc... Touch-up /Edit: Texturing, painting, carving Deformation Stitch together pieces Lots of other stuff similar to image processing Geometric Modeling Summer Semester 2010 Introduction 16 / 93
17 Example Example: The Stanford Digital Michelangelo Project [Levoy et al.: The Digital Michelangelo Project, Siggraph 2000] Geometric Modeling Summer Semester 2010 Introduction 17 / 93
18 Scan Registration [data set: Stanford 3D Scanning Repository] Geometric Modeling Summer Semester 2010 Introduction 18 / 93
19 Feature Tracking Fully Automatic: [Implementation: Martin Bokeloh (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 19 / 93
20 Scanning the World... Example: The Wägele Laser scanners (2D sheets of distance measurments) [Biber et al. 2005] A pull-through measurement device can acquire complete buildings in a few hours Geometric Modeling Summer Semester 2010 Introduction 20 / 93
21 This is what you get... Corridor CS Building University of Tübingen (6.5 GB) CS Building, Outside (nicer colors...)...lots of artifacts (the scanner does not really like windows) Geometric Modeling Summer Semester 2010 Introduction 21 / 93
22 Automatic Processing Example: Automatic Outlier Removal Geometric Modeling Summer Semester 2010 Introduction 22 / 93
23 Think Big More Problems: Occluded areas, shiny / transparent objects holes (lots of holes, actually) Huge amounts of data (really huge) City Scanning There are big companies trying to scan large areas Think Google Earth in full resolution How about a virtual online walk through New York, Tokyo, Saarbrücken? Lots of open research problems to get there Geometric Modeling Summer Semester 2010 Introduction 23 / 93
24 [data set: Institute for Cartography, Leibnitz University Hannover]
25 HUGE Data Sets The Largest Data Set I have On My Hard-Drive... Data set: Outdoor Scan (structure from video) of a part of the UNC campus ( pts / 63.5 GB), courtesy of J.-M. Frahm, University of North Carolina Geometric Modeling Summer Semester 2010 Introduction 25 / 93
26 Hole Filling Wei-Levoy Texture Synthesis Algorithm: [Implementation: Alexander Berner (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 26 / 93
27 Filling Holes [implementation: Alexander Berner (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 27 / 93
28 Filling Holes [implementation: Alexander Berner (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 28 / 93
29 Symmetry Detection [data set: M. Wacker, HTW Dresden] Geometric Modeling Summer Semester 2010 Introduction 29 / 93
30 Results Geometric Modeling Summer Semester 2010 Introduction 30 / 93
31 Results Geometric Modeling Summer Semester 2010 Introduction 31 / 93
32 Results Geometric Modeling Summer Semester 2010 Introduction 32 / 93
33 Line Feature Matching [data sets: Kartographisches Institut, Universität Hannover / M. Wacker, HTW Dresden] Geometric Modeling Summer Semester 2010 Introduction 33 / 93
34 Reconstruction by Symmetry overlay of 16 parts [data sets: Kartographisches Institut, Universität Hannover] Geometric Modeling Summer Semester 2010 Introduction 34 / 93
35 Inverse Procedural Modeling Geometric Modeling Summer Semester 2010 Introduction 35 / 93
36 Overview Our approach Take existing model Analyse shape structure Derive shape modification rules Output Input Geometric Modeling Summer Semester 2010 Introduction 36 / 93
37 Technique Overview Conceptual Steps: Symmetry detection Finding docking sites and dockers Combine into a shape grammar Input Symmetry Docking site Result Geometric Modeling Summer Semester 2010 Introduction 37 / 93
38 Results Geometric Modeling Summer Semester 2010 Introduction 38 / 93
39 Results ~ triangles Geometric Modeling Summer Semester 2010 Introduction 39 / 93
40 Results Geometric Modeling Summer Semester 2010 Introduction 40 / 93
41 Results Geometric Modeling Summer Semester 2010 Introduction 41 / 93
42 Deformable Shape Matching [data set: Stanford 3D Scanning Repository] Geometric Modeling Summer Semester 2010 Introduction 42 / 93
43 Problem Statement Deformable Matching Two shapes: original, deformed How to establish correspondences? Looking for global optimum Arbitrary pose Assumption Approximately isometric deformation [data set: S. König, TU Dresden] Geometric Modeling Summer Semester 2010 Introduction 43 / 93
44 Results [data sets: Stanford 3D Scanning Repository / Carsten Stoll] Geometric Modeling Summer Semester 2010 Introduction 44 / 93
45 Animation Reconstruction [data set: P. Phong, Stanford University] Geometric Modeling Summer Semester 2010 Introduction 45 / 93
46 Scanning Moving Geometry Real-time 3D scanners: Acquire geometry at video rates Capture 3D movies: performance capture Technique still immature, but very interesting applications, in particular special effects for movies [Davis et al. 2003] Geometric Modeling Summer Semester 2010 Introduction 46 / 93
47 Real-Time 3D Scanners space-time stereo courtesy of James Davis University of California at Santa Cruz color-coded structured light courtesy of Phil Fong Stanford University high-speed structured light courtesy of Stefan Gumhold TU Dresden Geometric Modeling Summer Semester 2010 Introduction 47 / 93 47
48 Reconstruction Dynamic geometry reconstruction Hole filling Remove noise and outliers Establish correspondences Need to know where every point on the object goes to over time Simplifies further editing Geometric Modeling Summer Semester 2010 Introduction 48 / 93
49 Animation Reconstruction Remove noise, outliers Fill-in holes (from all frames) Dense correspondences Geometric Modeling Summer Semester 2010 Introduction 49 / 93
50 Factorization t = 0 t = 1 t = 2 f f data points d t,i f (x, t) deformation field S x point on urshape S [data set courtesy of P. Phong, Stanford University] Geometric Modeling Summer Semester 2010 Introduction 50 / 93
51 Geometric Modeling Summer Semester 2010 Introduction 51 / 93
52 Geometric Modeling Summer Semester 2010 Introduction 52 / 93
53 Overview Topics
54 Overview: Geometric Modeling 2010 Mathematical Background (Recap) Linear algebra: vector spaces, function spaces, quadrics Analysis: multi-dim. calculus, differential geometry Numerics: quadratic and non-linear optimization Geometric Modeling Smooth curves: polynomial interpolation & approximation, Bezier curves, B-Splines, NURBS Smooth surfaces: spline surfaces, implicit functions, variational modeling Meshes: meshes, multi-resolution, subdivision Geometric Modeling Summer Semester 2010 Introduction 54 / 93
55 Overview: Geometric Modeling 2010 Geometry Processing 3D Scanning: Overview Registration: ICP, NDT Surface Reconstruction: smoothing, topology reconstruction, moving least-squares Editing: free-form deformation Preliminary List: Topics might change Not presented strictly in this order Geometric Modeling Summer Semester 2010 Introduction 55 / 93
56 Topics Overview Current List of Topics (subject to changes): Math Background: Linear Algebra, Analysis, Differential Geometry, Numerics, Topology Interpolation and Approximation Spline Curves Blossoming and Polar Forms Rational Splines Spline Surfaces Subdivision Surfaces Implicit Functions Variational Modeling Point Based Representations Multi Resolution Representations Surface Parametrization Geometric Modeling Summer Semester 2010 Introduction 56 / 93
57 Overview Modeling Techniques
58 Geometric Modeling What do we want to do? d empty space (typically 3 ) B geometric object B 3 Geometric Modeling Summer Semester 2010 Introduction 58 / 93
59 Fundamental Problem The Problem: d B infinite number of points my computer: 8GB of memory We need to encode a continuous model with a finite amount of information Geometric Modeling Summer Semester 2010 Introduction 59 / 93
60 Modeling Approaches Two Basic Approaches Discrete representations Fixed discrete bins Continuous representations Mathematical description Evaluate continuously Geometric Modeling Summer Semester 2010 Introduction 60 / 93
61 Discrete Representations You know this... Fixed Grid of values: (i 1,..., i d s ) d s (x 1,..., x d t ) d t Typical scenarios: d s = 2, d t = 3: Bitmap images d s = 3, d t = 1: Volume data (scalar fields) d s = 2, d t = 1: Depth maps (range images) PDEs: Finite Differences models Geometric Modeling Summer Semester 2010 Introduction 61 / 93
62 Modeling Approaches Two Basic Approaches Discrete representations Fixed discrete bins Continuous representations Mathematical description Evaluate continuously Geometric Modeling Summer Semester 2010 Introduction 62 / 93
63 Continuous Models Basic Principle: Procedural Modeling Query Parameters (a finite set of numbers from a continuous set) Algorithm(s) determines the class of objects that can be represented finite set of Shape Parameters determines the object shape Answer Geometric Modeling Summer Semester 2010 Introduction 63 / 93
64 Example: Continuous Model Example: Sphere Shape Parameters: center, radius (4 numbers) Algorithms: Ray Intersection (e.g. for display) Input: Ray (angle, position: 5 numbers) Output: {true, false} Inside/outside test (e.g. for rasterization) Input: Position (3 numbers) Output: {true, false} Parametrization (e.g. for display) Input: longitude, latitude (, ) Output: position (3 numbers) Geometric Modeling Summer Semester 2010 Introduction 64 / 93
65 So Many Questions... Several algorithms for the same representation: Parametrization compute surface points according to continuous parameters (Signed) distance computation distance to surface of points in space, inside/outside test Intersection with rays (rendering), other objects (collision detection) Conversion into other representations. Many more... In addition, we also need algorithms to construct and alter the models. Geometric Modeling Summer Semester 2010 Introduction 65 / 93
66 Continuous, Procedural Models Continuous representations An algorithm describes the shape The shape is determined by a finite number of continuous parameters The shape can be queried with a finite number of continuous parameters More involved (have to ask for information) But potentially infinite resolution (continuous model) Structural model complexity still limited by algorithm and parameters This lecture examines these representations and the corresponding algorithms Geometric Modeling Summer Semester 2010 Introduction 66 / 93
67 Classes of Models (Main) classes of models in this lecture: Primitive meshes Parametric models Implicit models Particle / point-based models Remarks Most models are hybrid (combine several of these) Representations can be converted (may be approximate) Some questions are much easier to answer for certain representations Geometric Modeling Summer Semester 2010 Introduction 67 / 93
68 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 68 / 93
69 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 69 / 93
70 Parametric Models v d s S d t f f (u, v) (u, v) Parametric Models u Function f maps from parameter domain to target space Evaluation of f gives one point on the model Geometric Modeling Summer Semester 2010 Introduction 70 / 93
71 input: 3D input: 2D input: 1D output: 1D output: 2D output: 3D t f(t) t y t y z u x x function graph plane curve space curve u y u y v v z x x plane warp surface w u v y z x space warp
72 input: 3D input: 2D input: 1D output: 1D output: 2D output: 3D t f(t) t y t y z u x x function graph plane curve space curve u y u y v v z x x plane warp surface w u v y z x space warp
73 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 73 / 93
74 Primitive Meshes Primitive Meshes Collection of geometric primitives Triangles Quadrilaterals More general primitives (spline patches) Typically, the primitives are parametric surfaces Composite model: Mesh encodes topology, rough shape Primitive parameter encode local geometry Triangle meshes rule the world ( triangle soup ) Geometric Modeling Summer Semester 2010 Introduction 74 / 93
75 Primitive Meshes Complex Topology for Parametric Models Mesh of parameter domains attached in a mesh Domain can have complex shape ( trimmed patches ) Separate mapping function f for each part (typically of the same class) Geometric Modeling Summer Semester 2010 Introduction 75 / 93
76 Meshes are Great Advantages of mesh-based modeling: Compact representation (usually) Can represent arbitrary topology Using the right parametric surfaces as parts, many important geometric objects can be represented exactly (e.g. NURBS: circles, cylinders, spheres CAD/CAM) Geometric Modeling Summer Semester 2010 Introduction 76 / 93
77 Meshes are not so great Problem with Meshes: Need to specify a mesh first, then edit geometry Problems for larger changes Mesh structure and shape need to be adjusted Mesh encodes object topology Changing object topology is painful Difficult to use for many applications (such as surface reconstruction) Rule of thumb: If the topology or the coarse scale shape changes drastically and frequently during computations, meshes are hard to use Drastic example: Fluid simulation (surface of splashing water) Geometric Modeling Summer Semester 2010 Introduction 77 / 93
78 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 78 / 93
79 Implicit Modeling General Formulation: Curve / Surface S = {x f(x) = 0} x d (d = 2,3), f(x) S is (usually) a d-1 dimensional object This means...: The surface is obtained implicitly as the set of points for which some given function vanishes (f(x) = 0) Alternative notation: S = f -1 (0) ( inverse yields a set) Geometric Modeling Summer Semester 2010 Introduction 79 / 93
80 Implicit Modeling Example: Circle: x 2 + y 2 = r 2 f r (x,y) = x 2 + y 2 - r 2 = 0 Sphere: x 2 + y 2 + z 2 = r 2 r 2 x 2 y 2 Special Case: Signed distance field Function value is signed distance to surface f ( x, y) sign( x 2 y 2 r 2 ) x 2 y 2 r 2 Negative means inside, positive means outside Geometric Modeling Summer Semester 2010 Introduction 80 / 93
81 Implicit Modeling Example: Circle: x 2 + y 2 = r 2 f r (x,y) = x 2 + y 2 - r 2 = 0 Sphere: x 2 + y 2 + z 2 = r 2 Signed squared distance field r 2 (has some useful properties, y 2 e.g. from a statistical point x 2 of view) Special Case: Signed distance field Function value is signed distance to surface f ( x, y) sign( x 2 y 2 r 2 ) x 2 y 2 r 2 Negative means inside, positive means outside Geometric Modeling Summer Semester 2010 Introduction 81 / 93
82 Implicit Modeling: Pros & Cons Advantages: More general than parametric techniques Topology can be changed easily (depends on how f is specified, though) Implicit representations are the standard technique for simulations with free boundaries. Also known as level-set methods. Typical example: Fluid simulation (evolving water-air interface) Geometric modeling: Surface reconstruction, blobby surfaces Geometric Modeling Summer Semester 2010 Introduction 82 / 93
83 Implicit Modeling: Pros & Cons Disadvantages: Need to solve inversion problem S = f -1 (0) Algorithms for display, conversion etc. tend to be more difficult and more expensive (inside/outside test is easy though for signed distance fields) Representing objects takes more memory (we will discuss standard representations later) Geometric Modeling Summer Semester 2010 Introduction 83 / 93
84 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 84 / 93
85 Particle Representations Particle / Point-based Representations Geometry is represented as a set of points / particles The particles form a (typically irregular) sample of the geometric object Need additional information to deal with the empty space around the particles additional assumptions Geometric Modeling Summer Semester 2010 Introduction 85 / 93
86 Particle Representations Helpful Information Each particle may carries a set of attributes Must have: Its position Additional geometric information: Particle density (sample spacing), surface normals Additionally: Color, physical quantities (mass, pressure, temperature),... This information can be used to improve the reconstruction of the geometric object described by the particles Geometric Modeling Summer Semester 2010 Introduction 86 / 93
87 The Wrath of Khan Why Star Trek is at fault... Particle methods first used in computer graphics to represent fuzzy phenomena (fire, clouds, smoke) Particle Systems a Technique for Modeling a Class of Fuzzy Objects *Reeves Probably most well-known example: Genesis sequence Geometric Modeling Summer Semester 2010 Introduction 87 / 93
88 Genesis Sequence [Reeves 1983] Geometric Modeling Summer Semester 2010 Introduction 88 / 93
89 Non-Fire Objects Particle Traces for Modeling Plants (also from [Reeves 1983]) Geometric Modeling Summer Semester 2010 Introduction 89 / 93
90 Geometric Modeling How became the geometric modeling crowd interested in this? 3D Scanners 3D scanning devices yield point clouds (often: measure distance to points in space, one at a time) Then you have to deal with the problem anyway Need algorithms to directly work on point clouds (this is the geometry name for particle system) Geometric Modeling Summer Semester 2010 Introduction 90 / 93
91 Geometric Modeling How became the geometric modeling crowd interested in this? Other Reasons: Similar advantages as implicit techniques Topology does not matter (for the good and for the bad) Topology is easy to change Multi-scale representations are easy to do (more details on multi-resolution techniques later) Often easier to use than implicit or parametric techniques Geometric Modeling Summer Semester 2010 Introduction 91 / 93
92 Multi-Scale Geometry w/points Geometric Modeling Summer Semester 2010 Introduction 92 / 93
93 Summary Summary Lots of different representations No silver bullet In theory, everything always works, but might be just too complicated/expensive Best choice depends on the application We will look on all of this... Focus on parametric techniques though Most common approach Geometric Modeling Summer Semester 2010 Introduction 93 / 93
Geometric Registration for Deformable Shapes 1.1 Introduction
Geometric Registration for Deformable Shapes 1.1 Introduction Overview Data Sources and Applications Problem Statement Overview Presenters Will Chang University of California at San Diego, USA Hao Li ETH
More informationIntroduction to Geometry. Computer Graphics CMU /15-662
Introduction to Geometry Computer Graphics CMU 15-462/15-662 Assignment 2: 3D Modeling You will be able to create your own models (This mesh was created in Scotty3D in about 5 minutes... you can do much
More informationStatistical Geometry Processing Winter Semester 2011/2012
Statistical Geometry Processing Winter Semester 2011/2012 Representations of Geometry Motivation 3 Geometric Modeling What do we want to do? d empty space (typically 3 ) B geometric object B 3 4 Fundamental
More information9. Three Dimensional Object Representations
9. Three Dimensional Object Representations Methods: Polygon and Quadric surfaces: For simple Euclidean objects Spline surfaces and construction: For curved surfaces Procedural methods: Eg. Fractals, Particle
More informationL1 - 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
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More informationCHAPTER 1 Graphics Systems and Models 3
?????? 1 CHAPTER 1 Graphics Systems and Models 3 1.1 Applications of Computer Graphics 4 1.1.1 Display of Information............. 4 1.1.2 Design.................... 5 1.1.3 Simulation and Animation...........
More informationPhysically-Based Modeling and Animation. University of Missouri at Columbia
Overview of Geometric Modeling Overview 3D Shape Primitives: Points Vertices. Curves Lines, polylines, curves. Surfaces Triangle meshes, splines, subdivision surfaces, implicit surfaces, particles. Solids
More informationCS354 Computer Graphics Surface Representation IV. Qixing Huang March 7th 2018
CS354 Computer Graphics Surface Representation IV Qixing Huang March 7th 2018 Today s Topic Subdivision surfaces Implicit surface representation Subdivision Surfaces Building complex models We can extend
More informationFrom curves to surfaces. Parametric surfaces and solid modeling. Extrusions. Surfaces of revolution. So far have discussed spline curves in 2D
From curves to surfaces Parametric surfaces and solid modeling CS 465 Lecture 12 2007 Doug James & Steve Marschner 1 So far have discussed spline curves in 2D it turns out that this already provides of
More informationModeling 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
More informationPolygon 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
More informationPolygon 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
More informationGeometric Modeling. Bing-Yu Chen National Taiwan University The University of Tokyo
Geometric Modeling Bing-Yu Chen National Taiwan University The University of Tokyo What are 3D Objects? 3D Object Representations What are 3D objects? The Graphics Process 3D Object Representations Raw
More informationCSG 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
More informationLecture 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
More information3D Modeling: Surfaces
CS 430/536 Computer Graphics I 3D Modeling: Surfaces Week 8, Lecture 16 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel
More information11/1/13. 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,
More informationImplicit Surfaces & Solid Representations COS 426
Implicit Surfaces & Solid Representations COS 426 3D Object Representations Desirable properties of an object representation Easy to acquire Accurate Concise Intuitive editing Efficient editing Efficient
More informationEngineering designs today are frequently
Basic CAD Engineering designs today are frequently constructed as mathematical solid models instead of solely as 2D drawings. A solid model is one that represents a shape as a 3D object having mass properties.
More informationComputergrafik. Matthias Zwicker. Herbst 2010
Computergrafik Matthias Zwicker Universität Bern Herbst 2010 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling Piecewise Bézier curves Each segment
More informationComputergrafik. Matthias Zwicker Universität Bern Herbst 2016
Computergrafik Matthias Zwicker Universität Bern Herbst 2016 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling 2 Piecewise Bézier curves Each
More informationObject representation
Object representation Geri s Game Pixar 1997 Subdivision surfaces Polhemus 3d scan Over 700 controls 2 Computer Graphics Quick test #1 Describe the picture Graphical systems, visualization and multimedia
More informationCurves and Surfaces Computer Graphics I Lecture 10
15-462 Computer Graphics I Lecture 10 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] September 30, 2003 Doug James Carnegie
More informationComputer Graphics 1. Chapter 2 (May 19th, 2011, 2-4pm): 3D Modeling. LMU München Medieninformatik Andreas Butz Computergraphik 1 SS2011
Computer Graphics 1 Chapter 2 (May 19th, 2011, 2-4pm): 3D Modeling 1 The 3D rendering pipeline (our version for this class) 3D models in model coordinates 3D models in world coordinates 2D Polygons in
More informationGeometry Processing & Geometric Queries. Computer Graphics CMU /15-662
Geometry Processing & Geometric Queries Computer Graphics CMU 15-462/15-662 Last time: Meshes & Manifolds Mathematical description of geometry - simplifying assumption: manifold - for polygon meshes: fans,
More informationDgp _ lecture 2. Curves
Dgp _ lecture 2 Curves Questions? This lecture will be asking questions about curves, their Relationship to surfaces, and how they are used and controlled. Topics of discussion will be: Free form Curves
More information3D 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
More informationAnimation & Rendering
7M836 Animation & Rendering Introduction, color, raster graphics, modeling, transformations Arjan Kok, Kees Huizing, Huub van de Wetering h.v.d.wetering@tue.nl 1 Purpose Understand 3D computer graphics
More informationPoint based Rendering
Point based Rendering CS535 Daniel Aliaga Current Standards Traditionally, graphics has worked with triangles as the rendering primitive Triangles are really just the lowest common denominator for surfaces
More informationCurves D.A. Forsyth, with slides from John Hart
Curves D.A. Forsyth, with slides from John Hart Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction
More informationComputer Graphics CS 543 Lecture 13a Curves, Tesselation/Geometry Shaders & Level of Detail
Computer Graphics CS 54 Lecture 1a Curves, Tesselation/Geometry Shaders & Level of Detail Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines
More informationGeometric Modeling Topics
Geometric Modeling Topics George Allen, george.allen@siemens.com Outline General background Convergent modeling Multi-material objects Giga-face lattices Page 2 Boundary Representation (b-rep) Topology
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More informationEF432. Introduction to spagetti and meatballs
EF432 Introduction to spagetti and meatballs CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~karan/courses/418/ Instructors: L2501, T 6-8pm
More informationReview. Stephen J. Guy
Review Stephen J. Guy Overview Pixar short Review last class Review course Area of Graphics Image Processing Rendering Modeling Animation Misc Area of Graphics Image Processing Rendering Modeling Animation
More informationShape Representation Basic problem We make pictures of things How do we describe those things? Many of those things are shapes Other things include
Shape Representation Basic problem We make pictures of things How do we describe those things? Many of those things are shapes Other things include motion, behavior Graphics is a form of simulation and
More information3D 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
More informationTopics and things to know about them:
Practice Final CMSC 427 Distributed Tuesday, December 11, 2007 Review Session, Monday, December 17, 5:00pm, 4424 AV Williams Final: 10:30 AM Wednesday, December 19, 2007 General Guidelines: The final will
More informationTopic 0. Introduction: What Is Computer Graphics? CSC 418/2504: Computer Graphics EF432. Today s Topics. What is Computer Graphics?
EF432 Introduction to spagetti and meatballs CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~karan/courses/418/ Instructors: L0101, W 12-2pm
More informationCSE528 Computer Graphics: Theory, Algorithms, and Applications
CSE528 Computer Graphics: Theory, Algorithms, and Applications Hong Qin State University of New York at Stony Brook (Stony Brook University) Stony Brook, New York 11794--4400 Tel: (631)632-8450; Fax: (631)632-8334
More informationOverview of 3D Object Representations
Overview of 3D Object Representations Thomas Funkhouser Princeton University C0S 426, Fall 2000 Course Syllabus I. Image processing II. Rendering III. Modeling IV. Animation Image Processing (Rusty Coleman,
More informationHomework 1: Implicit Surfaces, Collision Detection, & Volumetric Data Structures. Loop Subdivision. Loop Subdivision. Questions/Comments?
Homework 1: Questions/Comments? Implicit Surfaces,, & Volumetric Data Structures Loop Subdivision Shirley, Fundamentals of Computer Graphics Loop Subdivision SIGGRAPH 2000 course notes Subdivision for
More informationCS-184: Computer Graphics. Today
CS-84: Computer Graphics Lecture #5: Curves and Surfaces Prof. James O Brien University of California, Berkeley V25F-5-. Today General curve and surface representations Splines and other polynomial bases
More informationCurves & Surfaces. Last Time? Progressive Meshes. Selective Refinement. Adjacency Data Structures. Mesh Simplification. Mesh Simplification
Last Time? Adjacency Data Structures Curves & Surfaces Geometric & topologic information Dynamic allocation Efficiency of access Mesh Simplification edge collapse/vertex split geomorphs progressive transmission
More informationImage-Based Modeling and Rendering. Image-Based Modeling and Rendering. Final projects IBMR. What we have learnt so far. What IBMR is about
Image-Based Modeling and Rendering Image-Based Modeling and Rendering MIT EECS 6.837 Frédo Durand and Seth Teller 1 Some slides courtesy of Leonard McMillan, Wojciech Matusik, Byong Mok Oh, Max Chen 2
More information3D Object Representation. Michael Kazhdan ( /657)
3D Object Representation Michael Kazhdan (601.457/657) 3D Objects How can this object be represented in a computer? 3D Objects This one? H&B Figure 10.46 3D Objects This one? H&B Figure 9.9 3D Objects
More informationCurve and Surface Basics
Curve and Surface Basics Implicit and parametric forms Power basis form Bezier curves Rational Bezier Curves Tensor Product Surfaces ME525x NURBS Curve and Surface Modeling Page 1 Implicit and Parametric
More informationEF432. Introduction to spagetti and meatballs
EF432 Introduction to spagetti and meatballs CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~karan/courses/418/fall2015 Instructor: Karan
More informationLecture 1 Course Introduction
UMass Lowell Computer Science 91.580.201 Geometric Modeling Prof. Karen Daniels Spring, 2009 Lecture 1 Course Introduction Course Introduction What is Geometric Modeling? Adapted from: Geometric Modeling
More informationChapter 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
More informationCS123 INTRODUCTION TO COMPUTER GRAPHICS. Describing Shapes. Constructing Objects in Computer Graphics 1/15
Describing Shapes Constructing Objects in Computer Graphics 1/15 2D Object Definition (1/3) Lines and polylines: Polylines: lines drawn between ordered points A closed polyline is a polygon, a simple polygon
More information0. Introduction: What is Computer Graphics? 1. Basics of scan conversion (line drawing) 2. Representing 2D curves
CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~elf Instructor: Eugene Fiume Office: BA 5266 Phone: 416 978 5472 (not a reliable way) Email:
More informationPipeline Operations. CS 4620 Lecture Steve Marschner. Cornell CS4620 Spring 2018 Lecture 11
Pipeline Operations CS 4620 Lecture 11 1 Pipeline you are here APPLICATION COMMAND STREAM 3D transformations; shading VERTEX PROCESSING TRANSFORMED GEOMETRY conversion of primitives to pixels RASTERIZATION
More informationCentral issues in modelling
Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction includes: manual modelling; fitting to
More information11/1/13. Visualization. Scientific Visualization. Types of Data. Height Field. Contour Curves. Meshes
CSCI 420 Computer Graphics Lecture 26 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 2.11] Jernej Barbic University of Southern California Scientific Visualization
More informationVisualization. CSCI 420 Computer Graphics Lecture 26
CSCI 420 Computer Graphics Lecture 26 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 11] Jernej Barbic University of Southern California 1 Scientific Visualization
More informationIntroduction to the Mathematical Concepts of CATIA V5
CATIA V5 Training Foils Introduction to the Mathematical Concepts of CATIA V5 Version 5 Release 19 January 2009 EDU_CAT_EN_MTH_FI_V5R19 1 About this course Objectives of the course Upon completion of this
More informationGeometric 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
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationCurves & Surfaces. MIT EECS 6.837, Durand and Cutler
Curves & Surfaces Schedule Sunday October 5 th, * 3-5 PM * Review Session for Quiz 1 Extra Office Hours on Monday Tuesday October 7 th : Quiz 1: In class 1 hand-written 8.5x11 sheet of notes allowed Wednesday
More informationLecture 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
More informationSubdivision Surfaces. Homework 1: Questions on Homework? Last Time? Today. Tensor Product. What s an illegal edge collapse?
Homework 1: Questions/Comments? Subdivision Surfaces Questions on Homework? Last Time? What s an illegal edge collapse? Curves & Surfaces Continuity Definitions 2 3 C0, G1, C1, C 1 a b 4 Interpolation
More informationComputer Graphics Introduction. Taku Komura
Computer Graphics Introduction Taku Komura What s this course all about? We will cover Graphics programming and algorithms Graphics data structures Applied geometry, modeling and rendering Not covering
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #12: Curves and Surfaces Prof. James O Brien University of California, Berkeley V2007-F-12-1.0 Today General curve and surface representations Splines and other polynomial
More informationMaths at the Movies. Chris Budd
Maths at the Movies Chris Budd See maths in the movies in different ways Sometimes maths in the background Moriarty Some movies hate maths Some feature mathematicians Some films are about mathematicians
More informationSubdivision Surfaces. Homework 1: Questions/Comments?
Subdivision Surfaces Homework 1: Questions/Comments? 1 Questions on Homework? What s an illegal edge collapse? 1 2 3 a b 4 7 To be legal, the ring of vertex neighbors must be unique (have no duplicates)!
More informationGeometric Representations. Stelian Coros
Geometric Representations Stelian Coros Geometric Representations Languages for describing shape Boundary representations Polygonal meshes Subdivision surfaces Implicit surfaces Volumetric models Parametric
More informationCSE528 Computer Graphics: Theory, Algorithms, and Applications
CSE528 Computer Graphics: Theory, Algorithms, and Applications Hong Qin State University of New York at Stony Brook (Stony Brook University) Stony Brook, New York 11794--4400 Tel: (631)632-8450; Fax: (631)632-8334
More informationMODELING AND HIERARCHY
MODELING AND HIERARCHY Introduction Models are abstractions of the world both of the real world in which we live and of virtual worlds that we create with computers. We are all familiar with mathematical
More informationReading. 18. Projections and Z-buffers. Required: Watt, Section , 6.3, 6.6 (esp. intro and subsections 1, 4, and 8 10), Further reading:
Reading Required: Watt, Section 5.2.2 5.2.4, 6.3, 6.6 (esp. intro and subsections 1, 4, and 8 10), Further reading: 18. Projections and Z-buffers Foley, et al, Chapter 5.6 and Chapter 6 David F. Rogers
More informationVALLIAMMAI 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
More informationSRM INSTITUTE OF SCIENCE AND TECHNOLOGY
SRM INSTITUTE OF SCIENCE AND TECHNOLOGY DEPARTMENT OF INFORMATION TECHNOLOGY QUESTION BANK SUB.NAME: COMPUTER GRAPHICS SUB.CODE: IT307 CLASS : III/IT UNIT-1 2-marks 1. What is the various applications
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationShape Modeling with Point-Sampled Geometry
Shape Modeling with Point-Sampled Geometry Mark Pauly Richard Keiser Leif Kobbelt Markus Gross ETH Zürich ETH Zürich RWTH Aachen ETH Zürich Motivation Surface representations Explicit surfaces (B-reps)
More informationInstructor. Goals. Image Synthesis Examples. Applications. Foundations of Computer Graphics. Why Study 3D Computer Graphics?
Foundations of Computer Graphics Motivation: Why do we study 3D Graphics? http://www.cs.berkeley.edu/~ravir Instructor http://www.cs.berkeley.edu/~ravir PhD Stanford, 2002. PhD thesis developed Spherical
More informationWater. Notes. Free surface. Boundary conditions. This week: extend our 3D flow solver to full 3D water We need to add two things:
Notes Added a 2D cross-section viewer for assignment 6 Not great, but an alternative if the full 3d viewer isn t working for you Warning about the formulas in Fedkiw, Stam, and Jensen - maybe not right
More information3D Modeling I. CG08b Lior Shapira Lecture 8. Based on: Thomas Funkhouser,Princeton University. Thomas Funkhouser 2000
3D Modeling I CG08b Lior Shapira Lecture 8 Based on: Thomas Funkhouser,Princeton University Course Syllabus I. Image processing II. Rendering III. Modeling IV. Animation Image Processing (Rusty Coleman,
More information03 - Reconstruction. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Spring 17 - Daniele Panozzo
3 - Reconstruction Acknowledgements: Olga Sorkine-Hornung Geometry Acquisition Pipeline Scanning: results in range images Registration: bring all range images to one coordinate system Stitching/ reconstruction:
More informationMotivation. Freeform Shape Representations for Efficient Geometry Processing. Operations on Geometric Objects. Functional Representations
Motivation Freeform Shape Representations for Efficient Geometry Processing Eurographics 23 Granada, Spain Geometry Processing (points, wireframes, patches, volumes) Efficient algorithms always have to
More informationTNM079 Modeling & Animation Lecture 6 (Implicit surfaces)
TNM079 Modeling & Animation Lecture 6 (Implicit surfaces) Mark Eric Dieckmann, Media and Information Technology, ITN Linköpings universitet Campus Norrköping SE-60174 Norrköping May 4, 2016 Content of
More information3D 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
More informationVisualization Computer Graphics I Lecture 20
15-462 Computer Graphics I Lecture 20 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 12] November 20, 2003 Doug James Carnegie Mellon University http://www.cs.cmu.edu/~djames/15-462/fall03
More informationSurface Modeling. Polygon Tables. Types: Generating models: Polygon Surfaces. Polygon surfaces Curved surfaces Volumes. Interactive Procedural
Surface Modeling Types: Polygon surfaces Curved surfaces Volumes Generating models: Interactive Procedural Polygon Tables We specify a polygon surface with a set of vertex coordinates and associated attribute
More informationCS337 INTRODUCTION TO COMPUTER GRAPHICS. Describing Shapes. Constructing Objects in Computer Graphics. Bin Sheng Representing Shape 9/20/16 1/15
Describing Shapes Constructing Objects in Computer Graphics 1/15 2D Object Definition (1/3) Lines and polylines: Polylines: lines drawn between ordered points A closed polyline is a polygon, a simple polygon
More informationPipeline Operations. CS 4620 Lecture 14
Pipeline Operations CS 4620 Lecture 14 2014 Steve Marschner 1 Pipeline you are here APPLICATION COMMAND STREAM 3D transformations; shading VERTEX PROCESSING TRANSFORMED GEOMETRY conversion of primitives
More informationTriangle 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
More information6.837 Introduction to Computer Graphics Quiz 2 Thursday November 20, :40-4pm One hand-written sheet of notes allowed
6.837 Introduction to Computer Graphics Quiz 2 Thursday November 20, 2003 2:40-4pm One hand-written sheet of notes allowed Name: 1 2 3 4 5 6 7 / 4 / 15 / 5 / 5 / 12 / 2 / 7 Total / 50 1 Animation [ /4]
More informationCS GPU and GPGPU Programming Lecture 2: Introduction; GPU Architecture 1. Markus Hadwiger, KAUST
CS 380 - GPU and GPGPU Programming Lecture 2: Introduction; GPU Architecture 1 Markus Hadwiger, KAUST Reading Assignment #2 (until Feb. 17) Read (required): GLSL book, chapter 4 (The OpenGL Programmable
More informationIntroduction to Computer Graphics
Introduction to Computer Graphics James D. Foley Georgia Institute of Technology Andries van Dam Brown University Steven K. Feiner Columbia University John F. Hughes Brown University Richard L. Phillips
More informationGEOMETRIC TOOLS FOR COMPUTER GRAPHICS
GEOMETRIC TOOLS FOR COMPUTER GRAPHICS PHILIP J. SCHNEIDER DAVID H. EBERLY MORGAN KAUFMANN PUBLISHERS A N I M P R I N T O F E L S E V I E R S C I E N C E A M S T E R D A M B O S T O N L O N D O N N E W
More informationComputer 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
More informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More informationThe Traditional Graphics Pipeline
Last Time? The Traditional Graphics Pipeline Reading for Today A Practical Model for Subsurface Light Transport, Jensen, Marschner, Levoy, & Hanrahan, SIGGRAPH 2001 Participating Media Measuring BRDFs
More informationRay Casting of Trimmed NURBS Surfaces on the GPU
Ray Casting of Trimmed NURBS Surfaces on the GPU Hans-Friedrich Pabst Jan P. Springer André Schollmeyer Robert Lenhardt Christian Lessig Bernd Fröhlich Bauhaus University Weimar Faculty of Media Virtual
More informationTriangle 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
More informationInteractive Computer Graphics A TOP-DOWN APPROACH WITH SHADER-BASED OPENGL
International Edition Interactive Computer Graphics A TOP-DOWN APPROACH WITH SHADER-BASED OPENGL Sixth Edition Edward Angel Dave Shreiner Interactive Computer Graphics: A Top-Down Approach with Shader-Based
More information3D Modeling techniques
3D Modeling techniques 0. Reconstruction From real data (not covered) 1. Procedural modeling Automatic modeling of a self-similar objects or scenes 2. Interactive modeling Provide tools to computer artists
More informationGeometric 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
More informationComputer Graphics I Lecture 11
15-462 Computer Graphics I Lecture 11 Midterm Review Assignment 3 Movie Midterm Review Midterm Preview February 26, 2002 Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
More information