BEZIER CURVES AND SURFACES


 Janel Bryant
 3 years ago
 Views:
Transcription
1 Department of Applied Mathematics and Computational Sciences University of Cantabria UCCAGD Group COMPUTERAIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: BEZIER CURVES AND SURFACES Andrés Iglesias Web pages: Andrés Iglesias. See:
2 Andrés Iglesias. See: BEZIER CURVES n i= P i B n i (t) Bézier curves Let P={P,P,...,P n } be a set of points P i IR d, d=,. The Bézier curve associated with the set P is defined by: B n i (t) where represent the Bernstein polynomials, which are given by: ( ) Bi n (t) = n ( t) n i t i i i =,..., n n being the polynomial degree. Bézier curve with n=5 (six control or Bézier points) Bernstein polynomials B i (t)
3 Bézier curves Control polygon 6 Control points Bézier curve 7 control points n= n i= P i B n i (t) Andrés Iglesias. See:
4 Given by: Properties: Extreme values: Normalizing property: B n i (t) = Bernstein polynomials ( ) n i B in ()= B in ()= i=,...,n B n ()= B nn ()= B in ()= B in ()= n i= B n i (t)= ( t) n i t i i =,..., n Maxima: n the degree i the index t the variable Positivity: B in (t) in [,] Simmetry: Andrés Iglesias. See: B in (t) = B nin (t) B in (t) attains exactly one maximum on the interval [,], at t=i /n.
5 Properties of the Bézier curves The Bézier curve generally follows the shape of the control polygon, which consists of the segments joining the control points. D curve Control polygon Bézier scheme is useful for design. D curve Andrés Iglesias. See:
6 Properties of the Bézier curves Bézier curves exhibit global control: moving a control point alters the shape of the whole curve. Control point traslation. LOCAL vs. GLOBAL CONTROL Bsplines allow local control: only a part of the curve is modified when changing a control point. Control point traslation. (,) (,) (,) (,) The curve changes here The curve does not change here Andrés Iglesias. See:
7 Properties of the Bézier curves Interpolation. A Bézier curve always interpolates the end control points. Tangency. The endpoint tangent vectors are parallel to P  P and P n  P n Convex hull property. The curve is contained in the convex hull of its defining control points. Variation disminishing property. No straight line intersects a Bézier curve more times than it intersects its control polygon. Intersections Curve: Polygon: Curve: Polygon: Curve: Polygon: For a threedimensional Bézier curve, replaces the words straight line with the word plane. Andrés Iglesias. See:
8 Properties of the Bézier curves A given Bézier curve can be subdivided at a point t=t into two Bézier segments which join together at the point corresponding to the parameter value t=t. Lefthand side: control points Andrés Iglesias. See: Original curve: control points 5 6 t=. t=. t=. Subdivided curve: 7 control points Righthand side: control points
9 Properties of the Bézier curves Degree raising: any Bézier curve of degree n (with control points P i ) can be expressed in terms of a new basis of degree n+. The new control points Q i are given by: i Q i = P i + ( ) P n+ n+ i i=,...,n+ P  = P n+ = Original cubic curve: control points Final quartic curve: 5 control points 5 6 Andrés Iglesias. See: 5 6
10 Bézier curves Degree raising of the Bézier curve of degree n= to degree n= Andrés Iglesias. See:
11 Rational Bézier curves There are a number of important curves and surfaces which cannot be represented faithfully using polynomials, namely, circles, ellipses, hyperbolas, cylinders, cones, etc. All the conics can be well represented using rational functions, which are the ratio of two polynomials. Rational Bézier curve R(t) = n i= P i w i B n i (t) n i= w i B n i (t) w i weights If all w i =, we recover the Bézier curve. Andrés Iglesias. See: Farin, G.: Curves and Surfaces for CAGD, Academic Press, rd. Edition, 99 (Chapters and 5). Hoschek, J. and Lasser, D.: Fundamentals of CAGD, A.K. Peters, 99 (Chapter ). Anand, V.: Computer Graphics and Geometric Modeling for Engineers, John Wiley & Sons, 99 (Chapter ).
12 Andrés Iglesias. See: Changing the weights: Rational Bézier curves w i > > the curve approximates to P i w i < > the curve moves away from P i
13 Rational Bézier curves Influence of the weights: The effect of changing a weight is different from that of moving a control point. Moving a control point: a nonrational Bézier curve with a change in one control point. Original curve Final curve Changing a weight: a rational Bézier curve with one weight changed. Andrés Iglesias. See:
14 Andrés Iglesias. See: Rational Bézier curves Rational Bézier curves are useful to represent conics, which become an important tool in the aircraft industry. Let c(t) be a point on a conic. Then, there exist numbers w, w and w and twodimensional points P, P and P such that: c(t) = s = s < s > w P B (t) +w P B (t) +w P B (t) w B (t) +w B (t) +w B (t) If we take w = w = and we define : gives a parabolic arc gives an elliptic arc gives a hyperbolic arc w = s = w + w hyperbola w = parabola elipse w =
15 Example: the circle.75 (, ) Rational Bézier curves /.5.5 (/, /).75 (/, /).5 /.5 (,).5.5 (,) w = w = / w = (,) Andrés Iglesias. See: /
16 BEZIER SURFACES Let P={{P,P,...,P n }, {P,P,...,P n },..., {P m,p m,...,p mn }} be a set of points P ij IR (i=,,...,m ; j=,,...n) Bézier surfaces The Bézier surface associated with the set P is defined by: m n S(u, v) = P ij B m i (u)b n j (v) i= B m i (u) j= where and represent the Bernstein polynomials of degrees m and n and in the variables u and v, respectively. Bn j (v) x y z x y z x y z x y z x y z x y z Andrés Iglesias. See:
17 Bézier surfaces Note that along the isoparametric lines u=u and v=v, the surface reduces to Bézier curves: n m S(u,v) = with control points: j= b j = b j B n j (v) S(u, v ) = m i= P ij B m i (u ) i= c i = Andrés Iglesias. See: c i B m i (u) n j= P ij B n j (v ) Isoparametric lines u=u } v DOMAIN Z D space Y S(u,v) } u Isoparametric lines v=v X
18 From the equation: S(u, v) = Bézier surfaces is obtained from a control point and the product of two univariate Bernstein polynomials. Each product makes up a basis function of the surface. For instance: Function B (u). B (v): m i= n j= P ij B m i (u)b n j (v) it is clear that each term.8.6 B (u). B (v) B (v) B (u) Andrés Iglesias. See:
19 If a single surface does not approximate enough a given set of points, we may use several patches joined together. Bézier surfaces BLENDING SURFACES Two Bézier patches F and F are connected with C continuity by using a Bézier F patch. F F F BLENDING Andrés Iglesias. See:
20 Rational Bézier surfaces If we introduce weights w ij to a nonrational Bézier surface, we obtain a rational Bézier surface: m n S(u, v) = i= j= m i= j= P ij w ij B m i (u)bn j (v) n w ij B m i (u)b n j (v).5.5 w ij = Example: Andrés Iglesias. See: ( ) ( ) 7.
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: WireFrame Representation Object is represented as as a set of points
More informationQUADRATIC BÉZIER CURVES
OnLine Geometric Modeling Notes QUADRATIC BÉZIER CURVES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview The Bézier curve
More informationWe can display an object on a monitor screen in three different computermodel 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 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 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 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 informationparametric spline curves
parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape modeling (3D) different representation implicit curves parametric curves (mostly used) 2D and 3D curves are
More informationHomework 3 Model Solution Section
Homework 3 Model Solution Section 12.6 13.1. 12.6.3 Describe and sketch the surface + z 2 = 1. If we cut the surface by a plane y = k which is parallel to xzplane, the intersection is + z 2 = 1 on a plane,
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 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 informationEngineering Geometry
Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the righthand rule.
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 informationGymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz
SCHEME OF WORK Subject: Mathematics Year: Third grade, 3.X School year:../ List of topics Topics Time period 1. Revision (functions, plane geometry) September 2. Constructive geometry in the plane October
More information5. GEOMETRIC MODELING
5. GEOMETRIC MODELING Types of Curves and Their Mathematical Representation Types of Surfaces and Their Mathematical Representation Types of Solids and Their Mathematical Representation CAD/CAM Data Exchange
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 informationDegree Reduction of Interval SB Curves
International Journal of Video&Image Processing and Network Security IJVIPNSIJENS Vol:13 No:04 1 Degree Reduction of Interval SB Curves O. Ismail, Senior Member, IEEE Abstract Ball basis was introduced
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 information42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections
2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You
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 10.110.6] Goals How do we draw surfaces? Approximate with polygons Draw polygons
More informationCOMPUTERAIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: HIGH LEVEL COMPUTATION PROGRAMS (MATHEMATICA AND MATLAB) ) IN CAGD AND COMPUTER GRAPHICS
Department of Applied Mathematics and Computational Sciences University of Cantabria UCCAGD Group COMPUTERAIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: HIGH LEVEL COMPUTATION PROGRAMS (MATHEMATICA AND
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More information3.6 Cylinders and Quadric Surfaces
3.6 Cylinders and Quadric Surfaces Objectives I know the definition of a cylinder. I can name the 6 quadric surfaces, write their equation, and sketch their graph. Let s take stock in the types of equations
More informationCSE 167: Lecture 13: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011
CSE 167: Introduction to Computer Graphics Lecture 13: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011 Announcements Homework project #6 due Friday, Nov 18
More informationWe want to define smooth curves:  for defining paths of cameras or objects.  for defining 1D shapes of objects
lecture 10  cubic curves  cubic splines  bicubic surfaces We want to define smooth curves:  for defining paths of cameras or objects  for defining 1D shapes of objects We want to define smooth surfaces
More informationOn the Arc Length of Parametric Cubic Curves
Journal for Geometry and Graphics Volume 3 (1999), No. 1, 1 15 On the Arc Length of Parametric Cubic Curves Zsolt Bancsik, Imre Juhász Department of Descriptive Geometry, University of Miskolc, H3515
More informationParametric Representation of Curves and Surfaces
Parametric Representation of Crves and Srfaces How does the compter compte crves and srfaces? MAE 455 CompterAided Design and Drafting Types of Crve Eqations Implicit Form Explicit Form Parametric Form
More informationCOMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg
COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg September 28, 2016 ii T. W. Sederberg iii Preface I have taught a course at Brigham Young University titled Computer Aided Geometric Design since 1983.
More informationThe distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña
The distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña Abstract. Recently, Nairn, Peters, and Lutterkort bounded the distance between a Bézier curve and its control
More informationAlgorithms for RealTime Tool Path Generation
Algorithms for RealTime Tool Path Generation Gyula Hermann John von Neumann Faculty of Information Technology, Budapest Polytechnic H1034 Nagyszombat utca 19 Budapest Hungary, hermgyviif.hu Abstract:The
More informationDesignMentor: A Pedagogical Tool for Computer Graphics and Computer Aided Design
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 informationCommon Curriculum Map. Discipline: Math Course: College Algebra
Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a
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 information4. Factor polynomials over complex numbers, describe geometrically, and apply to realworld situations. 5. Determine and apply relationships among syn
I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of
More informationME 111: Engineering Drawing
ME 111: Engineering Drawing Lecture 4 08082011 Engineering Curves and Theory of Projection Indian Institute of Technology Guwahati Guwahati 781039 Eccentrici ty = Distance of the point from the focus
More informationOverview Mathematical Practices Congruence
Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationLocalization of touching points for interpolation of discrete circles
Annales Mathematicae et Informaticae 36 (2009) pp. 103 110 http://ami.ektf.hu Localization of touching points for interpolation of discrete circles Roland Kunkli Department of Computer Graphics and Image
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationLevel: High School: Geometry. Domain: Expressing Geometric Properties with Equations GGPE
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Translate between the geometric
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 informationName: Class: Date: Conics Multiple Choice PostTest. Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: Conics Multiple Choice PostTest Multiple Choice Identify the choice that best completes the statement or answers the question. 1 Graph the equation x = 4(y 2) 2 + 1. Then describe the
More informationaxis axis. axis. at point.
Chapter 5 Tangent Lines Sometimes, a concept can make a lot of sense to us visually, but when we try to do some explicit calculations we are quickly humbled We are going to illustrate this sort of thing
More informationCathedral Brasilia Part 1
Cathedral Brasilia Part 1 1 Introduction: The lesson involves the start of making a representative model an iconic cathedral in Brasilia. The Cathedral Brasilia, designed by Oscar Niemeyer, is a hyperboloid
More informationAlgebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2
1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect
More informationGeometry Vocabulary. Created by Dani Krejci referencing:
Geometry Vocabulary Created by Dani Krejci referencing: http://mrsdell.org/geometry/vocabulary.html point An exact location in space, usually represented by a dot. A This is point A. line A straight path
More information8.9 Intersection of Lines and Conics
8.9 Intersection of Lines and Conics The centre circle of a hockey rink has a radius of 4.5 m. A diameter of the centre circle lies on the centre red line. centre (red) line centre circle INVESTIGATE &
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationTExES Mathematics 7 12 (235) Test at a Glance
TExES Mathematics 7 12 (235) Test at a Glance See the test preparation manual for complete information about the test along with sample questions, study tips and preparation resources. Test Name Mathematics
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 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 (ba)/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 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 informationExample Degree Clip Impl. Int Sub. 1 3 2.5 1 10 15 2 3 1.8 1 5 6 3 5 1 1.7 3 5 4 10 1 na 2 4. Table 7.1: Relative computation times
Chapter 7 Curve Intersection Several algorithms address the problem of computing the points at which two curves intersect. Predominant approaches are the Bézier subdivision algorithm [LR80], the interval
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationMiddle Grades Mathematics 5 9
Middle Grades Mathematics 5 9 Section 25 1 Knowledge of mathematics through problem solving 1. Identify appropriate mathematical problems from realworld situations. 2. Apply problemsolving strategies
More informationCHAPTER 1 Splines and Bsplines an Introduction
CHAPTER 1 Splines and Bsplines 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 informationBirmingham City Schools
Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 PreAssessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationHIGH SCHOOL: GEOMETRY (Page 1 of 4)
HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course
More informationINTEGRATION FINDING AREAS
INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the
More informationRasterizing the outlines of fonts
ELECTRONIC PUBLISHING, VOL. 6(3), 171 181 (SEPTEMBER 1993) Rasterizing the outlines of fonts FIAZ HUSSAIN School of Computing and Mathematical Sciences De Montfort University, Kents Hill Campus, Hammerwood
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationPure Math 30: Explained!
www.puremath30.com 45 Conic Sections: There are 4 main conic sections: circle, ellipse, parabola, and hyperbola. It is possible to create each of these shapes by passing a plane through a three dimensional
More informationA matrix method for degreeraising of Bspline curves *
VOI. 40 NO. 1 SCIENCE IN CHINA (Series E) February 1997 A matrix method for degreeraising of Bspline curves * QIN Kaihuai (%*>/$) (Department of Computer Science and Technology, Tsinghua University,
More informationSphere centered at the origin.
A Quadratic surfaces In this appendix we will study several families of socalled quadratic surfaces, namely surfaces z = f(x, y) which are defined by equations of the type A + By 2 + Cz 2 + Dxy + Exz
More informationTRIGONOMETRY GRADES THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618
TRIGONOMETRY GRADES 1112 THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618 Board Approval Date: October 29, 2012 Michael Nitti Written by: EHS Mathematics Department Superintendent In accordance
More informationComal Independent School District PreAP PreCalculus Scope and Sequence
Comal Independent School District Pre PreCalculus Scope and Sequence Third Quarter Assurances. The student will plot points in the Cartesian plane, use the distance formula to find the distance between
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationGouvernement du Québec Ministère de l Éducation, 2004 0400910 ISBN 2550437012
Gouvernement du Québec Ministère de l Éducation, 2004 0400910 ISBN 2550437012 Legal deposit Bibliothèque nationale du Québec, 2004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation
More informationWhere parallel lines meet...
Where parallel lines meet...a geometric love story Colorado State University MATH DAY 2009 A simple question with an annoying answer... Q: How many points of intersection do two distinct lines in the have?
More informationAdvanced Algebra 2. I. Equations and Inequalities
Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers
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 01 Sets There are no statemandated Precalculus 02 Operations
More informationSolution. a) The line in question has parameterization γ(t) = (0, t, t). Plugging this into the equation of the surface yields
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Midterm # 1 (Tue 21 Feb 2012) Practice Exam Solution Guide Practice problems: The following assortment of problems
More informationAlgebra II. Larson, Boswell, Kanold, & Stiff (2001) Algebra II, Houghton Mifflin Company: Evanston, Illinois. TI 83 or 84 Graphing Calculator
Algebra II Text: Supplemental Materials: Larson, Boswell, Kanold, & Stiff (2001) Algebra II, Houghton Mifflin Company: Evanston, Illinois. TI 83 or 84 Graphing Calculator Course Description: The purpose
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationPreAlgebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio PreAlgebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationME 111: Engineering Drawing
ME 111: Engineering Drawing Lecture 3 05082011 SCALES AND Engineering Curves Indian Institute of Technology Guwahati Guwahati 781039 Definition A scale is defined as the ratio of the linear dimensions
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
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/essentialscagd c 2000 Farin & Hansford
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 informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More information12.1. VectorValued Functions. VectorValued Functions. Objectives. Space Curves and VectorValued Functions. Space Curves and VectorValued Functions
12 VectorValued Functions 12.1 VectorValued Functions Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Analyze and sketch a space curve given
More informationActivity 6. Inequalities, They Are Not Just Linear Anymore! Objectives. Introduction. Problem. Exploration
Objectives Graph quadratic inequalities Graph linearquadratic and quadratic systems of inequalities Activity 6 Introduction What do satellite dishes, car headlights, and camera lenses have in common?
More information52. Explain interrupt scheme for retrieving input data. Also explain any one algorithm for input device handling. 53. Explain the process of 3D
Computer Graphics 1. Derive the equation for the intercept form of the line. 2. Explain the frame buffer, point and pixels. 3. Describe the Digital Differential Analyzer (DDA) for line drawing. 4. Explain
More informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School  Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationarxiv: v1 [cs.gr] 13 Jan 2016
Implicit equations of nondegenerate rational Bezier quadric triangles A. Cantón, L. FernándezJambrina, E. Rosado María, and M.J. VázquezGallo arxiv:1601.03224v1 [cs.gr] 13 Jan 2016 Universidad Politécnica
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
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 informationContents. The Real Numbers. Linear Equations and Inequalities in One Variable
dug33513_fm.qxd 11/20/07 3:21 PM Page vii Preface Guided Tour: Features and Supplements Applications Index 1 2 The Real Numbers 1.1 1.2 1.3 1.4 1.5 1.6 1 Sets 2 The Real Numbers 9 Operations on the Set
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More information3 Drawing 2D shapes. Launch form Z.
3 Drawing 2D shapes Launch form Z. If you have followed our instructions to this point, five icons will be displayed in the upper left corner of your screen. You can tear the three shown below off, to
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationMidterm Exam I, Calculus III, Sample A
Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of
More informationAssignment 3. Solutions. Problems. February 22.
Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.
More informationIntersection of Ellipses
Intersection of Ellipses David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: October 10, 2000 Last Modified: June 23, 2015 Contents 1 Introduction
More informationM.L. Sampoli CLOSED SPLINE CURVES BOUNDING MAXIMAL AREA
Rend. Sem. Mat. Univ. Pol. Torino Vol. 61, 3 (2003) Splines and Radial Functions M.L. Sampoli CLOSED SPLINE CURVES BOUNDING MAXIMAL AREA Abstract. In this paper we study the problem of constructing a closed
More informationParametric Curves. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) YanBin 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 informationGraphical objects geometrical data handling inside DWG file
Graphical objects geometrical data handling inside DWG file Author: Vidmantas Nenorta, Kaunas University of Technology, Lithuania, vidmantas.nenorta@ktu.lt Abstract The paper deals with required (necessary)
More informationChapter 33D Modeling
Chapter 33D 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 informationStraight Line motion with rigid sets
Straight ine motion with rigid sets arxiv:40.4743v [math.mg] 9 Jan 04 Robert Connelly and uis Montejano January, 04 Abstract If one is given a rigid triangle in the plane or space, we show that the only
More information