Curves and Surfaces 1


 Anna Hodge
 9 months ago
 Views:
Transcription
1 Curves and Surfaces 1
2 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric BiCubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2
3 The Teapot 3
4 Representing Polygon Meshes explicit representation by a list of vertex coordinates pointers to a vertex list pointers to an edge list 4
5 Pointers to a Vertex List 5
6 Pointers to an Edge List 6
7 Ax+By+Cz+D=0 Plane Equation And (A, B, C) means the normal vector so, given points P 1, P 2, and P 3 on the plane (A, B, C) =P 1 P 2 P 1 P 3 What happened if (A, B, C) =(0, 0, 0)? The distance from a vertex (x, y, z) to the plane is 7
8 8 Parametric Cubic Curves The cubic polynomials that define a curve segment are of the form
9 9 Parametric Cubic Curves The curve segment can be rewrite as where
10 10 Continuity between curve segments
11 Tangent Vector 11
12 Continuity between curve segments G 0 geometric continuity two curve segments join together G 1 geometric continuity the directions (but not necessarily the magnitudes) of the two segments tangent vectors are equal at a join point 12
13 Continuity between curve segments C 1 continuous the tangent vectors of the two cubic curve segments are equal (both directions and magnitudes) at the segments join point C n continuous the direction and magnitude of through the nth derivative are equal at the join point 13
14 14 Continuity between curve segments
15 15 Continuity between curve segments
16 Three Types of Parametric Cubic Curves Hermite Curves defined by two endpoints and two endpoint tangent vectors Bézier Curves defined by two endpoints and two control points which control the endpoint tangent vectors Splines defined by four control points 16
17 Parametric Cubic Curves Rewrite the coefficient matrix as where M is a 4 4 basis matrix, G is called the geometry matrix so 17
18 18 Parametric Cubic Curves where function is called the blending
19 19 Hermite Curves Given the endpoints P 1 and P 4 and tangent vectors at R 1 and R 4 What are Hermite basis matrix M H Hermite geometry vector G H Hermite blending functions B H By definition
20 20 Hermite Curves Since
21 21 Hermite Curves so and
22 Four control points Bezier Curves Two endpoints, two direction points Length of lines from each endpoint to its direction point representing the speed with which the curve sets off towards the direction point Fig. 4.8,
23 Bézier Curves Given the endpoints and and two control points and which determine the endpoints tangent vectors, such that What is Bézier basis matrix M B Bézier geometry vector G B Bézier blending functions B B 23
24 by definition then Bézier Curves so 24
25 25 Bézier Curves and
26 Subdividing Bézier Curves How to draw the curve? How to convert it to be linesegments? 26
27 Bezier Curves Constructing a Bezier curve Fig Finding midpoints of lines 27
28 Bezier Curves 28
29 Convex Hull 29
30 Spline the polynomial coefficients for natural cubic splines are dependent on all n control points has one more degree of continuity than is inherent in the Hermite and Bézier forms moving any one control point affects the entire curve the computation time needed to invert the matrix can interfere with rapid interactive reshaping of a curve 30
31 BSpline 31
32 Uniform NonRational BSplines cubic BSpline has m+1 control points has m2 cubic polynomial curve segments uniform the knots are spaced at equal intervals of the parameter t nonrational not rational cubic polynomial curves 32
33 Uniform NonRational BSplines Curve segment Q i is defined by points thus BSpline geometry matrix if then 33
34 34 Uniform NonRational BSplines so BSpline basis matrix BSpline blending functions
35 NonUniform NonRational BSplines the knotvalue sequence is a nondecreasing sequence allow multiple knot and the number of identical parameter is the multiplicity so Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5) 35
36 NonUniform NonRational BSplines Where is the jthorder blending function for weighting control point p i 36
37 Knot Multiplicity & Continuity Since Q(t i ) is within the convex hull of P i3, P i2, and P i1 If t i =t i+1, Q(t i ) is within the convex hull of P i3, P i2, and P i1 and the convex hull of P i2, P i1, and P i, so it will lie on P i2 P i1 If t i =t i+1 =t i+2, Q(t i ) will lie on p i1 If t i =t i+1 =t i+2 =t i+3, Q(t i ) will lie on both P i1 and P i, and the curve becomes broken 37
38 Knot Multiplicity & Continuity multiplicity 1 : C 2 continuity multiplicity 2 : C 1 continuity multiplicity 3 : C 0 continuity multiplicity 4 : no continuity 38
39 NURBS: NonUniform Rational BSplines rational x(t), y(t) and z(t) are defined as the ratio of two cubic polynomials rational cubic polynomial curve segments are ratios of polynomials can be Bézier, Hermite, or BSplines 39
40 Parametric BiCubic Surfaces Parametric cubic curves are so parametric bicubic surfaces are If we allow the points in G to vary in 3D along some path, then since G i (t) are cubics 40
41 41 Parametric BiCubic Surfaces so
42 Hermite Surfaces 42
43 Bézier Surfaces 43
44 BSpline Surfaces 44
45 Normals to Surfaces 45
46 46 Quadric Surfaces implicit surface equation an alternative representation
47 advantages Quadric Surfaces computing the surface normal testing whether a point is on the surface computing z given x and y calculating intersections of one surface with another 47
Curves and Surfaces Computer Graphics I Lecture 9
15462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.110.6] February 19, 2002 Frank Pfenning Carnegie
More informationBezier Curves, BSplines, NURBS
Bezier Curves, BSplines, NURBS Example Application: Font Design and Display Curved objects are everywhere There is always need for: mathematical fidelity high precision artistic freedom and flexibility
More informationRepresenting 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 BSplines Drawing Curves Applications of Bezier splines
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 informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.haoli.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationBspline Curves. Smoother than other curve forms
Curves and Surfaces Bspline 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,
More informationComputer Graphics. Curves and Surfaces. Hermite/Bezier Curves, (B)Splines, and NURBS. By Ulf Assarsson
Computer Graphics Curves and Surfaces Hermite/Bezier Curves, (B)Splines, and NURBS By Ulf Assarsson Most of the material is originally made by Edward Angel and is adapted to this course by Ulf Assarsson.
More information2D 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
More informationIntro 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
More informationCurves. 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
More informationCurves and Surface I. Angel Ch.10
Curves and Surface I Angel Ch.10 Representation of Curves and Surfaces Piecewise linear representation is inefficient  line segments to approximate curve  polygon mesh to approximate surfaces  can
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 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 informationCS770/870 Spring 2017 Curve Generation
CS770/870 Spring 2017 Curve Generation Primary resources used in preparing these notes: 1. Foley, van Dam, Feiner, Hughes, Phillips, Introduction to Computer Graphics, AddisonWesley, 1993. 2. Angel, Interactive
More informationRational Bezier Curves
Rational Bezier Curves Use of homogeneous coordinates Rational spline curve: define a curve in one higher dimension space, project it down on the homogenizing variable Mathematical formulation: n P(u)
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 informationCOMPUTER 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
More informationCSE 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
More informationCSE 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:
More information(Spline, Bezier, BSpline)
(Spline, Bezier, BSpline) 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
More informationLECTURE #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 Bspline
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 informationCurves 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 Piecewisecubic curves, Bsplines Surface
More informationApproximation of 3DParametric Functions by Bicubic Bspline Functions
International Journal of Mathematical Modelling & Computations Vol. 02, No. 03, 2012, 211220 Approximation of 3DParametric Functions by Bicubic Bspline Functions M. Amirfakhrian a, a Department of Mathematics,
More informationCurves & Surfaces. MIT EECS 6.837, Durand and Cutler
Curves & Surfaces Schedule Sunday October 5 th, * 35 PM * Review Session for Quiz 1 Extra Office Hours on Monday Tuesday October 7 th : Quiz 1: In class 1 handwritten 8.5x11 sheet of notes allowed Wednesday
More informationAn introduction to NURBS
An introduction to NURBS Philippe Lavoie January 20, 1999 A three dimensional (3D) object is composed of curves and surfaces. One must find a way to represent these to be able to model accurately an object.
More informationCOMP3421. 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
More informationAdvanced 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),
More information15.10 Curve Interpolation using Uniform Cubic BSpline Curves. CS Dept, UK
1 An analysis of the problem: To get the curve constructed, how many knots are needed? Consider the following case: So, to interpolate (n +1) data points, one needs (n +7) knots,, for a uniform cubic Bspline
More informationProperties of Blending Functions
Chapter 5 Properties of Blending Functions We have just studied how the Bernstein polynomials serve very nicely as blending functions. We have noted that a degree n Bézier curve always begins at P 0 and
More informationECE 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
More informationSubdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational BSpline Demo 2 Problems with NURBS A single
More informationSpline Methods Draft. Tom Lyche and Knut Mørken
Spline Methods Draft Tom Lyche and Knut Mørken January 5, 2005 2 Contents 1 Splines and Bsplines an Introduction 3 1.1 Convex combinations and convex hulls.................... 3 1.1.1 Stable computations...........................
More informationBezier Curves. An Introduction. Detlef Reimers
Bezier Curves An Introduction Detlef Reimers detlefreimers@gmx.de http://detlefreimers.de September 1, 2011 Chapter 1 Bezier Curve Basics 1.1 Linear Interpolation This section will give you a basic introduction
More informationKnot Insertion and Reparametrization of Interval Bspline Curves
International Journal of Video&Image Processing and Network Security IJVIPNSIJENS Vol:14 No:05 1 Knot Insertion and Reparametrization of Interval Bspline Curves O. Ismail, Senior Member, IEEE Abstract
More informationMathematical Tools in Computer Graphics with C# Implementations Table of Contents
Mathematical Tools in Computer Graphics with C# Implementations by Hardy Alexandre, WilliHans Steeb, World Scientific Publishing Company, Incorporated, 2008 Table of Contents List of Figures Notation
More informationCS452/552; EE465/505. Color Display Issues
CS452/552; EE465/505 Color Display Issues 416 15 2 Outline! Color Display Issues Color Systems Dithering and Halftoning! Splines Hermite Splines Bezier Splines CatmullRom Splines Read: Angel, Chapter
More informationFreeform Curves on Spheres of Arbitrary Dimension
Freeform Curves on Spheres of Arbitrary Dimension Scott Schaefer and Ron Goldman Rice University 6100 Main St. Houston, TX 77005 sschaefe@rice.edu and rng@rice.edu Abstract Recursive evaluation procedures
More informationIntroduction to Geometry. Computer Graphics CMU /15662
Introduction to Geometry Computer Graphics CMU 15462/15662 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 informationOutline. The de Casteljau Algorithm. Properties of Piecewise Linear Interpolations. 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
More informationDirect 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
More informationAdvanced TextureMapping Curves and Curved Surfaces. PreLecture Business. Texture Modes. Texture Modes. Review quiz
Advanced TextureMapping Curves and Curved Surfaces Preecture Business loadtexture example midterm handed bac, code posted (still) get going on pp3! more on texturing review quiz CS148: Intro to CG Instructor:
More informationCATIA V4/V5 Interoperability Project 2 : Migration of V4 surface : Influence of the transfer s settings Genta Yoshioka
CATIA V4/V5 Interoperability Project 2 : Migration of V4 surface : Influence of the transfer s settings Genta Yoshioka Version 1.0 03/08/2001 CATIA Interoperability Project Office CIPO IBM Frankfurt, Germany
More informationEfficient Ray Tracing of Trimmed NURBS Surfaces
MaxPlanckInstitut für Informatik Computer Graphics Group Saarbrücken, Germany Efficient Ray Tracing of Trimmed NURBS Surfaces Master Thesis in Computer Science Computer Science Department University
More informationObjectives. Continue discussion of shading Introduce modified Phong model Consider computation of required vectors
Objectives Continue discussion of shading Introduce modified Phong model Consider computation of required vectors 1 Lambertian Surface Perfectly diffuse reflector Light scattered equally in all directions
More informationFunctions bs3_curve Aa thru Lz
Chapter 18. Functions bs3_curve Aa thru Lz Topic: Ignore bs3_curve_accurate_derivs Action: Gets the number of derivatives that bs3_curve_evaluate can calculate. Prototype: int bs3_curve_accurate_derivs
More information3D Modeling techniques
3D Modeling techniques 0. Reconstruction From real data (not covered) 1. Procedural modeling Automatic modeling of a selfsimilar objects or scenes 2. Interactive modeling Provide tools to computer artists
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 informationMA 323 Geometric Modelling Course Notes: Day 21 Three Dimensional Bezier Curves, Projections and Rational Bezier Curves
MA 323 Geometric Modelling Course Notes: Day 21 Three Dimensional Bezier Curves, Projections and Rational Bezier Curves David L. Finn Over the next few days, we will be looking at extensions of Bezier
More informationCSCI 4620/8626. Coordinate Reference Frames
CSCI 4620/8626 Computer Graphics Graphics Output Primitives Last update: 20140203 Coordinate Reference Frames To describe a picture, the worldcoordinate reference frame (2D or 3D) must be selected.
More informationShading II. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico
Shading II Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico 1 Objectives Continue discussion of shading Introduce modified Phong model
More informationCurves 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
More informationMA 323 Geometric Modelling Course Notes: Day 14 Properties of Bezier Curves
MA 323 Geometric Modelling Course Notes: Day 14 Properties of Bezier Curves David L. Finn In this section, we discuss the geometric properties of Bezier curves. These properties are either implied directly
More informationSPIRAL TRANSITION CURVES AND THEIR APPLICATIONS. Zulfiqar Habib and Manabu Sakai. Received August 21, 2003
Scientiae Mathematicae Japonicae Online, e2004, 25 262 25 SPIRAL TRANSITION CURVES AND THEIR APPLICATIONS Zulfiqar Habib and Manabu Sakai Received August 2, 200 Abstract. A method for family of G 2 planar
More informationLinear Precision for Parametric Patches
Department of Mathematics Texas A&M University March 30, 2007 / Texas A&M University Algebraic Geometry and Geometric modeling Geometric modeling uses polynomials to build computer models for industrial
More informationFathi ElYafi Project and Software Development Manager Engineering Simulation
An Introduction to Geometry Design Algorithms Fathi ElYafi Project and Software Development Manager Engineering Simulation 1 Geometry: Overview Geometry Basics Definitions Data Semantic Topology Mathematics
More informationLagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers
In this section we present Lagrange s method for maximizing or minimizing a general function f(x, y, z) subject to a constraint (or side condition) of the form g(x, y, z) = k. Figure 1 shows this curve
More information3.2 THREE DIMENSIONAL OBJECT REPRESENTATIONS
3.1 THREE DIMENSIONAL CONCEPTS We can rotate an object about an axis with any spatial orientation in threedimensional space. Twodimensional rotations, on the other hand, are always around an axis that
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 informationPS Geometric Modeling Homework Assignment Sheet I (Due 20Oct2017)
Homework Assignment Sheet I (Due 20Oct2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist
More informationEquation of tangent plane: for implicitly defined surfaces section 12.9
Equation of tangent plane: for implicitly defined surfaces section 12.9 Some surfaces are defined implicitly, such as the sphere x 2 + y 2 + z 2 = 1. In general an implicitly defined surface has the equation
More informationFigure 5.1: Spline and ducks.
Chapter 5 BSPLINE CURVES Most shapes are simply too complicated to define using a single Bézier curve. A spline curve is a sequence of curve segments that are connected together to form a single continuous
More informationBSplines and NURBS Week 5, Lecture 9
CS 430/585 Computer Graphics I BSplines an NURBS Week 5, Lecture 9 Davi Breen, William Regli an Maxim Peysakhov Geometric an Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationGeometry. Chapter 5. Types of Curves and Surfaces
Chapter 5. Geometry Geometry refers to the physical items represented by the model (such as points, curves, and surfaces), independent of their spatial or topological relationships. The ACIS free form
More informationAmerican International Journal of Research in Science, Technology, Engineering & Mathematics
American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 38349, ISSN (Online): 383580, ISSN (CDROM): 38369
More informationAdaptive Tessellation for Trimmed NURBS Surface
Adaptive Tessellation for Trimmed NURBS Surface Ma YingLiang and Terry Hewitt 2 Manchester Visualization Centre, University of Manchester, Manchester, M3 9PL, U.K. may@cs.man.ac.uk 2 W.T.Hewitt@man.ac.uk
More informationChapter 43D Modeling
Chapter 43D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume and Pointbased Graphics 1 The 3D rendering
More informationA second order algorithm for orthogonal projection onto curves and surfaces
A second order algorithm for orthogonal projection onto curves and surfaces Shimin Hu and Johannes Wallner Dept. of Computer Science and Technology, Tsinghua University, Beijing, China shimin@tsinghua.edu.cn;
More informationPythagorean  Hodograph Curves: Algebra and Geometry Inseparable
Rida T. Farouki Pythagorean  Hodograph Curves: Algebra and Geometry Inseparable With 204 Figures and 15 Tables 4y Springer Contents 1 Introduction 1 1.1 The Lure of Analytic Geometry 1 1.2 Symbiosis of
More informationMA 323 Geometric Modelling Course Notes: Day 10 Higher Order Polynomial Curves
MA 323 Geometric Modelling Course Notes: Day 10 Higher Order Polynomial Curves David L. Finn December 14th, 2004 Yesterday, we introduced quintic Hermite curves as a higher order variant of cubic Hermite
More informationKeyword: Quadratic Bézier Curve, Bisection Algorithm, Biarc, Biarc Method, Hausdorff Distances, Tolerance Band.
Department of Computer Science Approximation Methods for Quadratic Bézier Curve, by Circular Arcs within a Tolerance Band Seminar aus Informatik Univ.Prof. Dr. Wolfgang Pree Seyed Amir Hossein Siahposhha
More informationThe Journal of MacroTrends in Technology and Innovation
MACROJOURNALS The Journal of MacroTrends in Technology and Innovation Automatic Knot Adjustment Using Dolphin Echolocation Algorithm for BSpline Curve Approximation Hasan Ali AKYÜREK*, Erkan ÜLKER**,
More informationComputer graphics (cs602) Final term mcqs fall 2013 Libriansmine
Computer graphics (cs602) Final term mcqs fall 2013 Libriansmine Question # 1 Total Marks: 1 Consider the following problem from lighting: A point (P1) is at (0, 0, 0) with normal equal to 1/(2*sqrt(2))*(sqrt(2),
More informationChapter 2: Rhino Objects
The fundamental geometric objects in Rhino are points, curves, surfaces, polysurfaces, extrusion objects, and polygon mesh objects. Why NURBS modeling NURBS (nonuniform rational Bsplines) are mathematical
More informationCity Research Online. Permanent City Research Online URL:
Slabaugh, G.G., Unal, G.B., Fang, T., Rossignac, J. & Whited, B. Variational Skinning of an Ordered Set of Discrete D Balls. Lecture Notes in Computer Science, 4975(008), pp. 450461. doi: 10.1007/978354079468_34
More informationAn Introduction to Bezier Curves, BSplines, and Tensor Product Surfaces with History and Applications
An Introduction to Bezier Curves, BSplines, and Tensor Product Surfaces with History and Applications Benjamin T. Bertka University of California Santa Cruz May 30 th, 2008 1 History Before computer graphics
More informationSpline Surfaces, Subdivision Surfaces
CSC3100 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
More informationSurfaces for CAGD. FSP Tutorial. FSPSeminar, Graz, November
Surfaces for CAGD FSP Tutorial FSPSeminar, 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
More informationKochanekBartels Cubic Splines (TCB Splines)
KochanekBartels Cubic Splines (TCB Splines) David Eberly, Geometric Tools, Redmond WA 9805 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More informationThe Essentials of CAGD
The Essentials of CAGD Chapter 6: Bézier Patches Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentialscagd c 2 Farin & Hansford The
More informationCS Object Representation. Aditi Majumder, CS 112 Slide 1
CS 112  Object Representation Aditi Majumder, CS 112 Slide 1 What is Graphics? Modeling Computer representation of the 3D world Analysis For efficient rendering For catering the model to different applications..
More informationNormals 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,
More informationArcLength Parameterized Spline Curves for RealTime Simulation
ArcLength Parameterized Spline Curves for RealTime Simulation Hongling Wang, Joseph Kearney, and Kendall Atkinson Abstract. Parametric curves are frequently used in computer animation and virtual environments
More informationBlending curves. Albert Wiltsche
Journal for Geometry and Graphics Volume 9 (2005), No. 1, 67 75. Blenng curves Albert Wiltsche Institute of Geometry, Graz University of Technology Kopernikusgasse 24, A8010 Graz, Austria email: wiltsche@tugraz.at
More informationLighting and Shading II. Angel and Shreiner: Interactive Computer Graphics 7E AddisonWesley 2015
Lighting and Shading II 1 Objectives Continue discussion of shading Introduce modified Phong model Consider computation of required vectors 2 Ambient Light Ambient light is the result of multiple interactions
More informationFilling Holes with Bspline Surfaces
Journal for Geometry and Graphics Volume 6 (22), No. 1, 83 98. Filling Holes with Bspline Surfaces Márta SzilvásiNagy Department of Geometry, Budapest University of Technology and Economics Egry József
More informationCompatibility of Data Transfer between CAD Applications
58 H. KUCHYŇKOVÁ, COMPATIBILITY OF DATA TRANSFER BETWEEN CAD APPLICATIONS Compatibility of Data Transfer between CAD Applications Hana KUCHYŇKOVÁ 1 1 Dept. of Power Electrical and Electronic Engineering,
More informationSpline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong
More informationNURBS: NonUniform Rational BSplines AUI Course Denbigh Starkey
NURBS: NonUniform Rational BSplines AUI Course Denbigh Starkey 1. Background 2 2. Definitions 3 3. Using NURBS to define a circle 4 4. Homogeneous coordinates & control points at infinity 9 5. Constructing
More informationFairing Wireframes in Industrial Surface Design
Fairing Wireframes in Industrial Surface Design YuKun Lai, YongJin Liu, Yu Zang, ShiMin Hu Tsinghua National Laboratory for Information Science and Technology, Department of Computer Science and Technology,
More informationBEZIER SURFACE GENERATION OF THE PATELLA
BEZIER SURFACE GENERATION OF THE PATELLA A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering By Dale A. Patrick BSEE University of Toledo, 7
More informationComputer Graphics 1. Chapter 2 (May 19th, 2011, 24pm): 3D Modeling. LMU München Medieninformatik Andreas Butz Computergraphik 1 SS2011
Computer Graphics 1 Chapter 2 (May 19th, 2011, 24pm): 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 informationToday. Bsplines. Bsplines. Bsplines. Computergrafik. Curves NURBS Surfaces. Bilinear patch Bicubic Bézier patch Advanced surface modeling
Comptergrafik Matthias Zwicker Uniersität Bern Herbst 29 Cres Srfaces Parametric srfaces Bicbic Bézier patch Adanced srface modeling Piecewise Bézier cres Each segment spans for control points Each segment
More informationApproximate computation of curves on Bspline surfaces
ComputerAided Design ( ) www.elsevier.com/locate/cad Approximate computation of curves on Bspline surfaces YiJun Yang a,b,c,, Song Cao a,c, JunHai Yong a,c, Hui Zhang a,c, JeanClaude Paul a,c,d, JiaGuang
More informationCurve fitting using linear models
Curve fitting using linear models Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark September 28, 2012 1 / 12 Outline for today linear models and basis functions polynomial regression
More informationApproximating CatmullClark Subdivision Surfaces with Bicubic Patches
Approximating CatmullClark Subdivision Surfaces with Bicubic Patches Charles Loop Microsoft Research Scott Schaefer Texas A&M University April 24, 2007 Technical Report MSRTR200744 Microsoft Research
More informationBlending Two Parametric Quadratic Bezier Curves
2015, TextRoad Publication ISSN 20904304 Journal of Basic and Applied Scientific Research www.textroad.com Blending Two Parametric Quadratic Bezier Curves Nurul Husna Hassan, Mazwin Tan, Norasrani Ramli,
More informationLeastSquares Fitting of Data with BSpline Curves
LeastSquares Fitting of Data with BSpline Curves David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More informationCURVILINEAR MESH GENERATION IN 3D
CURVILINEAR MESH GENERATION IN 3D Saikat Dey, Robert M. O'Bara 2 and Mark S. Shephard 2 SFA Inc. / Naval Research Laboratory, Largo, MD., U.S.A., dey@cosmic.nrl.navy.mil 2 Scientific Computation Research
More information