CS354 Computer Graphics Surface Representation III. Qixing Huang March 5th 2018
|
|
|
- Samson Dean
- 5 years ago
- Views:
Transcription
1 CS354 Computer Graphics Surface Representation III Qixing Huang March 5th 2018
2 Today s Topic Bspline curve operations (Brief) Knot Insertion/Deletion Subdivision (Focus) Subdivision curves Subdivision surfaces
3 Bspline Polynomials of order k that are stitched together with C k-1 continuity Image from
![Knot insertion Break a curve segments into two segments [t1, t3]](/docs-images/85/91910291/images/4-0.jpg "to [t1, t2] and [t2, t3] C k continuity at knot t2 Becomes C k-1")
4 Knot insertion Break a curve segments into two segments [t1, t3] to [t1, t2] and [t2, t3] C k continuity at knot t2 Becomes C k-1 continuity after moving the control point Image from
5 Knot insertion Image from
6 Knot removal Image from
7 Subdivision Curves/Surfaces Slide Credit: Mirela-Ben Chen
8 Problems with NURBS A single NURBS patch is either a disk a tube or a torus Must use many NURBS patches to model complex geometry When deforming a surface made of NURBS patches,cra cks arise at the seams
9 Subdivision Subdivision defines a smooth curve or surfac eaes the limit of a sequence of successive refi nements
10 Subdivision Rules How the connectivity changes How the geometry changes Old points New points
11 Subdivision Surfaces Generalization of spline curves / surfaces Arbitrary control meshes Successive refinement (subdivision) Converges to smooth limit surface Connection between splines and meshes
12 Subdivision Surfaces Generalization of spline curves / surfaces Arbitrary control meshes Successive refinement (subdivision) Converges to smooth limit surface Connection between splines and meshes
13 Example: Geri s Game (Pixar) Subdivision used for Geri s hands and head Clothing Tie and shoes
14 Example: Geri s Game (Pixar) Woody s hand (NURBS) Geri s hand (subdivision)
15 Example: Geri s Game (Pixar) Sharp and semi-sharp features
16 Example: Games Supported in hardware in DirectX 11
17 Subdivision curves Given a control polygon......find a smooth curve related to that polygon
18 Subdivision curve types Approximating Interpolating Corner Cutting
19 Approximating
20 Approximating Splitting step: split each edge in two
21 Approximating Averaging step: relocate each (original) vertex according to some (simple) rule...
22 Approximating Start over...
23 Approximating...splitting...
24 Approximating...averaging...
25 Approximating...and so on...
26 Approximating If the rule is designed carefully the control polygons will converge to a smooth limit curve!
27 Equivalent to Insert single new point at mid-edge Filter entire set of points Catmull-Clark rule: Filter with (1/8, 6/8, 1/8)
28 Corner Cutting Subdivision rule: Insert two new vertices at ¼ and ¾ of each edge Remove the old vertices Connect the new vertices
29 B-spline curves Piecewise polynomial of degree n
30 B-spline curves Distinguish between odd and even points Linear B-spline Odd coefficients (1/2, 1/2) Even coefficient (1)
31 B-spline curves Quadratic B Spline (Chaikin) Odd coefficients (¼, ¾) Even coefficients (¾, ¼) Cubic B-Spline (Catmull-Clark) Odd coefficients (4/8, 4/8) Even coefficients (1/8, 6/8, 1/8)
32 Cubic B-Spline
33 Cubic B-Spline
34 Cubic B-Spline
35 Cubic B-Spline
36 Cubic B-Spline
37 Cubic B-Spline
38 Cubic B-Spline
39 Cubic B-Spline
40 Cubic B-Spline
41 Cubic B-Spline
42 Cubic B-Spline
43 Cubic B-Spline
44 Cubic B-Spline
45 Cubic B-Spline
46 Cubic B-Spline
47 B-Spline Curves
48 Interpolating (4-point scheme) Keep old vertices Generate new vertices by fitting cubic curve to old vertices C 1 continuous limit curve
49 Interpolating
50 Interpolating
51 Interpolating
52 Interpolating
53 Interpolating
54 Local Subdivision Matrix Example: Cubic B-Splines Invariant neighborhood size: 5
55 Analysis of Subdivision Analysis via eigen-decomposition of matrix S Compute the eigenvalues and eigenvectors Let be real and X a complete set of eigenvectors
56 Limit Behavior - Position Any vector is linear combination of eigenvectors: Apply subdivision matrix:
57 Limit Behavior - Position
58 Limit Behavior - Tangent
59 Limit Behavior - Tangent
60 Example: Cubic Splines Subdivision matrix & rules Eigenvalues
61 Example: Cubic Splines Eigenvectors Limit position and tangent
62 Questions?
Subdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2 Problems with NURBS A single
Subdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2 Problems with NURBS A single
Subdivision Curves and Surfaces
Subdivision Surfaces or How to Generate a Smooth Mesh?? Subdivision Curves and Surfaces Subdivision given polyline(2d)/mesh(3d) recursively modify & add vertices to achieve smooth curve/surface Each iteration
Subdivision overview
Subdivision overview CS4620 Lecture 16 2018 Steve Marschner 1 Introduction: corner cutting Piecewise linear curve too jagged for you? Lop off the corners! results in a curve with twice as many corners
Subdivision. Outline. Key Questions. Subdivision Surfaces. Advanced Computer Graphics (Spring 2013) Video: Geri s Game (outside link)
Advanced Computer Graphics (Spring 03) CS 83, Lecture 7: Subdivision Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs83/sp3 Slides courtesy of Szymon Rusinkiewicz, James O Brien with material from Denis
Subdivision Surfaces
Subdivision Surfaces CS 4620 Lecture 31 Cornell CS4620 Fall 2015 1 Administration A5 due on Friday Dreamworks visiting Thu/Fri Rest of class Surfaces, Animation, Rendering w/ prior instructor Steve Marschner
Subdivision curves and surfaces. Brian Curless CSE 557 Fall 2015
Subdivision curves and surfaces Brian Curless CSE 557 Fall 2015 1 Reading Recommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications, 1996, section 6.1-6.3, 10.2,
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
CS354 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
Physically-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
Recursive Subdivision Surfaces for Geometric Modeling
Recursive Subdivision Surfaces for Geometric Modeling Weiyin Ma City University of Hong Kong, Dept. of Manufacturing Engineering & Engineering Management Ahmad Nasri American University of Beirut, Dept.
Subdivision Curves and Surfaces: An Introduction
Subdivision Curves and Surfaces: An Introduction Corner Cutting De Casteljau s and de Boor s algorithms all use corner-cutting procedures. Corner cutting can be local or non-local. A cut is local if it
Curve Corner Cutting
Subdivision ision Techniqueses Spring 2010 1 Curve Corner Cutting Take two points on different edges of a polygon and join them with a line segment. Then, use this line segment to replace all vertices
Approximate Catmull-Clark Patches. Scott Schaefer Charles Loop
Approximate Catmull-Clark Patches Scott Schaefer Charles Loop Approximate Catmull-Clark Patches Scott Schaefer Charles Loop Catmull-Clark Surface ACC-Patches Polygon Models Prevalent in game industry Very
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
UNIVERSITY OF CALGARY. Subdivision Surfaces. Advanced Geometric Modeling Faramarz Samavati
Subdivision Surfaces Surfaces Having arbitrary Topologies Tensor Product Surfaces Non Tensor Surfaces We can t find u-curves and v-curves in general surfaces General Subdivision Coarse mesh Subdivision
3D 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
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
Subdivision Surfaces. Homework 1: Last Time? Today. Bilinear Patch. Tensor Product. Spline Surfaces / Patches
Homework 1: Questions/Comments? Subdivision Surfaces Last Time? Curves & Surfaces Continuity Definitions Spline Surfaces / Patches Tensor Product Bilinear Patches Bezier Patches Trimming Curves C0, G1,
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
Subdivision surfaces. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Subdivision surfaces University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Reading Recommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications,
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
Advanced Geometric Modeling CPSC789
Advanced Geometric Modeling CPSC789 Fall 2004 General information about the course CPSC 789 Advanced Geometric Modeling Fall 2004 Lecture Time and Place ENF 334 TR 9:30 10:45 Instructor : Office: MS 618
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,
Subdivision 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)!
Subdivision 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
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
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
Subdivision curves. University of Texas at Austin CS384G - Computer Graphics
Subdivision curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Reading Recommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications,
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
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,
Advanced Graphics. Subdivision Surfaces. Alex Benton, University of Cambridge Supported in part by Google UK, Ltd
Advanced Graphics Subdivision Surfaces Alex Benton, University of Cambridge A.Benton@damtp.cam.ac.uk Supported in part by Google UK, Ltd NURBS patches aren t the greatest NURBS patches are nxm, forming
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
Surfaces for CAGD. FSP Tutorial. FSP-Seminar, Graz, November
Surfaces for CAGD FSP Tutorial FSP-Seminar, Graz, November 2005 1 Tensor Product Surfaces Given: two curve schemes (Bézier curves or B splines): I: x(u) = m i=0 F i(u)b i, u [a, b], II: x(v) = n j=0 G
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,
Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang 1 Guoping Wang 2 1 University of Science and Technology of China 2 Peking University, China SIAM Conference on Geometric and Physical Modeling Doo-Sabin
Spline Surfaces, Subdivision Surfaces
CS-C3100 Computer Graphics Spline Surfaces, Subdivision Surfaces vectorportal.com Trivia Assignment 1 due this Sunday! Feedback on the starter code, difficulty, etc., much appreciated Put in your README
Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation
Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation Karl-Heinz Brakhage RWTH Aachen, 55 Aachen, Deutschland, Email: brakhage@igpm.rwth-aachen.de Abstract In this paper
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
Subdivision of Curves and Surfaces: An Overview
Subdivision of Curves and Surfaces: An Overview Ben Herbst, Karin M Hunter, Emile Rossouw Applied Mathematics, Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland,
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
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
QUADRATIC UNIFORM B-SPLINE CURVE REFINEMENT
On-Line Geometric Modeling Notes QUADRATIC UNIFORM B-SPLINE CURVE REFINEMENT Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview
Interpolatory 3-Subdivision
EUROGRAPHICS 2000 / M. Gross and F.R.A. Hopgood (Guest Editors) Volume 19 (2000), Number 3 Interpolatory 3-Subdivision U. Labsik G. Greiner Computer Graphics Group University of Erlangen-Nuremberg Am Weichselgarten
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
Advanced Computer Graphics
Advanced Computer Graphics Lecture 2: Modeling (1): Polygon Meshes Bernhard Jung TU-BAF, Summer 2007 Overview Computer Graphics Icon: Utah teapot Polygon Meshes Subdivision Polygon Mesh Optimization high-level:
Grid Generation and Grid Conversion by Subdivision Schemes
Grid Generation and Grid Conversion by Subdivision Schemes Karl Heinz Brakhage Institute for Geometry and Applied Mathematics RWTH Aachen University D-55 Aachen brakhage@igpm.rwth-aachen.de Abstract In
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches Charles Loop Microsoft Research Scott Schaefer Texas A&M University April 24, 2007 Technical Report MSR-TR-2007-44 Microsoft Research
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches CHARLES LOOP Microsoft Research and SCOTT SCHAEFER Texas A&M University We present a simple and computationally efficient algorithm
CS-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
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
Shape optimization of smooth surfaces with arbitrary topology
International conference on Innovative Methods in Product Design June 15 th 17 th, 2011, Venice, Italy Shape optimization of smooth surfaces with arbitrary topology Przemysław Kiciak (a) (a) Institut Matematyki
Curves & 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
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
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
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
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
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
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
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
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
u 0+u 2 new boundary vertex
Combined Subdivision Schemes for the design of surfaces satisfying boundary conditions Adi Levin School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Email:fadilev@math.tau.ac.ilg
Subdivision on Arbitrary Meshes: Algorithms and Theory
Subdivision on Arbitrary Meshes: Algorithms and Theory Denis Zorin New York University 719 Broadway, 12th floor, New York, USA E-mail: dzorin@mrl.nyu.edu Subdivision surfaces have become a standard geometric
Example: Loop Scheme. Example: Loop Scheme. What makes a good scheme? recursive application leads to a smooth surface.
Example: Loop Scheme What makes a good scheme? recursive application leads to a smooth surface 200, Denis Zorin Example: Loop Scheme Refinement rule 200, Denis Zorin Example: Loop Scheme Two geometric
Honeycomb Subdivision
Honeycomb Subdivision Ergun Akleman and Vinod Srinivasan Visualization Sciences Program, Texas A&M University Abstract In this paper, we introduce a new subdivision scheme which we call honeycomb subdivision.
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
Free-Form Deformation and Other Deformation Techniques
Free-Form Deformation and Other Deformation Techniques Deformation Deformation Basic Definition Deformation: A transformation/mapping of the positions of every particle in the original object to those
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
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
Efficient GPU Rendering of Subdivision Surfaces. Tim Foley,
Efficient GPU Rendering of Subdivision Surfaces Tim Foley, 2017-03-02 Collaborators Activision Wade Brainerd Stanford Matthias Nießner NVIDIA Manuel Kraemer Henry Moreton 2 Subdivision surfaces are a powerful
Geometric modeling 1
Geometric Modeling 1 Look around the room. To make a 3D model of a room requires modeling every single object you can see. Leaving out smaller objects (clutter) makes the room seem sterile and unrealistic
Bézier Splines. B-Splines. B-Splines. CS 475 / CS 675 Computer Graphics. Lecture 14 : Modelling Curves 3 B-Splines. n i t i 1 t n i. J n,i.
Bézier Splines CS 475 / CS 675 Computer Graphics Lecture 14 : Modelling Curves 3 n P t = B i J n,i t with 0 t 1 J n, i t = i=0 n i t i 1 t n i No local control. Degree restricted by the control polygon.
Advanced Modeling 2. Katja Bühler, Andrej Varchola, Eduard Gröller. March 24, x(t) z(t)
Advanced Modeling 2 Katja Bühler, Andrej Varchola, Eduard Gröller March 24, 2014 1 Parametric Representations A parametric curve in E 3 is given by x(t) c : c(t) = y(t) ; t I = [a, b] R z(t) where x(t),
09 - Designing Surfaces. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo
9 - Designing Surfaces Triangular surfaces A surface can be discretized by a collection of points and triangles Each triangle is a subset of a plane Every point on the surface can be expressed as an affine
G 2 Interpolation for Polar Surfaces
1 G 2 Interpolation for Polar Surfaces Jianzhong Wang 1, Fuhua Cheng 2,3 1 University of Kentucky, jwangf@uky.edu 2 University of Kentucky, cheng@cs.uky.edu 3 National Tsinhua University ABSTRACT In this
Spline Notes. Marc Olano University of Maryland, Baltimore County. February 20, 2004
Spline Notes Marc Olano University of Maryland, Baltimore County February, 4 Introduction I. Modeled after drafting tool A. Thin strip of wood or metal B. Control smooth curved path by running between
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
Computer 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
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:
CS 475 / CS Computer Graphics. Modelling Curves 3 - B-Splines
CS 475 / CS 675 - Computer Graphics Modelling Curves 3 - Bézier Splines n P t = i=0 No local control. B i J n,i t with 0 t 1 J n,i t = n i t i 1 t n i Degree restricted by the control polygon. http://www.cs.mtu.edu/~shene/courses/cs3621/notes/spline/bezier/bezier-move-ct-pt.html
Introduction 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
Non-Uniform Recursive Doo-Sabin Surfaces
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang a,b,c,, Guoping Wang d,e a School of Computer Science and Technology, University of Science and Technology of China, PR China b Key Laboratory of
Subdivision Surfaces. Course Syllabus. Course Syllabus. Modeling. Equivalence of Representations. 3D Object Representations
Subdivision Surfaces Adam Finkelstein Princeton University COS 426, Spring 2003 Course Syllabus I. Image processing II. Rendering III. Modeling IV. Animation Image Processing (Rusty Coleman, CS426, Fall99)
Remark. Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 331
Remark Reconsidering the motivating example, we observe that the derivatives are typically not given by the problem specification. However, they can be estimated in a pre-processing step. A good estimate
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
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
MA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces
MA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces David L. Finn Today, we continue our discussion of subdivision surfaces, by first looking in more detail at the midpoint method and
Distance Functions 1
Distance Functions 1 Distance function Given: geometric object F (curve, surface, solid, ) Assigns to each point the shortest distance from F Level sets of the distance function are trimmed offsets F p
REAL-TIME SMOOTH SURFACE CONSTRUCTION ON THE GRAPHICS PROCESSING UNIT
REAL-TIME SMOOTH SURFACE CONSTRUCTION ON THE GRAPHICS PROCESSING UNIT By TIANYUN NI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
A subdivision scheme for hexahedral meshes
A subdivision scheme for hexahedral meshes Chandrajit Bajaj Department of Computer Sciences, University of Texas Scott Schaefer Department of Computer Science, Rice University Joe Warren Department of
Advanced Graphics. Beziers, B-splines, and NURBS. Alex Benton, University of Cambridge Supported in part by Google UK, Ltd
Advanced Graphics Beziers, B-splines, and NURBS Alex Benton, University of Cambridge A.Benton@damtp.cam.ac.uk Supported in part by Google UK, Ltd Bezier splines, B-Splines, and NURBS Expensive products
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn School of Mathematical Sciences Tel Aviv University Michael S. Floater Department of Informatics University of
In this course we will need a set of techniques to represent curves and surfaces in 2-d and 3-d. Some reasons for this include
Parametric Curves and Surfaces In this course we will need a set of techniques to represent curves and surfaces in 2-d and 3-d. Some reasons for this include Describing curves in space that objects move
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
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
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
Images from 3D Creative Magazine. 3D Modelling Systems
Images from 3D Creative Magazine 3D Modelling Systems Contents Reference & Accuracy 3D Primitives Transforms Move (Translate) Rotate Scale Mirror Align 3D Booleans Deforms Bend Taper Skew Twist Squash
Normals of subdivision surfaces and their control polyhedra
Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,
(Spline, Bezier, B-Spline)
(Spline, Bezier, B-Spline) Spline Drafting terminology Spline is a flexible strip that is easily flexed to pass through a series of design points (control points) to produce a smooth curve. Spline curve
Object 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