CSE 167: Lecture 13: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011
|
|
- Colleen Gilbert
- 8 years ago
- Views:
Transcription
1 CSE 167: Introduction to Computer Graphics Lecture 13: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011
2 Announcements Homework project #6 due Friday, Nov 18 at 1:30pm To be presented in lab 260 No grading this Friday, Nov 11 (Veterans Day) Last day for late grading project 5: Thursday, Nov 10 Final project presentations date+time confirmed: December 2 nd, 1-3pm, CSE
3 Lecture Overview Polynomial Curves cont d Bézier Curves Introduction Drawing Bézier curves Piecewise Bézier curves 3
4 Polynomial Curves Quadratic: (2 nd order) Cubic: (3 rd order) We usually define the curve for 0 t 1 4
5 Control Points Polynomial coefficients a, b, c, d can be interpreted as control points Remember: a, b, c, d have x,y,z components each Unfortunately, they do not intuitively describe the shape of the curve Goal: intuitive control points 5
6 Control Points How many control points? Two points define a line (1 st order) Three points define a quadratic curve (2 nd order) Four points define a cubic curve (3 rd order) k+1 points define a k-order curve Let s start with a line 6
7 First Order Curve Based on linear interpolation (LERP) Weighted average between two values Value could be a number, vector, color, Interpolate between points p 0 and p 1 with parameter t Defines a curve that is straight (first-order spline) t=0 corresponds to p 0 t=1 corresponds to p 1 t=0.5 corresponds to midpoint 7 p 0. 0<t<1 t=0. p 1 t=1 x(t) = Lerp( t, p 0, p 1 )= ( 1 t)p 0 + t p 1
8 Linear Interpolation Three equivalent ways to write it Expose different properties 1. Regroup for points p weighted sum of control points 2. Regroup for t polynomial in t 3. Matrix form 8
9 Weighted Average x(t) = (1 t)p 0 + (t)p 1 = B 0 (t) p 0 + B 1 (t)p 1, where B 0 (t) = 1 t and B 1 (t) = t Weights are a function of t Sum is always 1, for any value of t Also known as blending functions 9
10 Linear Polynomial x(t) = (p p 0 ) t + p vector point Curve is based at point p 0 Add the vector, scaled by t p 0..5(p 1 -p 0 ). p 1 -p 0. 10
11 Matrix Form Geometry matrix Geometric basis Polynomial basis In components 11
12 Tangent For a straight line, the tangent is constant Weighted average Polynomial Matrix form 12
13 Lecture Overview Polynomial Curves cont d Bézier Curves Introduction Drawing Bézier curves Piecewise Bézier curves 13
14 Bézier Curves Are a higher order extension of linear interpolation p 1 p 1 p 1 p 2 p 1 p 3 p 0 p 0 p 0 p 2 Linear Quadratic Cubic 14
15 Bézier Curves Give intuitive control over curve with control points Endpoints are interpolated, intermediate points are approximated Convex Hull property Variation-Diminishing property Many demo applets online, for example: Demo: ezier/bezier.html 15
16 Cubic Bézier Curve Most common case Defined by four control points: Two interpolated endpoints (points are on the curve) Two points control the tangents at the endpoints Points x on curve defined as function of parameter t p 0 x(t) p 1 p 2 16 p 3
17 Algorithmic Construction Algorithmic construction De Casteljau algorithm, developed at Citroen in 1959, named after its inventor Paul de Casteljau (pronounced Cast-all- Joe ) Developed independently from Bézier s work: Bézier created the formulation using blending functions, Casteljau devised the recursive interpolation algorithm 17
18 De Casteljau Algorithm A recursive series of linear interpolations Works for any order Bezier function, not only cubic Not very efficient to evaluate Other forms more commonly used But: Gives intuition about the geometry Useful for subdivision 18
19 De Casteljau Algorithm Given: Four control points A value of t (here t 0.25) p 1 p 0 p 2 p 3 19
20 De Casteljau Algorithm p 1 q 1 q 0 (t) = Lerp( t,p 0,p ) 1 p 0 q 0 ( ) q 1 (t) = Lerp t,p 1,p 2 p 2 q 2 (t) = Lerp( t,p 2,p ) 3 q 2 p 3 20
21 De Casteljau Algorithm q 1 q r 0 0 r 1 ( ) r 0 (t) = Lerp t,q 0 (t),q 1 (t) r 1 (t) = Lerp( t,q 1 (t),q 2 (t)) q 2 21
22 De Casteljau Algorithm r 0 x r 1 x(t) = Lerp ( t,r 0 (t),r 1 (t)) 22
23 De Casteljau Algorithm p 1 p 0 x p 2 Applets Demo: p 3
24 Recursive Linear Interpolation x = Lerp t,r 0,r 1 ( ) r = Lerp ( t,q,q ) ( ) r 1 = Lerp t,q 1,q 2 q 0 = Lerp( t,p 0,p ) 1 q 1 = Lerp( t,p 1,p ) 2 q 2 = Lerp( t,p 2,p ) 3 p 0 p 1 p 2 p 3 p 1 q 0 r 0 p 2 x q 1 r 1 p 3 q 2 p 4 24
25 Expand the LERPs q 0 (t) = Lerp( t,p 0,p 1 )= ( 1 t)p 0 + tp 1 q 1 (t) = Lerp( t,p 1,p 2 )= ( 1 t)p 1 + tp 2 q 2 (t) = Lerp( t,p 2,p 3 )= ( 1 t)p 2 + tp 3 r 0 (t) = Lerp ( t,q 0 (t),q 1 (t) )= 1 t r 1 (t) = Lerp( t,q 1 (t),q 2 (t))= 1 t ( ) ( )p 0 + tp 1 )+ t ( 1 t )p 1 + tp 2 ( )p 1 + tp 2 )+ t ( 1 t)p 2 + tp 3 ( ) 1 t ( ) 1 t x(t) = Lerp t,r 0 (t),r 1 (t) = ( 1 t) ( 1 t) ( 1 t)p 0 + tp 1 )+ t ( 1 t)p 1 + tp 2 +t ( 1 t) ( 1 t)p 1 + tp 2 )+ t ( 1 t)p 2 + tp 3 25 ( ) ( ) ( ) ( ) ( ) ( )
26 Weighted Average of Control Points Regroup for p: x(t) = 1 t ( )p 0 + tp 1 )+ t ( 1 t)p 1 + tp 2 ( )p 1 + tp 2 )+ t( 1 t)p 2 + tp 3 ) ( ) ( 1 t) 1 t +t 1 t ( ) ( ) ( ) 1 t ( ) x(t) = 1 t ( ) 3 p 0 + 3( 1 t) 2 tp t ( )t 2 p 2 + t 3 p 3 B 0 (t ) B (t ) x(t) = t 3 + 3t 2 3t + 1 p 0 + 3t 3 6t 2 + 3t ( ) ( ) + 3t 3 + 3t p + t 3 2 { p 3 B 2 (t ) ( ) B 3 (t ) ( ) p 1 26
27 Cubic Bernstein Polynomials x(t) = B 0 ( t)p 0 + B 1 ( t)p 1 + B 2 ( t)p 2 + B 3 ( t)p 3 The cubic Bernstein polynomials : B 0 ( t)= t 3 + 3t 2 3t + 1 B 1 ( t)= 3t 3 6t 2 + 3t B 2 ( t)= 3t 3 + 3t 2 ( t)= t 3 B 3 B i (t) = 1 Weights B i (t) add up to 1 for any t 27
28 General Bernstein Polynomials B 1 0 B 1 1 ( t)= t + 1 B 2 0 ( t)= t 2 2t + 1 B 3 0 ( t)= t 3 + 3t 2 3t + 1 ( t)= t B 2 1 ( t)= 2t 2 + 2t B 3 1 ( t)= 3t 3 6t 2 + 3t B 2 2 ( t)= t 2 B 3 2 ( t)= 3t 3 + 3t 2 B 3 3 ( t)= t 3 28 B i n ( t)= n i ( 1 t)n i t B n i ( t) = 1 n ( ) i i = n! i! ( n i)! n! = factorial of n (n+1)! = n! x (n+1)
29 General Bézier Curves nth-order Bernstein polynomials form nth-order Bézier curves B i n ( t)= n ( 1 t)n i t i n i=0 x( t)= B n i ( t)p i ( ) i 29
30 Bézier Curve Properties Overview: Convex Hull property Variation Diminishing property Affine Invariance 30
31 Definitions Convex hull of a set of points: Polyhedral volume created such that all lines connecting any two points lie completely inside it (or on its boundary) Convex combination of a set of points: Weighted average of the points, where all weights between 0 and 1, sum up to 1 Any convex combination always of a set of points lies within the convex hull 31
32 Convex Hull Property A Bézier curve is a convex combination of the control points (by definition, see Bernstein polynomials) Bézier curve is always inside the convex hull Makes curve predictable Allows culling, intersection testing, adaptive tessellation Demo: p 1 p 3 p 0 p 2 32
33 Variation Diminishing Property If the curve is in a plane, this means no straight line intersects a Bézier curve more times than it intersects the curve's control polyline Curve is not more wiggly than control polyline 33
34 Affine Invariance Transforming Bézier curves Two ways to transform: Transform the control points, then compute resulting spline points Compute spline points then transform them Either way, we get the same points Curve is defined via affine combination of points Invariant under affine transformations Convex hull property remains true 34
35 Cubic Polynomial Form Start with Bernstein form: x(t) = ( t 3 + 3t 2 3t + 1)p 0 + ( 3t 3 6t 2 + 3t)p 1 + ( 3t 3 + 3t 2 )p 2 + ( t 3 )p 3 Regroup into coefficients of t : x(t) = ( p 0 + 3p 1 3p 2 + p 3 )t 3 + ( 3p 0 6p 1 + 3p 2 )t 2 + ( 3p 0 + 3p 1 )t + ( p 0 )1 x(t) = at 3 + bt 2 + ct + d ( ) ( ) ( ) ( ) a = p 0 + 3p 1 3p 2 + p 3 b = 3p 0 6p 1 + 3p 2 c = 3p 0 + 3p 1 d = p 0 Good for fast evaluation Precompute constant coefficients (a,b,c,d) Not much geometric intuition 35
36 Cubic Matrix Form x(t) = a r t 3 r b c r t 2 d t 1 r a = p 0 + 3p 1 3p 2 + p 3 r b = ( 3p 0 6p 1 + 3p 2 ) r c = ( 3p 0 + 3p 1 ) d = p 0 ( ) ( ) t x(t) = [ p 0 p 1 p 2 p 3 ] t t { T G Bez B Bez Other types of cubic splines use different basis matrices B Bez 36
37 Cubic Matrix Form In 3D: 3 parallel equations for x, y and z: 37 [ ] x x (t) = p 0 x p 1x p 2 x p 3x x y (t) = p 0 y p 1y p 2 y p 3y x z (t) = p 0z p 1z p 2z p 3z t t t t t t t t t
38 Matrix Form Bundle into a single matrix x(t) = t 3 p 0 x p 1x p 2 x p 3x p 0 y p 1y p 2 y p 3y t t p 0z p 1z p 2z p 3z x(t) = G Bez B Bez T x(t) = C T Efficient evaluation Pre-compute C Take advantage of existing 4x4 matrix hardware support 38
39 Lecture Overview Polynomial Curves cont d Bézier Curves Introduction Drawing Bézier curves Piecewise Bézier curves 39
40 Drawing Bézier Curves Draw line segments or individual pixels Approximate the curve as a series of line segments (tessellation) Uniform sampling Adaptive sampling Recursive subdivision 40
41 Uniform Sampling Approximate curve with N straight segments N chosen in advance Evaluate x i = x( t i ) where t i = i N for i = 0, 1,K, N Connect the points with lines Too few points? Poor approximation Curve is faceted Too many points? x i = a r i3 N + b r i2 3 N + c r i 2 N + d Slow to draw too many line segments Segments may draw on top of each other x(t) x 0 x 1 x 2 x 3 x 4 41
42 Adaptive Sampling Use only as many line segments as you need Fewer segments where curve is mostly flat More segments where curve bends Segments never smaller than a pixel x(t) 42
43 Recursive Subdivision Any cubic curve segment can be expressed as a Bézier curve Any piece of a cubic curve is itself a cubic curve Therefore: Any Bézier curve can be broken down into smaller Bézier curves 43
44 De Casteljau Subdivision p 1 p 0 q r 0 0 r x 1 p 2 De Casteljau construction points are the control points of two Bézier sub-segments q 2 p 3 44
45 Adaptive Subdivision Algorithm Use De Casteljau construction to split Bézier segment For each half If flat enough : draw line segment Else: recurse Curve is flat enough if hull is flat enough Test how far the approximating control points are from a straight segment If less than one pixel, the hull is flat enough 45
46 Drawing Bézier Curves With OpenGL Indirect OpenGL support for drawing curves: Define evaluator map (glmap) Draw line strip by evaluating map (glevalcoord) Optimize by pre-computing coordinate grid (glmapgrid and glevalmesh) More details about OpenGL implementation: /opengl_nurbs.pdf 46
47 Lecture Overview Polynomial Curves cont d Bézier Curves Introduction Drawing Bézier curves Piecewise Bézier curves 47
48 More Control Points Cubic Bézier curve limited to 4 control points Cubic curve can only have one inflection (point where curve changes direction of bending) Need more control points for more complex curves k-1 order Bézier curve with k control points Hard to control and hard to work with Intermediate points don t have obvious effect on shape Changing any control point changes the whole curve Want local support: each control point only influences nearby portion of curve 48
49 Piecewise Curves Sequence of simple (low-order) curves, end-to-end Known as a piecewise polynomial curve Sequence of line segments Piecewise linear curve Sequence of cubic curve segments Piecewise cubic curve (here piecewise Bézier) 49
50 Mathematical Continuity C 0 continuity: Curve segments are connected C 1 continuity: C 0 & 1st-order derivatives agree at joints Tangents/normals are C 0 continuous Important for smooth shading C 2 continuity: C 1 & 2nd-order derivatives agree at joints Tangents/normals are C 1 continuous Important for high quality reflections
51 Global Parameterization Given N curve segments x 0 (t), x 1 (t),, x N-1 (t) Each is parameterized for t from 0 to 1 Define a piecewise curve Global parameter u from 0 to N x 0 (u), 0 u 1 x(u) = x 1 (u 1), 1 u 2 M M x N 1 (u ( N 1)), N 1 u N x(u) = x i (u i), where i = u (and x(n) = x N 1 (1)) Alternate: solution u also goes from 0 to 1 x(u) = x i (Nu i), where i = Nu 51
52 Piecewise-Linear Curve Given N+1 points p 0, p 1,, p N Define curve x(u) = Lerp(u i,p i,p i+1 ), i u i + 1 = (1 u + i)p i + (u i)p i+1, i = u p 1 x(1.5) x(2.9) p 3 p p 4 p 6 2 x(5.25) p p 5 0 N+1 points define N linear segments x(i)=p i C 0 continuous by construction C 1 at p i when p i -p i-1 = p i+1 -p i 52
53 Piecewise Bézier curve Given 3N + 1 points p 0,p 1,K,p 3N Define N Bézier segments: x 0 (t) = B 0 (t)p 0 + B 1 (t)p 1 + B 2 (t)p 2 + B 3 (t)p 3 x 1 (t) = B 0 (t)p 3 + B 1 (t)p 4 + B 2 (t)p 5 + B 3 (t)p 6 M x N 1 (t) = B 0 (t)p 3N 3 + B 1 (t)p 3N 2 + B 2 (t)p 3N 1 + B 3 (t)p 3N p 7 p 8 p 6 x 2 (t) p 9 p 0 p p 2 1 x 0 (t) p 3 x 1 (t) x 3 (t) p 10 p 11 p 12 p 4 p 5 53
54 Piecewise Bézier Curve Parameter in 0<=u<=3N x(u) = x 0 ( 1 u), 0 u 3 3 x 1 ( 1 u 1), 3 u 6 3 M M x N 1 ( 1 3 u (N 1)), 3N 3 u 3N 1 x(u) = x i ( u i 3 ), where i = 1 u 3 x(8.75) x 0 (t) x 1 (t) x 2 (t) x 3 (t) u=0 x(3.5) u=12 54
55 Piecewise Bézier Curve 3N+1 points define N Bézier segments x(3i)=p 3i C 0 continuous by construction C 1 continuous at p 3i when p 3i - p 3i-1 = p 3i+1 - p 3i C 2 is harder to achieve p 4 p 1 p 2 P 3 p 6 p 1 p 2 P 3 p 6 p 4 p 5 p 0 p 5 p 0 C 1 discontinuous C 1 continuous 55
56 Piecewise Bézier Curves Used often in 2D drawing programs Inconveniences Must have 4 or 7 or 10 or 13 or (1 plus a multiple of 3) control points Some points interpolate, others approximate Need to impose constraints on control points to obtain C 1 continuity C 2 continuity more difficult Solutions User interface using Bézier handles Generalization to B-splines or NURBS 56
57 Bézier Handles Segment end points (interpolating) presented as curve control points Midpoints (approximating points) presented as handles Can have option to enforce C 1 continuity Adobe Illustrator 57
58 Next Lecture Parametric surfaces 58
Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points
More informationComputer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include
More informationGeometric Modelling & Curves
Geometric Modelling & Curves Geometric Modeling Creating symbolic models of the physical world has long been a goal of mathematicians, scientists, engineers, etc. Recently technology has advanced sufficiently
More informationWe can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
More informationModeling Curves and Surfaces
Modeling Curves and Surfaces Graphics I Modeling for Computer Graphics!? 1 How can we generate this kind of objects? Umm!? Mathematical Modeling! S Do not worry too much about your difficulties in mathematics,
More information(Refer Slide Time: 1:42)
Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture - 10 Curves So today we are going to have a new topic. So far
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More informationCHAPTER 1 Splines and B-splines an Introduction
CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
More informationCurves and Surfaces. Goals. How do we draw surfaces? How do we specify a surface? How do we approximate a surface?
Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] Goals How do we draw surfaces? Approximate with polygons Draw polygons
More informationTopics in Computer Graphics Chap 14: Tensor Product Patches
Topics in Computer Graphics Chap 14: Tensor Product Patches fall, 2011 University of Seoul School of Computer Science Minho Kim Table of contents Bilinear Interpolation The Direct de Casteljau Algorithm
More informationBEZIER CURVES AND SURFACES
Department of Applied Mathematics and Computational Sciences University of Cantabria UC-CAGD Group COMPUTER-AIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: BEZIER CURVES AND SURFACES Andrés Iglesias e-mail:
More informationDegree Reduction of Interval SB Curves
International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:13 No:04 1 Degree Reduction of Interval SB Curves O. Ismail, Senior Member, IEEE Abstract Ball basis was introduced
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationDhiren Bhatia Carnegie Mellon University
Dhiren Bhatia Carnegie Mellon University University Course Evaluations available online Please Fill! December 4 : In-class final exam Held during class time All students expected to give final this date
More informationHow To Use Design Mentor
DesignMentor: A Pedagogical Tool for Computer Graphics and Computer Aided Design John L. Lowther and Ching Kuang Shene Programmers: Yuan Zhao and Yan Zhou (ver 1) Budirijanto Purnomo (ver 2) Michigan Technological
More informationMA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
More information3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials
3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines
More informationComputer Animation: Art, Science and Criticism
Computer Animation: Art, Science and Criticism Tom Ellman Harry Roseman Lecture 4 Parametric Curve A procedure for distorting a straight line into a (possibly) curved line. The procedure lives in a black
More informationIntroduction to Computer Graphics. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2012
CSE 167: Introduction to Computer Graphics Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2012 Today Course organization Course overview 2 Course Staff Instructor Jürgen Schulze,
More informationNURBS Drawing Week 5, Lecture 10
CS 430/536 Computer Graphics I NURBS Drawing Week 5, Lecture 10 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationB2.53-R3: COMPUTER GRAPHICS. NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions.
B2.53-R3: COMPUTER GRAPHICS NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions. 2. PART ONE is to be answered in the TEAR-OFF ANSWER
More informationNURBS Drawing Week 5, Lecture 10
CS 430/536 Computer Graphics I NURBS Drawing Week 5, Lecture 10 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationA Short Introduction to Computer Graphics
A Short Introduction to Computer Graphics Frédo Durand MIT Laboratory for Computer Science 1 Introduction Chapter I: Basics Although computer graphics is a vast field that encompasses almost any graphical
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationIntroduction to Computer Graphics Marie-Paule Cani & Estelle Duveau
Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction
More informationA Geometric Characterization of Parametric Cubic Curves
A Geometric Characterization of Parametric Cubic Curves MAUREEN C. STONE Xerox PARC and TONY D. DEROSE University of Washington In this paper, we analyze planar parametric cubic curves to determine conditions
More informationCorollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality
Corollary For equidistant knots, i.e., u i = a + i (b-a)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality 120202: ESM4A - Numerical Methods
More informationH.Calculating Normal Vectors
Appendix H H.Calculating Normal Vectors This appendix describes how to calculate normal vectors for surfaces. You need to define normals to use the OpenGL lighting facility, which is described in Chapter
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationA matrix method for degree-raising of B-spline curves *
VOI. 40 NO. 1 SCIENCE IN CHINA (Series E) February 1997 A matrix method for degree-raising of B-spline curves * QIN Kaihuai (%*>/$) (Department of Computer Science and Technology, Tsinghua University,
More informationThe Essentials of CAGD
The Essentials of CAGD Chapter 2: Lines and Planes Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentials-cagd c 2000 Farin & Hansford
More informationPERFORMANCE OF BÉZIER CURVES RENDERING IN WEB BROWSERS
PERFORMANCE OF BÉZIER CURVES RENDERING IN WEB BROWSERS Abstract By Ammar Hattab APMA2821Q Project Report Prof. Mark Ainsworth Spring 2013 Brown University Bézier curves are used everywhere in graphics,
More informationNumerical algorithms for curve approximation and novel user oriented interactive tools
UNIVERSITÀ DEGLI STUDI DI BARI Dottorato di Ricerca in Matematica XXI Ciclo A.A. 2008/2009 Settore Scientifico-Disciplinare: MAT/08 Analisi Numerica Tesi di Dottorato Numerical algorithms for curve approximation
More informationVALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203.
VALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203. DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year & Semester : III Year, V Semester Section : CSE - 1 & 2 Subject Code : CS6504 Subject
More informationCSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Linda Shapiro Today Registration should be done. Homework 1 due 11:59 pm next Wednesday, January 14 Review math essential
More informationCPIT-285 Computer Graphics
Department of Information Technology B.S.Information Technology ABET Course Binder CPIT-85 Computer Graphics Prepared by Prof. Alhasanain Muhammad Albarhamtoushi Page of Sunday December 4 0 : PM Cover
More informationJim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More informationFinite Element Formulation for Plates - Handout 3 -
Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the
More informationComputer Applications in Textile Engineering. Computer Applications in Textile Engineering
3. Computer Graphics Sungmin Kim http://latam.jnu.ac.kr Computer Graphics Definition Introduction Research field related to the activities that includes graphics as input and output Importance Interactive
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationCOMP175: Computer Graphics. Lecture 1 Introduction and Display Technologies
COMP175: Computer Graphics Lecture 1 Introduction and Display Technologies Course mechanics Number: COMP 175-01, Fall 2009 Meetings: TR 1:30-2:45pm Instructor: Sara Su (sarasu@cs.tufts.edu) TA: Matt Menke
More informationConstrained curve and surface fitting
Constrained curve and surface fitting Simon Flöry FSP-Meeting Strobl (June 20, 2006), floery@geoemtrie.tuwien.ac.at, Vienna University of Technology Overview Introduction Motivation, Overview, Problem
More informationMonash University Clayton s School of Information Technology CSE3313 Computer Graphics Sample Exam Questions 2007
Monash University Clayton s School of Information Technology CSE3313 Computer Graphics Questions 2007 INSTRUCTIONS: Answer all questions. Spend approximately 1 minute per mark. Question 1 30 Marks Total
More informationThe Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal
More informationA Mathematica Package for CAGD and Computer Graphics
A Mathematica Package for CAGD and Computer Graphics Andrés Iglesias 1, Flabio Gutiérrez 1,2 and Akemi Gálvez 1 1 Departament of Applied Mathematics and Computational Sciences University of Cantabria,
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations
More informationSeminar. Path planning using Voronoi diagrams and B-Splines. Stefano Martina stefano.martina@stud.unifi.it
Seminar Path planning using Voronoi diagrams and B-Splines Stefano Martina stefano.martina@stud.unifi.it 23 may 2016 This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International
More informationCOMP-557: Fundamentals of Computer Graphics McGill University, Fall 2010
COMP-557: Fundamentals of Computer Graphics McGill University, Fall 2010 Class times 2:25 PM - 3:55 PM Mondays and Wednesdays Lecture room Trottier Building 2120 Instructor Paul Kry, kry@cs.mcgill.ca Course
More informationHow To Draw A 3D Virtual World In 3D Space (Computer Graphics)
2 Computer Graphics What You Will Learn: The objectives of this chapter are quite ambitious; you should refer to the references cited in each Section to get a deeper explanation of the topics presented.
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More information1. Abstract 2. Introduction 3. Algorithms and Techniques
MS PROJECT Virtual Surgery Piyush Soni under the guidance of Dr. Jarek Rossignac, Brian Whited Georgia Institute of Technology, Graphics, Visualization and Usability Center Atlanta, GA piyush_soni@gatech.edu,
More informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More informationPiecewise Cubic Splines
280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computer-assisted design), CAM (computer-assisted manufacturing), and
More informationClassification of Fingerprints. Sarat C. Dass Department of Statistics & Probability
Classification of Fingerprints Sarat C. Dass Department of Statistics & Probability Fingerprint Classification Fingerprint classification is a coarse level partitioning of a fingerprint database into smaller
More informationMinkowski Sum of Polytopes Defined by Their Vertices
Minkowski Sum of Polytopes Defined by Their Vertices Vincent Delos, Denis Teissandier To cite this version: Vincent Delos, Denis Teissandier. Minkowski Sum of Polytopes Defined by Their Vertices. Journal
More informationSubdivision for Modeling and Animation
SIGGRAPH 99 Course Notes Subdivision for Modeling and Animation Organizers: Denis Zorin, New York University Peter Schröder, California Institute of Technology Lecturers Denis Zorin Media Research Laboratory
More informationSect. 1.3: Factoring
Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion
More informationCS 4620 Practicum Programming Assignment 6 Animation
CS 4620 Practicum Programming Assignment 6 Animation out: Friday 14th November 2014 due: : Monday 24th November 2014 1 Introduction In this assignment, we will explore a common topic in animation: key
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationComputational Geometry. Lecture 1: Introduction and Convex Hulls
Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,
More informationGAME ENGINE DESIGN. A Practical Approach to Real-Time Computer Graphics. ahhb. DAVID H. EBERLY Geometrie Tools, Inc.
3D GAME ENGINE DESIGN A Practical Approach to Real-Time Computer Graphics SECOND EDITION DAVID H. EBERLY Geometrie Tools, Inc. ahhb _ jfw H NEW YORK-OXFORD-PARIS-SAN DIEGO fl^^h ' 4M arfcrgsbjlilhg, SAN
More informationHow To Write An Isgeometric Finite Element Analysis
NTNU Norwegian University of Science and Technology Faculty of Information Technology, Mathematics and Electrical Engineering MASTER S THESIS for stud.techn. Kjetil André Johannessen Faculty of Information
More informationPlanar Curve Intersection
Chapter 7 Planar Curve Intersection Curve intersection involves finding the points at which two planar curves intersect. If the two curves are parametric, the solution also identifies the parameter values
More informationArrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
More informationIntroduction to Computer Graphics
Introduction to Computer Graphics Torsten Möller TASC 8021 778-782-2215 torsten@sfu.ca www.cs.sfu.ca/~torsten Today What is computer graphics? Contents of this course Syllabus Overview of course topics
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More information67 204 Mathematics for Business Analysis I Fall 2007
67 204 Mathematics for Business Analysis I Fall 2007 Instructor Asõkā Rāmanāyake Office: Swart 223 Office Hours: Monday 12:40 1:40 Wednesday 8:00 9:00 Thursday 9:10 11:20 If you cannot make my office hours,
More informationGenerate Cryptographic key using generated 3D- Digital Image
Dr.Hala B. Abdul Wahab*, Dr.Suhad M. Kadhum* & Ekhlas K. Gbashi* Received on: 20/ 1 /2008 Accepted on: 4/9 /2008 Abstract Every few years, computer security has to re-invent itself. New technologies and
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationPoint Lattices in Computer Graphics and Visualization how signal processing may help computer graphics
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch
More informationAn Overview of the Finite Element Analysis
CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry
More informationParametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
More informationCOMPUTER GRAPHICS IMPORTANT QUESTION AND ANSWERS. Computer graphics
Computer graphics 1. Define Computer graphics. Computer graphics remains one of the most existing and rapidly growing computer fields. Computer graphics may be defined as a pictorial representation or
More informationAlgorithms for Real-Time Tool Path Generation
Algorithms for Real-Time Tool Path Generation Gyula Hermann John von Neumann Faculty of Information Technology, Budapest Polytechnic H-1034 Nagyszombat utca 19 Budapest Hungary, hermgyviif.hu Abstract:The
More informationLecture Notes, CEng 477
Computer Graphics Hardware and Software Lecture Notes, CEng 477 What is Computer Graphics? Different things in different contexts: pictures, scenes that are generated by a computer. tools used to make
More informationInteractive Computer Graphics
Interactive Computer Graphics A Top-Down Approach Using OpenGL FIFTH EDITION EDWARD ANGEL UNIVERSITY OF NEW MEXICO PEARSON Addison Wesley Boston San Francisco New York London Toronto Sydney Tokyo Singapore
More informationLinear Models and Conjoint Analysis with Nonlinear Spline Transformations
Linear Models and Conjoint Analysis with Nonlinear Spline Transformations Warren F. Kuhfeld Mark Garratt Abstract Many common data analysis models are based on the general linear univariate model, including
More informationanimation animation shape specification as a function of time
animation animation shape specification as a function of time animation representation many ways to represent changes with time intent artistic motion physically-plausible motion efficiency control typically
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics Computer Animation Adapted from notes by Yong Cao Virginia Tech 1 Outline Principles of Animation Keyframe Animation Additional challenges in animation 2 Classic animation Luxo
More informationINTRODUCTION TO RENDERING TECHNIQUES
INTRODUCTION TO RENDERING TECHNIQUES 22 Mar. 212 Yanir Kleiman What is 3D Graphics? Why 3D? Draw one frame at a time Model only once X 24 frames per second Color / texture only once 15, frames for a feature
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationCourse Overview. CSCI 480 Computer Graphics Lecture 1. Administrative Issues Modeling Animation Rendering OpenGL Programming [Angel Ch.
CSCI 480 Computer Graphics Lecture 1 Course Overview January 14, 2013 Jernej Barbic University of Southern California http://www-bcf.usc.edu/~jbarbic/cs480-s13/ Administrative Issues Modeling Animation
More informationBirmingham City Schools
Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 Pre-Assessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationPackage MBA. February 19, 2015. Index 7. Canopy LIDAR data
Version 0.0-8 Date 2014-4-28 Title Multilevel B-spline Approximation Package MBA February 19, 2015 Author Andrew O. Finley , Sudipto Banerjee Maintainer Andrew
More informationPre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationSummary: Transformations. Lecture 14 Parameter Estimation Readings T&V Sec 5.1-5.3. Parameter Estimation: Fitting Geometric Models
Summary: Transformations Lecture 14 Parameter Estimation eadings T&V Sec 5.1-5.3 Euclidean similarity affine projective Parameter Estimation We will talk about estimating parameters of 1) Geometric models
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. M.Sc. in Advanced Computer Science. Friday 18 th January 2008.
COMP60321 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE M.Sc. in Advanced Computer Science Computer Animation Friday 18 th January 2008 Time: 09:45 11:45 Please answer any THREE Questions
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationDiscrete Optimization
Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using
More informationDiablo Valley College Catalog 2014-2015
Mathematics MATH Michael Norris, Interim Dean Math and Computer Science Division Math Building, Room 267 Possible career opportunities Mathematicians work in a variety of fields, among them statistics,
More information