# Computer Graphics CS 543 Lecture 13a Curves, Tesselation/Geometry Shaders & Level of Detail

Size: px
Start display at page:

Transcription

1 Computer Graphics CS 54 Lecture 1a Curves, Tesselation/Geometry Shaders & Level of Detail Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI)

2 So Far Dealt with straight lines and flat surfaces Real world objects include curves Need to develop: Representations of curves Tools to render curves

3 Curve Representation: Explicit One variable expressed in terms of another Example: z f ( x, y) Works if one x-value for each y value Example: does not work for a sphere z x y Rarely used in CG because of this limitation

4 Curve Representation: Implicit Represent D curve or D surface as zeros of a formula Example: sphere representation x y z 1 0 May limit classes of functions used Polynomial: function which can be expressed as linear combination of integer powers of x, y, z Degree of algebraic function: highest power in function Example: mx 4 has degree of 4

5 Curve Representation: Parametric Represent D curve as functions, 1 parameter D surface as functions, parameters Example: parametric sphere ( x( u), y( u)) ( x( u, v), y( u, v), z( u, v)) x(, ) y(, ) z(, ) cos cos cos sin sin

6 Choosing Representations Different representation suitable for different applications Implicit representations good for: Computing ray intersection with surface Determing if point is inside/outside a surface Parametric representation good for: Dividing surface into small polygonal elements for rendering Subdivide into smaller patches Sometimes possible to convert one representation into another

7 Continuity Consider parametric curve P ( u) ( x( u), y( u), z( u)) T We would like smoothest curves possible Mathematically express smoothness as continuity (no jumps) Defn: if kth derivatives exist, and are continuous, curve has kth order parametric continuity denoted C k

8 Continuity 0 th order means curve is continuous 1 st order means curve tangent vectors vary continuously, etc

9 Interactive Curve Design Mathematical formula unsuitable for designers Prefer to interactively give sequence of points (control points) Write procedure: Input: sequence of points Output: parametric representation of curve

10 Interactive Curve Design 1 approach: curves pass through control points (interpolate) Example: Lagrangian Interpolating Polynomial Difficulty with this approach: Polynomials always have wiggles For straight lines wiggling is a problem Our approach: approximate control points (Bezier, B-Splines) called De Casteljau s algorithm

11 De Casteljau Algorithm Consider smooth curve that approximates sequence of control points [p0,p1,.] p( u) up (1 u) p0 1 0 u 1 Blending functions: u and (1 u) are non-negative and sum to one

12 De Casteljau Algorithm Now consider points line segments, P0 to P1 and P1 to P p 01 ( u) (1 u) p0 up1 p11 ( u) (1 u) p1 up

13 De Casteljau Algorithm Substituting known values of p ( ) and p ( ) 11 u 01 u p( u) (1 u) p01 up11( u) ( 1u) p0 (u(1 u)) p1 u p b 0( u ) b ( ) 1 u b ( u) Blending functions for degree Bezier curve b 0 ( u) (1 u) b1( u) u(1 u) b u) ( u Note: blending functions, non-negative, sum to 1

14 De Casteljau Algorithm Extend to 4 control points P0, P1, P, P p ( u) (1 u) p0 (u(1 u) ) p1 (u (1 u)) p u b ( u) b ( u) b ( u 0 1 ) b ( u ) Final result above is Bezier curve of degree

15 De Casteljau Algorithm p ( u) (1 u) p0 (u(1 u) ) p1 (u (1 u)) p u b ( u) b ( u) b ( u 0 1 ) b ( u ) Blending functions are polynomial functions called Bernstein s polynomials b b b b 0 1 ( u) ( u) ( u) ( u) (1 u) u u u(1 u) (1 u)

16 De Casteljau Algorithm p ( u) (1 u) p0 (u(1 u) ) p1 (u (1 u)) p u 1 1 Writing coefficient of blending functions gives Pascal s triangle control points 4 control points control points

17 De Casteljau Algorithm In general, blending function for k Bezier curve has form k ki i bik ( u) (1 u) u i Example b ( u) (1 u) u (1 u)

18 De Casteljau Algorithm Can express cubic parametric curve in matrix form 1 0 ],, [1, ) ( p p p p M u u u u p B where M B

19 Subdividing Bezier Curves OpenGL renders flat objects To render curves, approximate with small linear segments Subdivide surface to polygonal patches Bezier curves useful for elegant, recursive subdivision

20 Subdividing Bezier Curves Let (P0 P) denote original sequence of control points Recursively interpolate with u = ½ as below Sequences (P00,P01,P0,P0) and (P0,P1,P1,0) define Bezier curves also Bezier Curves can either be straightened or curved recursively in this way

21 Bezier Surfaces Bezier surfaces: interpolate in two dimensions This called Bilinear interpolation Example: 4 control points, P00, P01, P10, P11, parameters u and v Interpolate between P00 and P01 using u P10 and P11 using u P00 and P10 using v P01 and P11 using v p( u, v) (1 v)((1 u) p00 up01) v((1 u) p10 up11)

22 Bezier Surfaces Expressing in terms of blending functions p( u, v) b p 01 ( v) b01( u) p00 b01( v) b11b01( u) p01 b11( v) b11( u) 11 Generalizing p( u, v) i0 j0 b i, ( v) bj,( u) p i, j

23 Problems with Bezier Curves Bezier curves are elegant but too many control points To achieve smoother curve = more control points = higher order polynomial = more calculations Global support problem: All blending functions are non-zero for all values of u All control points contribute to all parts of the curve Means after modelling complex surface (e.g. a ship), if one control point is moves, recalculate everything!

24 B-spline blending functions, order B-Splines B-splines designed to address Bezier shortcomings B-Spline given by blending control points Local support: Each spline contributes in limited range Only non-zero splines contribute in a given range of u p( u) m i0 B ( u) i p i

25 NURBS Encompasses both Bezier curves/surfaces and B-splines Non-uniform Rational B-splines (NURBS) Rational function is ratio of two polynomials Some curves can be expressed as rational functions but not as simple polynomials No known exact polynomial for circle Rational parametrization of unit circle on xy-plane: 1 u x( u) 1 u u y( u) 1 u z( u) 0

26 NURBS We can apply homogeneous coordinates to bring in w Useful property of NURBS: preserved under transformation E.g. Rotate sphere defined as NURBS, still a sphere 1 ) ( 0 ) ( ) ( 1 ) ( u w u u z u u y u u x

27 Tesselation tesselation Far = Less detailed mesh Near = More detailed mesh Simplification Previously: Pre-generate mesh versions offline Tesselation shader unit new to GPU in DirectX 10 (007) Subdivide faces to yield finer detail, generate new vertices, primitives Mesh simplification/tesselation on GPU = Real time LoD Tesselation: Demo

28 Tessellation Shaders Can subdivide curves, surfaces on the GPU

29 Where Does Tesselation Shader Fit? Fixed number of vertices in/out Can change number of vertices

30 Geometry Shader After Tesselation shader. Can Handle whole primitives Generate new primitives Generate no primitives (cull)

31 Level of Detail Use simpler versions of objects if they make smaller contributions to the image LOD algorithms have three parts: Generation: Models of different details are generated Selection: Chooses which model should be used depending on criteria Switching: Changing from one model to another Can be used for models, textures, shading and more

32 Level of Detail

33 LOD Switching Discrete Geometry LODs LOD is switched suddenly from one frame to the next Blend LODs Two LODs are blended together over time New LOD is faded by increasing alpha value from 0 to 1 More expensive than rendering one LOD Faded LODs are drawn last to avoid distant objects drawing over the faded LOD

34 LOD Switching (cont.) Alpha LOD Alpha value of object is lowered as distance increases Experience is more continuous Performance is only felt when object disappears Requires sorting of scene based on transparency

35 LOD Selection Determining which LOD to render and which to blend Range-Based: LOD choice based on distance

36 Time-Critical LOD Rendering Using LOD to ensure constant frame rates Predictive algorithm Selects the LOD based on which objects are visible Heuristics: Maximize Constraint:

37 References Hill and Kelley, chapter 11 Angel and Shreiner, Interactive Computer Graphics, 6 th edition, Chapter 10 Shreiner, OpenGL Programming Guide, 8 th edition

### Curves and Surfaces Computer Graphics I Lecture 9

15-462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] February 19, 2002 Frank Pfenning Carnegie

### Curves 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

### Information Coding / Computer Graphics, ISY, LiTH. Splines

28(69) Splines Originally a drafting tool to create a smooth curve In computer graphics: a curve built from sections, each described by a 2nd or 3rd degree polynomial. Very common in non-real-time graphics,

### CHAPTER 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...........

### Curve 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

### Bezier Curves, B-Splines, NURBS

Bezier Curves, B-Splines, NURBS Example Application: Font Design and Display Curved objects are everywhere There is always need for: mathematical fidelity high precision artistic freedom and flexibility

### Curves. Computer Graphics CSE 167 Lecture 11

Curves Computer Graphics CSE 167 Lecture 11 CSE 167: Computer graphics Polynomial Curves Polynomial functions Bézier Curves Drawing Bézier curves Piecewise Bézier curves Based on slides courtesy of Jurgen

### Intro to Curves Week 4, Lecture 7

CS 430/536 Computer Graphics I Intro to Curves Week 4, Lecture 7 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University

### Computergrafik. 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

### CSE 167: Introduction to Computer Graphics Lecture #13: Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2017

CSE 167: Introduction to Computer Graphics Lecture #13: Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2017 Announcements Project 4 due Monday Nov 27 at 2pm Next Tuesday:

### Computergrafik. 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

### 2D Spline Curves. CS 4620 Lecture 13

2D Spline Curves CS 4620 Lecture 13 2008 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes [Boeing] that is, without discontinuities So far we can make things with corners

### Intro to Curves Week 1, Lecture 2

CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University Outline Math review Introduction to 2D curves

### Lecture IV Bézier Curves

Lecture IV Bézier Curves Why Curves? Why Curves? Why Curves? Why Curves? Why Curves? Linear (flat) Curved Easier More pieces Looks ugly Complicated Fewer pieces Looks smooth What is a curve? Intuitively:

### Design considerations

Curves Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in

### CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2

CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 Outline Math review Introduction to 2D curves

### CSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016

CSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016 Announcements Project 3 due tomorrow Midterm 2 next

### CSE 167: Introduction to Computer Graphics Lecture 12: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2013

CSE 167: Introduction to Computer Graphics Lecture 12: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2013 Announcements Homework assignment 5 due tomorrow, Nov

### Interactive 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

### Until 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

### 2D Spline Curves. CS 4620 Lecture 18

2D Spline Curves CS 4620 Lecture 18 2014 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes that is, without discontinuities So far we can make things with corners (lines,

### CS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside

CS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside Blending Functions Blending functions are more convenient basis than monomial basis canonical form (monomial

### Curves and Surfaces 1

Curves and Surfaces 1 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2 The Teapot 3 Representing

### Computer 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/

### Splines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes

CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster

### B-spline Curves. Smoother than other curve forms

Curves and Surfaces B-spline Curves These curves are approximating rather than interpolating curves. The curves come close to, but may not actually pass through, the control points. Usually used as multiple,

### Central 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

### 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

### Parametric Curves. University of Texas at Austin CS384G - Computer Graphics

Parametric Curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Parametric Representations 3 basic representation strategies: Explicit: y = mx + b Implicit: ax + by + c

### Curved PN Triangles. Alex Vlachos Jörg Peters

1 Curved PN Triangles Alex Vlachos AVlachos@ati.com Jörg Peters jorg@cise.ufl.edu Outline 2 Motivation Constraints Surface Properties Performance Demo Quick Demo 3 Constraints 4 Software Developers Must

### Computer 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

### Curves and Surfaces. Computer Graphics COMP 770 (236) Spring Instructor: Brandon Lloyd

Curves and Surfaces Computer Graphics COMP 770 (236) Spring 2007 Instructor: Brandon Lloyd 4/11/2007 Final projects Surface representations Smooth curves Subdivision Todays Topics 2 Final Project Requirements

### INF3320 Computer Graphics and Discrete Geometry

INF3320 Computer Graphics and Discrete Geometry More smooth Curves and Surfaces Christopher Dyken, Michael Floater and Martin Reimers 10.11.2010 Page 1 More smooth Curves and Surfaces Akenine-Möller, Haines

### Sung-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

### Computer 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

### Parametric curves. Brian Curless CSE 457 Spring 2016

Parametric curves Brian Curless CSE 457 Spring 2016 1 Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics

### Direct Rendering of Trimmed NURBS Surfaces

Direct Rendering of Trimmed NURBS Surfaces Hardware Graphics Pipeline 2/ 81 Hardware Graphics Pipeline GPU Video Memory CPU Vertex Processor Raster Unit Fragment Processor Render Target Screen Extended

### Curves 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

### Parametric Curves. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Parametric Curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Parametric Representations 3 basic representation strategies: Explicit: y = mx + b Implicit: ax + by + c

### 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

### Introduction to Computer Graphics

Introduction to Computer Graphics 2016 Spring National Cheng Kung University Instructors: Min-Chun Hu 胡敏君 Shih-Chin Weng 翁士欽 ( 西基電腦動畫 ) Data Representation Curves and Surfaces Limitations of Polygons Inherently

### Homework #2. Hidden Surfaces, Projections, Shading and Texture, Ray Tracing, and Parametric Curves

Computer Graphics Instructor: Brian Curless CSE 457 Spring 2013 Homework #2 Hidden Surfaces, Projections, Shading and Texture, Ray Tracing, and Parametric Curves Assigned: Sunday, May 12 th Due: Thursday,

### 3D 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

### COMPUTER AIDED ENGINEERING DESIGN (BFF2612)

COMPUTER AIDED ENGINEERING DESIGN (BFF2612) BASIC MATHEMATICAL CONCEPTS IN CAED by Dr. Mohd Nizar Mhd Razali Faculty of Manufacturing Engineering mnizar@ump.edu.my COORDINATE SYSTEM y+ y+ z+ z+ x+ RIGHT

### Fall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.

Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve

### Homework #2. Shading, Projections, Texture Mapping, Ray Tracing, and Bezier Curves

Computer Graphics Instructor: Brian Curless CSEP 557 Autumn 2016 Homework #2 Shading, Projections, Texture Mapping, Ray Tracing, and Bezier Curves Assigned: Wednesday, Nov 16 th Due: Wednesday, Nov 30

### TSBK03 Screen-Space Ambient Occlusion

TSBK03 Screen-Space Ambient Occlusion Joakim Gebart, Jimmy Liikala December 15, 2013 Contents 1 Abstract 1 2 History 2 2.1 Crysis method..................................... 2 3 Chosen method 2 3.1 Algorithm

### Computer Graphics CS 543 Lecture 1 (Part I) Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics CS 543 Lecture 1 (Part I) Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) About This Course Computer graphics: algorithms, mathematics, data structures..

### Introduction 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

### GLOBAL EDITION. Interactive Computer Graphics. A Top-Down Approach with WebGL SEVENTH EDITION. Edward Angel Dave Shreiner

GLOBAL EDITION Interactive Computer Graphics A Top-Down Approach with WebGL SEVENTH EDITION Edward Angel Dave Shreiner This page is intentionally left blank. Interactive Computer Graphics with WebGL, Global

### Computer Graphics (CS 543) Lecture 1 (Part 1): Introduction to Computer Graphics

Computer Graphics (CS 543) Lecture 1 (Part 1): Introduction to Computer Graphics Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) What is Computer Graphics (CG)? Computer

### From 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

### Objects 2: Curves & Splines Christian Miller CS Fall 2011

Objects 2: Curves & Splines Christian Miller CS 354 - Fall 2011 Parametric curves Curves that are defined by an equation and a parameter t Usually t [0, 1], and curve is finite Can be discretized at arbitrary

### 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

### For each question, indicate whether the statement is true or false by circling T or F, respectively.

True/False For each question, indicate whether the statement is true or false by circling T or F, respectively. 1. (T/F) Rasterization occurs before vertex transformation in the graphics pipeline. 2. (T/F)

### Surface 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

### COMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg

COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg January 31, 2011 ii T. W. Sederberg iii Preface This semester is the 24 th time I have taught a course at Brigham Young University titled, Computer Aided

### COMP3421. Global Lighting Part 2: Radiosity

COMP3421 Global Lighting Part 2: Radiosity Recap: Global Lighting The lighting equation we looked at earlier only handled direct lighting from sources: We added an ambient fudge term to account for all

### Rendering Curves and Surfaces. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico

Rendering Curves and Surfaces Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Objectives Introduce methods to draw curves - Approximate

### Interactive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1

Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade

### Splines. Connecting the Dots

Splines or: Connecting the Dots Jens Ogniewski Information Coding Group Linköping University Before we start... Some parts won t be part of the exam Basically all that is not described in the book. More

### Curves & 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

### Module Contact: Dr Stephen Laycock, CMP Copyright of the University of East Anglia Version 1

UNIVERSITY OF EAST ANGLIA School of Computing Sciences Main Series PG Examination 2013-14 COMPUTER GAMES DEVELOPMENT CMPSME27 Time allowed: 2 hours Answer any THREE questions. (40 marks each) Notes are

### Review of Tuesday. ECS 175 Chapter 3: Object Representation

Review of Tuesday We have learnt how to rasterize lines and fill polygons Colors (and other attributes) are specified at vertices Interpolation required to fill polygon with attributes 26 Review of Tuesday

### CS123 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

### Computer Graphics Fundamentals. Jon Macey

Computer Graphics Fundamentals Jon Macey jmacey@bournemouth.ac.uk http://nccastaff.bournemouth.ac.uk/jmacey/ 1 1 What is CG Fundamentals Looking at how Images (and Animations) are actually produced in

### ECE 600, Dr. Farag, Summer 09

ECE 6 Summer29 Course Supplements. Lecture 4 Curves and Surfaces Aly A. Farag University of Louisville Acknowledgements: Help with these slides were provided by Shireen Elhabian A smile is a curve that

### Pipeline 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

### CSE 167: Introduction to Computer Graphics Lecture #10: View Frustum Culling

CSE 167: Introduction to Computer Graphics Lecture #10: View Frustum Culling Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2015 Announcements Project 4 due tomorrow Project

### 08 - Designing Approximating Curves

08 - Designing Approximating Curves Acknowledgement: Olga Sorkine-Hornung, Alexander Sorkine-Hornung, Ilya Baran Last time Interpolating curves Monomials Lagrange Hermite Different control types Polynomials

### The 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

### Spring 2012 Final. CS184 - Foundations of Computer Graphics. University of California at Berkeley

Spring 2012 Final CS184 - Foundations of Computer Graphics University of California at Berkeley Write your name HERE: Write your login HERE: Closed book. You may not use any notes or printed/electronic

### CS-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

### Computer Graphics (CS 543) Lecture 9 (Part 2): Clipping. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics (CS 543) Lecture 9 (Part 2): Clipping Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) OpenGL Stages After projection, several stages before objects drawn

### CS 130 Exam I. Fall 2015

S 3 Exam I Fall 25 Name Student ID Signature You may not ask any questions during the test. If you believe that there is something wrong with a question, write down what you think the question is trying

### CS559 Computer Graphics Fall 2015

CS559 Computer Graphics Fall 2015 Practice Final Exam Time: 2 hrs 1. [XX Y Y % = ZZ%] MULTIPLE CHOICE SECTION. Circle or underline the correct answer (or answers). You do not need to provide a justification

### Problem Max. Points Total 80

University of California San Diego Department of Computer Science CSE167: Introduction to Computer Graphics Fall Quarter 2017 Midterm Examination #2 Tuesday, November 21 st, 2017 Instructor: Dr. Jürgen

### Computer Graphics (CS 543) Lecture 10: Soft Shadows (Maps and Volumes), Normal and Bump Mapping

Computer Graphics (CS 543) Lecture 10: Soft Shadows (Maps and Volumes), Normal and Bump Mapping Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Shadow Buffer Theory Observation:

### Ray 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

### CS337 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

### Pipeline 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

### Curves and Surface I. Angel Ch.10

Curves and Surface I Angel Ch.10 Representation of Curves and Surfaces Piece-wise linear representation is inefficient - line segments to approximate curve - polygon mesh to approximate surfaces - can

### Curves and Surfaces 2

Curves and Surfaces 2 Computer Graphics Lecture 17 Taku Komura Today More about Bezier and Bsplines de Casteljau s algorithm BSpline : General form de Boor s algorithm Knot insertion NURBS Subdivision

### Painter s HSR Algorithm

Painter s HSR Algorithm Render polygons farthest to nearest Similar to painter layers oil paint Viewer sees B behind A Render B then A Depth Sort Requires sorting polygons (based on depth) O(n log n) complexity

### Scene Management. Video Game Technologies 11498: MSc in Computer Science and Engineering 11156: MSc in Game Design and Development

Video Game Technologies 11498: MSc in Computer Science and Engineering 11156: MSc in Game Design and Development Chap. 5 Scene Management Overview Scene Management vs Rendering This chapter is about rendering

### CS130 : Computer Graphics Curves. Tamar Shinar Computer Science & Engineering UC Riverside

CS130 : Computer Graphics Curves Tamar Shinar Computer Science & Engineering UC Riverside Design considerations local control of shape design each segment independently smoothness and continuity ability

### Curves and Curved Surfaces. Adapted by FFL from CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006

Curves and Curved Surfaces Adapted by FFL from CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006 Outline for today Summary of Bézier curves Piecewise-cubic curves, B-splines Surface

### LECTURE #6. Geometric Modelling for Engineering Applications. Geometric modeling for engineering applications

LECTURE #6 Geometric modeling for engineering applications Geometric Modelling for Engineering Applications Introduction to modeling Geometric modeling Curve representation Hermite curve Bezier curve B-spline

### Parametric curves. Reading. Curves before computers. Mathematical curve representation. CSE 457 Winter Required:

Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Parametric curves CSE 457 Winter 2014 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics and Geometric

### 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

### CS452/552; EE465/505. Clipping & Scan Conversion

CS452/552; EE465/505 Clipping & Scan Conversion 3-31 15 Outline! From Geometry to Pixels: Overview Clipping (continued) Scan conversion Read: Angel, Chapter 8, 8.1-8.9 Project#1 due: this week Lab4 due:

### Outline. Properties of Piecewise Linear Interpolations. The de Casteljau Algorithm. Recall: Linear Interpolation

CS 430/585 Computer Graphics I Curve Drawing Algorithms Week 4, Lecture 8 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel

### Know 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

### Representing Curves Part II. Foley & Van Dam, Chapter 11

Representing Curves Part II Foley & Van Dam, Chapter 11 Representing Curves Polynomial Splines Bezier Curves Cardinal Splines Uniform, non rational B-Splines Drawing Curves Applications of Bezier splines

### Intro to Modeling Modeling in 3D

Intro to Modeling Modeling in 3D Polygon sets can approximate more complex shapes as discretized surfaces 2 1 2 3 Curve surfaces in 3D Sphere, ellipsoids, etc Curved Surfaces Modeling in 3D ) ( 2 2 2 2

### Dgp _ 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

### Shape modeling Modeling technique Shape representation! 3D Graphics Modeling Techniques

D Graphics http://chamilo2.grenet.fr/inp/courses/ensimag4mmgd6/ Shape Modeling technique Shape representation! Part : Basic techniques. Projective rendering pipeline 2. Procedural Modeling techniques Shape

### The Free-form Surface Modelling System

1. Introduction The Free-form Surface Modelling System Smooth curves and surfaces must be generated in many computer graphics applications. Many real-world objects are inherently smooth (fig.1), and much