CGT 581 G Geometric Modeling Curves
|
|
- Stuart Barton
- 5 years ago
- Views:
Transcription
1 CGT 581 G Geometric Modeling Curves Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Curves What is a curve? Mathematical definition 1) The continuous image of an interval in an space 2) A continuous map of a 1 space to an space Curves What is a curve? Definition from physics: curve is a trajectory of a moving point This is usually used in CG Curves Any point has two neighbors (no branching) Endpoints have only one neighbor Some curves do not have endpoints Infinite curves Closed curves Periodic curves Space filling curves are 2D. 1
2 Curve representation Explicit: (!) cannot represent closed curves (and other issues) Implicit:, 0 many useful properties usually evaluated numerically, is a scalar function (one number) Curve representation Parametric:,, The curve is a function of one parameter It is a trajectory of a moving point The meaning of the parameter is the time It returns three (3D) or more (nd) values that identify the coordinates of the point in the n dimensional space Curve representation Trajectory (curve) is thus represented as a set of points in space: Parametric Representation Example Sinus curve, sin Circle sin, cos Their evaluation is straightforward How sparse are they? Parabola, 2 2
3 Parametric Representation Example Different density. Take two curves:, 0 and t,0, 0 1 they represent the same shape of the curve, but,0 1 They represent the same curve with different parameterization does not mean midpoint of a curve segment! Parametric Representation The parameter usually , 0, 0 is the starting point of the curve Also denoted by 1 1, 1, 1 the end point of the curve Also denoted by Parametric Equation of a Line Defined by two points and its equation,, ; there are two special cases: Parametric Equation of a Line Can be also thought of as a blending function 1 That is a linear interpolation between and with the parameter 3
4 Parametric Curve Derivative The derivative of a curve w.r.t the parameter,, Parametric Curve Derivative Example: derivatives of the functions from the previous examples: sinus curve, sin 1, cos circle sin, cos cos, sin parabola 2, 2, 1 Parametric Curve Derivative Note: Curve is a set of points Derivative of a curve is a vector. Why? Take the limit process Δ lim Δ Δ Δ Δ Δ Δ Δ Parametric Curve Derivative The derivative is the tangent vector to the function in the given point Tangent is a line passing through the given point in the direction of the tangent vector 4
5 Tangent & Tangent Vector Example Example: Having a curve , Evaluate the tangent in the point 0.5. Solution: 1. Get the point Evaluate derivative 3. Evaluate tangent vector Tangent is: Tangent & Tangent Vector Example ad , ,0 ad , ad , 3/2 ad 4. 1,0 3, 3/2 by the way: 1, P(s) Q(t) Curvature Measures how a curve deviates from being flat. The tangent vector and the second derivative in a point define osculating plane Curvature The osculating circle lies in and touches at The radius of the osculating curve is The curvature of at is Helmut Pottman, Architectural geometry Three consecutive vertices possess circumscribed circle which lies in the osculating plane the osculating circle also 1 5
6 Curvature Note Curvature of an inflexion point is zero ( Parametric curve continuity Motivation: Expression of the entire shape from only one curve is usually impossible Curves are connected at their ends Every part of the curve is called a segment The point that two segments share is a knot (joint) The parametric continuity is denoted by C Parametric curves continuity knot segment Free form curves continuity 0 If two segments meet in a point, they are continuously connected or simply connected Denoted by 0 Two segments 1 and 2 are continuously connected iff (One segment starts where the other ends) 6
7 Free form curves continuity 1 Two segments 1 and 2 are parametrically continuously connected iff 1) Shared knot 2) 1 0 Derivatives in the knot are equal Does 1 imply 0? Free form curves continuity 1 So called 1 parametric continuity requires two segments to be 0 and their derivatives to be equal in the knot Denoted by 1 Two segments have identical tangent in the knot (not only the tangent vectors!) Magnitude and the direction are identical (If we move from the one curve to the other, speed and the direction is unchanged (continuous)) Continuity 0 vs. 1 Continuity 0 vs. Continuity 0 : Shared knot Q 1 (1)=Q 2 (0) Q 1 (1) Continuity 1 : shared knot, shared tangent, and the tangent vectors are nonzero Q 1 (t) Q 1 (t) Q 2 (0) Q 2 (t) Q 1 (1)=Q 2 (0) Q 1 (1)=Q 2 (0) Q 2 (0) 7
8 Continuity 2, 3, Continuity 2, 3, A curve is said to have continuity iff it has continuous derivatives of order 0, 1,, in every point Two curves meet continuously iff they have continuous derivatives of order 0, 1,, n in every point Note: polynomial functions are Continuity Continuity 2, 3,, Sometimes denoted refers to curves that include discontinuities Example: 8
9 Curves continuity (contd.) Example: 1 sin /2, cos / , /2 0 1 Are they parametrically continuous? Curves continuity (contd.) Solution: ad 1) 1 1 sin /2, cos /2 1, , 0 this is OK 2) 1 /2 cos /2, /2sin /2 2 0, /2 1 1 /2 cos /2, /2sin /2 0, /2 that is OK as well Curves continuity (contd.) These two segments form a curve that is parametrically continuous in the knot. Curves continuity (contd.) Example: Let s have curves: 2, 2, 0 1 2, 2, 0 1 Is the curve formed by them (if any) 1? 9
10 Curves continuity (contd.) Solution: 1) 1 2,2 0 2,2 2) 2,2 1,1 The curves definitively meet smoothly, but they are NOT parametrically continuous. Tangent vectors have different magnitudes! 2,2 1, Geometric Continuity 1 Geometric continuity means visual smoothness. The moving point changes the velocity but not the direction. Two curves are geometrically continuous if their derivatives in knot are positively linearly dependent. Geometric Continuity 1 Two segments 1 and 2 are geometrically continuously connected iff Geometric Continuity 1 1) they share a knot 2) , 0 their derivatives in the knot are linearly dependent Does 1 imply 1? 10
11 Geometric Continuity The direction of the second derivative are identical. Geometric Continuity 1) 1 0 they share a knot 2) 1 0, 0 derivatives in the knot are linearly dependent 3) 1 0, 0 accelerations in the knot are linearly dependent exist if the two connecting curves also share center of curvature at the joint.,, summary curves share knot curves also share tangent curves also share a common center of curvature at the joint. Note: geometric continuity exist if the curves can be re parameterized to have. Continuity summary Continuity (zero order continuity): 0 two curves meet in the knot Parametric continuity (first order continuity): 1 the curves meet in knot and their derivatives are equal (the joint is smooth) Geometric continuity: 1 the curves meet in knot and their derivatives are linearly positively dependent (the joint point is visually smooth) 11
12 Readings Architectural Geometry, Pottman et al Interactive Computer Graphics 5th edition, Ed. Angel pp Real Time Rendering 2nd edition, Moller, T.A., Haines, E., Geometric Modeling, 2nd edition, M.E.Mortenson 12
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
More informationParametric 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
More informationDesign 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
More informationCGT 581 G Geometric Modeling Surfaces (part I)
CGT 581 G Geometric Modeling Surfaces (part I) Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Polygonal Representation The common representation is a mesh of triangles
More informationParametric 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
More informationCurve and Surface Basics
Curve and Surface Basics Implicit and parametric forms Power basis form Bezier curves Rational Bezier Curves Tensor Product Surfaces ME525x NURBS Curve and Surface Modeling Page 1 Implicit and Parametric
More informationSplines. 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
More informationLecture 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:
More informationFall 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
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 informationCurves 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
More informationCurves 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
More informationCurve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur
Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur Email: jrkumar@iitk.ac.in Curve representation 1. Wireframe models There are three types
More informationIntroduction 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
More informationCurves 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
More informationMathematically, the path or the trajectory of a particle moving in space in described by a function of time.
Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 45 : Curves in space [Section 45.1] Objectives In this section you will learn the following : Concept of curve in space. Parametrization
More informationBezier 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
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More information3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More 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 informationPrecalculus 2 Section 10.6 Parametric Equations
Precalculus 2 Section 10.6 Parametric Equations Parametric Equations Write parametric equations. Graph parametric equations. Determine an equivalent rectangular equation for parametric equations. Determine
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 informationTopic 5.1: Line Elements and Scalar Line Integrals. Textbook: Section 16.2
Topic 5.1: Line Elements and Scalar Line Integrals Textbook: Section 16.2 Warm-Up: Derivatives of Vector Functions Suppose r(t) = x(t) î + y(t) ĵ + z(t) ˆk parameterizes a curve C. The vector: is: r (t)
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More informationObjects 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
More informationPARAMETERIZATIONS OF PLANE CURVES
PARAMETERIZATIONS OF PLANE CURVES Suppose we want to plot the path of a particle moving in a plane. This path looks like a curve, but we cannot plot it like we would plot any other type of curve in the
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More 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 informationCurves and Surfaces Computer Graphics I Lecture 10
15-462 Computer Graphics I Lecture 10 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] September 30, 2003 Doug James Carnegie
More informationA Curve Tutorial for Introductory Computer Graphics
A Curve Tutorial for Introductory Computer Graphics Michael Gleicher Department of Computer Sciences University of Wisconsin, Madison October 7, 2003 Note to 559 Students: These notes were put together
More informationGeometric Modeling of Curves
Curves Locus of a point moving with one degree of freedom Locus of a one-dimensional parameter family of point Mathematically defined using: Explicit equations Implicit equations Parametric equations (Hermite,
More informationReading. Parametric surfaces. Surfaces of revolution. Mathematical surface representations. Required:
Reading Required: Angel readings for Parametric Curves lecture, with emphasis on 11.1.2, 11.1.3, 11.1.5, 11.6.2, 11.7.3, 11.9.4. Parametric surfaces Optional Bartels, Beatty, and Barsky. An Introduction
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationIntro 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
More informationAppendix E Calculating Normal Vectors
OpenGL Programming Guide (Addison-Wesley Publishing Company) Appendix E Calculating Normal Vectors This appendix describes how to calculate normal vectors for surfaces. You need to define normals to use
More informationt dt ds Then, in the last class, we showed that F(s) = <2s/3, 1 2s/3, s/3> is arclength parametrization. Therefore,
13.4. Curvature Curvature Let F(t) be a vector values function. We say it is regular if F (t)=0 Let F(t) be a vector valued function which is arclength parametrized, which means F t 1 for all t. Then,
More informationTrajectory planning in Cartesian space
Robotics 1 Trajectory planning in Cartesian space Prof. Alessandro De Luca Robotics 1 1 Trajectories in Cartesian space! in general, the trajectory planning methods proposed in the joint space can be applied
More informationMotivation. Parametric Curves (later Surfaces) Outline. Tangents, Normals, Binormals. Arclength. Advanced Computer Graphics (Fall 2010)
Advanced Computer Graphics (Fall 2010) CS 283, Lecture 19: Basic Geometric Concepts and Rotations Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/fa10 Motivation Moving from rendering to simulation,
More informationTutorial 4. Differential Geometry I - Curves
23686 Numerical Geometry of Images Tutorial 4 Differential Geometry I - Curves Anastasia Dubrovina c 22 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves Differential Geometry
More informationFlank Millable Surface Design with Conical and Barrel Tools
461 Computer-Aided Design and Applications 2008 CAD Solutions, LLC http://www.cadanda.com Flank Millable Surface Design with Conical and Barrel Tools Chenggang Li 1, Sanjeev Bedi 2 and Stephen Mann 3 1
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 informationInformation 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,
More informationFor 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)
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 informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More informationCOMPUTER 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
More informationMath 32, August 20: Review & Parametric Equations
Math 3, August 0: Review & Parametric Equations Section 1: Review This course will continue the development of the Calculus tools started in Math 30 and Math 31. The primary difference between this course
More informationLecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10
Lecture 25: Bezier Subdivision And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 1. Divide and Conquer If we are going to build useful
More informationInterpolating/approximating pp gcurves
Chap 3 Interpolating Values 1 Outline Interpolating/approximating pp gcurves Controlling the motion of a point along a curve Interpolation of orientation Working with paths Interpolation between key frames
More informationMath 348 Differential Geometry of Curves and Surfaces
Math 348 Differential Geometry of Curves and Surfaces Lecture 3 Curves in Calculus Xinwei Yu Sept. 12, 2017 CAB 527, xinwei2@ualberta.ca Department of Mathematical & Statistical Sciences University of
More informationLet be a function. We say, is a plane curve given by the. Let a curve be given by function where is differentiable with continuous.
Module 8 : Applications of Integration - II Lecture 22 : Arc Length of a Plane Curve [Section 221] Objectives In this section you will learn the following : How to find the length of a plane curve 221
More informationComputer Animation. Rick Parent
Algorithms and Techniques Interpolating Values Animation Animator specified interpolation key frame Algorithmically controlled Physics-based Behavioral Data-driven motion capture Motivation Common problem:
More informationA second order algorithm for orthogonal projection onto curves and surfaces
A second order algorithm for orthogonal projection onto curves and surfaces Shi-min Hu and Johannes Wallner Dept. of Computer Science and Technology, Tsinghua University, Beijing, China shimin@tsinghua.edu.cn;
More informationCurves 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
More informationGeometric Primitives. Chapter 5
Chapter 5 Geometric Primitives In this chapter, we discuss the basic geometric primitives we will use to represent the world in which our graphic objects live. As discussed at the beginning of this class,
More informationDgp _ lecture 2. Curves
Dgp _ lecture 2 Curves Questions? This lecture will be asking questions about curves, their Relationship to surfaces, and how they are used and controlled. Topics of discussion will be: Free form Curves
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #12: Curves and Surfaces Prof. James O Brien University of California, Berkeley V2007-F-12-1.0 Today General curve and surface representations Splines and other polynomial
More informationarxiv:cs/ v1 [cs.gr] 22 Mar 2005
arxiv:cs/0503054v1 [cs.gr] 22 Mar 2005 ANALYTIC DEFINITION OF CURVES AND SURFACES BY PARABOLIC BLENDING by A.W. Overhauser Mathematical and Theoretical Sciences Department Scientific Laboratory, Ford Motor
More informationTO DUY ANH SHIP CALCULATION
TO DUY ANH SHIP CALCULATION Ship Calculattion (1)-Space Cuvers 3D-curves play an important role in the engineering, design and manufature in Shipbuilding. Prior of the development of mathematical and computer
More informationFundamentals of Computer Graphics. Lecture 3 Parametric curve and surface. Yong-Jin Liu.
Fundamentals of Computer Graphics Lecture 3 Parametric curve and surface Yong-Jin Liu liuyongjin@tsinghua.edu.cn Smooth curve and surface Design criteria of smooth curve and surface Smooth and continuity
More informationComputer Graphics CS 543 Lecture 13a Curves, Tesselation/Geometry Shaders & Level of Detail
Computer Graphics CS 54 Lecture 1a Curves, Tesselation/Geometry Shaders & Level of Detail Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines
More informationCS130 : 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
More information(Discrete) Differential Geometry
(Discrete) Differential Geometry Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties
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 informationCURVATURE ANALYSIS TUTORIAL
CURVATURE ANALYSIS TUTORIAL NICHOLAS HEATHCOTT JOHN FERGUSON This tutorial will provide tools for analyzing the curvature of your geometry, be it linear, planar, or a solid form. This means both the degree
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 informationENGI Parametric & Polar Curves Page 2-01
ENGI 3425 2. Parametric & Polar Curves Page 2-01 2. Parametric and Polar Curves Contents: 2.1 Parametric Vector Functions 2.2 Parametric Curve Sketching 2.3 Polar Coordinates r f 2.4 Polar Curve Sketching
More informationCS-184: Computer Graphics. Today
CS-84: Computer Graphics Lecture #5: Curves and Surfaces Prof. James O Brien University of California, Berkeley V25F-5-. Today General curve and surface representations Splines and other polynomial bases
More informationParametric 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
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 informationInteractive 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
More informationAssignment 4: Mesh Parametrization
CSCI-GA.3033-018 - Geometric Modeling Assignment 4: Mesh Parametrization In this exercise you will Familiarize yourself with vector field design on surfaces. Create scalar fields whose gradients align
More informationParametric Representation of Scroll Geometry with Variable Wall Thickness. * Corresponding Author: ABSTRACT 1.
1268, Page 1 Parametric Representation of Scroll Geometry with Variable Wall Thickness Bryce R. Shaffer 1 * and Eckhard A. Groll 2 1 Air Squared Inc. Broomfield, CO, USA 2 Purdue University, Mechanical
More informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. PARAMETRIC EQUATIONS
More informationGlossary of dictionary terms in the AP geometry units
Glossary of dictionary terms in the AP geometry units affine linear equation: an equation in which both sides are sums of terms that are either a number times y or a number times x or just a number [SlL2-D5]
More informationProf. Fanny Ficuciello Robotics for Bioengineering Trajectory planning
Trajectory planning to generate the reference inputs to the motion control system which ensures that the manipulator executes the planned trajectories path and trajectory joint space trajectories operational
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 informationIntro 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
More informationParametric 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
More informationResearch Article A Family of Even-Point Ternary Approximating Schemes
International Scholarly Research Network ISRN Applied Mathematics Volume, Article ID 97, pages doi:.5//97 Research Article A Family of Even-Point Ternary Approximating Schemes Abdul Ghaffar and Ghulam
More informationShape Control of Cubic H-Bézier Curve by Moving Control Point
Journal of Information & Computational Science 4: 2 (2007) 871 878 Available at http://www.joics.com Shape Control of Cubic H-Bézier Curve by Moving Control Point Hongyan Zhao a,b, Guojin Wang a,b, a Department
More information3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane?
Math 4 Practice Problems for Midterm. A unit vector that is perpendicular to both V =, 3, and W = 4,, is (a) V W (b) V W (c) 5 6 V W (d) 3 6 V W (e) 7 6 V W. In three dimensions, the graph of the equation
More informationCS 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
More informationIn 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
More informationKnot Insertion and Reparametrization of Interval B-spline Curves
International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:05 1 Knot Insertion and Reparametrization of Interval B-spline Curves O. Ismail, Senior Member, IEEE Abstract
More informationMATH 200 (Fall 2016) Exam 1 Solutions (a) (10 points) Find an equation of the sphere with center ( 2, 1, 4).
MATH 00 (Fall 016) Exam 1 Solutions 1 1. (a) (10 points) Find an equation of the sphere with center (, 1, 4). (x ( )) + (y 1) + (z ( 4)) 3 (x + ) + (y 1) + (z + 4) 9 (b) (10 points) Find an equation of
More informationFirst of all, we need to know what it means for a parameterize curve to be differentiable. FACT:
CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of
More informationBackground for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
More informationCircle inversion fractals are based on the geometric operation of inversion of a point with respect to a circle, shown schematically in Fig. 1.
MSE 350 Creating a Circle Inversion Fractal Instructor: R.G. Erdmann In this project, you will create a self-inverse fractal using an iterated function system (IFS). 1 Background: Circle Inversion Circle
More informationSection Parametrized Surfaces and Surface Integrals. (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals
Section 16.4 Parametrized Surfaces and Surface Integrals (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals MATH 127 (Section 16.4) Parametrized Surfaces and Surface Integrals
More informationName: Date: 1. Match the equation with its graph. Page 1
Name: Date: 1. Match the equation with its graph. y 6x A) C) Page 1 D) E) Page . Match the equation with its graph. ( x3) ( y3) A) C) Page 3 D) E) Page 4 3. Match the equation with its graph. ( x ) y 1
More informationComputer Graphics / Animation
Computer Graphics / Animation Artificial object represented by the number of points in space and time (for moving, animated objects). Essential point: How do you interpolate these points in space and time?
More informationColumbus State Community College Mathematics Department Public Syllabus. Course and Number: MATH 1172 Engineering Mathematics A
Columbus State Community College Mathematics Department Public Syllabus Course and Number: MATH 1172 Engineering Mathematics A CREDITS: 5 CLASS HOURS PER WEEK: 5 PREREQUISITES: MATH 1151 with a C or higher
More informationGeometry Processing & Geometric Queries. Computer Graphics CMU /15-662
Geometry Processing & Geometric Queries Computer Graphics CMU 15-462/15-662 Last time: Meshes & Manifolds Mathematical description of geometry - simplifying assumption: manifold - for polygon meshes: fans,
More informationVisualization Computer Graphics I Lecture 20
15-462 Computer Graphics I Lecture 20 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 12] April 15, 2003 Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More 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 B-spline
More informationA New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces
A New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces Mridula Dube 1, Urvashi Mishra 2 1 Department of Mathematics and Computer Science, R.D. University, Jabalpur, Madhya Pradesh, India 2
More informationEECS 487: Interactive Computer Graphics f
Interpolating Key Vales EECS 487: Interactive Compter Graphics f Keys Lectre 33: Keyframe interpolation and splines Cbic splines The key vales of each variable may occr at different frames The interpolation
More informationShape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011
CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell 2/15/2011 1 Motivation Geometry processing: understand geometric characteristics,
More information