Geometric Modelling & Curves
|
|
- Baldric Bridges
- 7 years ago
- Views:
Transcription
1 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 to make computer modeling of physical geometry feasible. 1
2 Modelers for Engineering Modeling for engineering applications require higher accuracy of representation. Engineering models are used in computer-based design, manufacturing and analysis. Geometric modeling simply means that design concepts are digitally inputted into software subsequently displays them in either 2-D or 3-D forms. a general term applied to 3-D computer-aided design techniques. Geometric models are computational (symbol) structures that capture the spatial aspects of the objects of interest for an application. 2
3 Geometric model = Geometry Topology Feature model = Geometric model Design intent Design intent = Constraints Rules 5 Components of geometric modeling system B-rep, CSG 6 3
4 What is Geometric modeling? The process of constructing a complete mathematical description (geometric database) to model a physical entity or system. Geometric Modeling GM is a general term applied to 3D computeraided design techniques. There are three main types of geometrical modelling used, namely: line or wireframe modelling, surface modelling, solid modelling. 4
5 How does geometric modeling fit into a modern design sequence? Computer-based geometric modeling is used to: visualize, analyze, document, produce a product or process. Geometric modeling Stress Analysis Engineering Analysis Geometric modeling Thermal Analysis After-life Analysis Geometric modeling is a basic engineering tool. Visualization Design Process Production planning Engineering drawing CNC programming Serves as the backbone of design Shadows the design process 5
6 Geometric modeling produce an appropriate database used for input into specialized engineering software tools to perform tasks in an integrated design sequence. support the move to a seamless work environment where the flow of data is continuous and need not recreated at each stage of design development. Geometric modeling 6
7 Shape modeling Reconstruction Feature Analysis Simplification Mesh generation Curve Shape Surface Medial axis Segmentation Dimension detection Sample Mesh Surface Volume Geometry A typical solid model is defined by volumes, areas, lines, and keypoints. Volumes are bounded by areas. They represent solid objects. Areas are bounded by lines. They represent faces of solid objects, or planar or shell objects. Lines are bounded by keypoints. They represent edges of objects. Keypoints are locations in 3-D space. They represent vertices of objects. Volumes Areas Lines & Keypoints 7
8 ...Geometry - Preprocessing There is a built-in hierarchy among solid model entities. Keypoints are the foundation entities. Lines are built from the keypoints, areas from lines, and volumes from areas. This hierarchy holds true regardless of how the solid model is created. not allow you to delete or modify a lower-order entity if it is attached to a higher-order entity. (Certain types of modifications are allowed discussed later.) Volumes Areas Lines Keypoints I ll just change this line Volumes Areas Lines Keypoints OOPs! Lines Keypoints Areas Mathematical Representation of Curves The user constructs a geometric model of an object on a CAD/CAM system by inputting the object data as required by the modeling technique via user interface. The software converts such data into a mathematical representation. 8
9 Mathematical Representation of Curves Geometric modeling to CAD/CAM is important; it is a mean to enable useful engineering analyses and judgment. In computer-based modeling and analysis, geometric models determine their relevance to design, analysis and manufacturing. Mathematical Representation of Curves Methods of defining points Explicit methods Implicit methods Absolute cartesian coordinates A digitize d Y P(x,y,z) d X Z Absolute cylindrical coordinates R P(R,,z) E 1 E 2 E 1, E 2 E 2 E 1 E 1 E 2 9
10 Mathematical Representation of Curves Methods of defining points Explicit methods Implicit methods Incremental cartesian coordinates Δz P 1 (xδx,yδy,z-δz) Y C Δx Δy P 0 (x,y,z) X Z Incremental cylindrical coordinates P 1 (RΔR, Δ,z) ΔR Δ R P 0 (R,,z) I 1 C I 2 C Mathematical Representation of Curves Methods of defining lines Methods Points defined by any method Illustration Incremental cylindrical coordinates Y w X w Z w 10
11 Mathematical Representation of Curves Methods of defining lines Methods Parallel or perpendical to an existing line Illustration Tangent to existing entities Mathematical Representation of Curves Methods of defining arcs and circles Methods Radius or diameter and center Illustration R R 2 1 Three points defined by any method 11
12 Mathematical Representation of Curves Methods of defining arcs and circles Methods Center and a point on the circle Tangent to line, pass through a given point, and with a given radius Illustration 2 1 R Mathematical Representation of Curves Methods of defining ellipses Methods Center and axes lengths Four points A B P c Illustration Two conjugate diameters 12
13 Mathematical Representation of Curves Methods of defining parabolas Methods Vertex and focus Illustration Three points P 1 P v P 2 Mathematical Representation of Curves Methods of defining synthetic curves Methods Cubic spline Bezier curves B-Spline curves P 0 P 0 P 0 Illustration P n P n P n Approximate a given set of data points Interpolate a given set of data points 13
14 Curve Representation All forms of geometric modeling require the ability to define curves. Linear curves (1 st order) may be defined simply through their endpoints. Must have a means for the representation for curves of a higher order: conics free form or space curves Curve Representation Some terms we will use: Tangent vector: Vector tangent to the slope of a curve at a given point. Normal vector : Vector perpendicular to the slope of a curve at a given point. 14
15 Parametric curves; Non-parametric and parametric forms Analytical representations of curves Parabola: y = b x 2 C Ellipse: x 2 /a 2 y 2 /b 2 = 1 Hyperbola: x y = k Implicit form: f (x, y) = 0 Explicit form: y = f (x) hard to represent multi-valued function Analytical representations Analytical forms are not suitable for CAD because; Equation is dependent on coordinate sys. Curves are unbounded Implicit form is inconvenient for finding points on the curve Difficult for transformation (rotation, pan) 15
16 Curve Representation Curves may be defined using different equation formats. explicit Y = f(x), Z = g(x) implicit f(x,y,z) = 0 parametric X = X(t), Y = Y(t), Z = Z(t) The explicit and implicit formats have serious disadvantages for use in computer-based modeling Parametric form Equations are de-coupled x = f (u) Matrix form: p (u) = [ u 3 u 2 u 1 ] [ A ] 16
17 Curve Representation Parametric: X = X(t), Y = Y(t), Z = Z(t); 0 t 1 (typ) Substituting a value for t gives a corresponding position along curve Overcomes problems associated with implicit and explicit methods Most commonly used representation scheme in modelers Parametric example Recall the parametric line representation Parametric representation of a line. The parameter u, is varied from 0 to 1 to define all points along the line. X = X(u) Y = Y(u) P 2 u P 1 17
18 Parametric Line Line defined in terms of its endpoints Positions along the line are based upon the parameter value For example, the midpoint of a line occurs at u = 0.5 Parametric Line This means a parametric line can be defined by: L(u) = [x(u), y(u), z(u)] = A (B - A)u where A and B and the line endpoints. e.g. A line from point A = (2,4,1) to point B = (7,5,5) can be represented as: x(u) = 2 (7-2)u = 2 5u y(u) = 4 (5-4)u = 4 u z(u) = 1 (5-1)u = 1 4u 18
19 Parametric cubic curves Algebraic form Geometric form: blending fn * geometric (boundary) conditions Blending function: p (u) = [ F 1 F 2 F 3 F 4 ] [ p(0), p(1), p u (0), p u (1) ] Magnitude and direction of tangent vectors Cubic Hermite blending function Boundary conditions 19
20 Blending functions Parametric definition Expanding the 2D parametric technique we used for a line to 3D, two parameters (u and v) are used. P 4 P 2 P 3 u P 1 v 20
21 Parametric definition Points along edge P 1 P 2 have the form of P(u,0), along P 3 P 4, P(u,1) and so on. P 4 P(1,v) P(u,1) P 2 P 3 P(u,0) u P 1 v P(0,v) Parametric definition By varying value of u and v, any point on the surface or the edge of the face may be defined. P 2 P 4 (u 1,v 1 ) P 3 u P 1 v 21
22 Parametric definition Another basic example would be that of a conic (circle) Two parameter curves are : X = cos (u) Y = sin (u) with range π/4 u π/4 Parametric definition Cos(u) Sin (u) π/4 π/4 π/4 π/4 Graph of X = cos (u) Graph of Y = sin (u) 22
23 Parametric definition Y u = π/4 X 0 1 u = π/4 Combined curve is a quarter circle Parametric definition Controls for this curve Shape (based upon parametric equation) Location (based upon center point) Size arc (based upon parameter range) radius ( a coefficient to unit value) Similar list could be formed for other conics 23
24 Curve Use in Design Engineering design requires ability to express complex curve shapes (beyond conics). examples are the bounding curves for: turbine blades ship hulls automotive body panels also curves of intersection between surfaces Representing Complex Curves Typically represented a series of simpler curves (each defined by a single equation) pieced together at their endpoints (piecewise construction). Simpler curves may be linear or polynomial Equations for simpler curves based upon control points (data pts. used to define the curve) 24
25 Use of control points General curve shape may be generated using methods of: Interpolation (also known as Curve fitting ) curve will pass though control points Approximation curve will pass near control points may interpolate the start and end points Control Points Defining Curves The following example shows an: Interpolating (passes through control points) Piecewise linear curve curve defined by multiple segments, in this case linear 25
26 Interpolating Curve Piecewise linear Linear segments used to approximate smooth shape Segments joints known as KNOTS Requires too many datapoints for most shape approximations Representation not flexible enough to editing Piecewise linear Piecewise polynomial (composite curves) Segments defined by polynomial functions Again, segments join at KNOTS Most common polynomial used is cubic (3 rd order) Segment shape controlled by two or more adjacent control points. Piecewise linear Interpolation curve (cubic) 26
27 Knot points Locations where segments join referred to as knots Knots may or may not coincide with control points in interpolating curves, typically they DO NOT coincide. Curve continuity concern is continuity at knots (where curve segments join) continuity conditions: point continuity (no slope or curvature restriction) tangent continuity (same slope at knot) curvature continuity (same slope and curvature at knot) 27
28 Piecewise curves Curve segments Composite curves Continuity conditions Continuity is symbolically represented by capital C with a superscript representing level. Curve continuity C 0 continuity, point/position continuity continuity of endpoint only, or continuity of position C 1 continuity, tangent continuity tangent continuity or first derivative of position C 2 continuity, curvature continuity: Hydrodynamic character, Light reflection curvature continuity or second derivative of position 28
29 Composite curves: continuity Point continuity Tangent continuity Curvature continuity Interpolation curves Interpolating piecewise polynomial curve Typically possess curvature continuity Shape defined by: endpoint and control point location tangent vectors at knots* curvature at knots* *often calculated internally by software 29
30 Approximation techniques Developed to permit greater design flexibility in the generation of freeform curves. Two very common methods in modern CAD systems, Bezier and B-Spline. Approximation techniques employ control points (set of vertices that approximate the curve) curves do not pass directly through points (except possibly at start and end) intermediate points affect shape as if exerting a pull on the curve allow user to to set shape by pulling out curve using control point location 30
31 Curves Both Bézier curves and B-splines are polynomial parametric curves. Polynomial parametric forms can not represent some simple curves such as circles. Bézier curves and B-splines are generalized to rational Bézier curves and Non-Uniform Rational B-splines, or NURBS for short. Rational Bézier curves are more powerful than Bézier curves since the former now can represent circles and ellipses. Similarly, NURBS are more powerful than B-splines. Continuity The relationship among these B-spline NURBS Bezier Rational Bezier Recursive Subdivision Algorithm Dr. M. Abid Bezier curves P. Bezier of Renault, P. de Casteljau of Citroen Intuitive interaction: Direct manipulation Approximated curve vs. Interpolated curve Control points 31
32 Bezier curve defined by 4 pts Pull by coincident control points 32
33 Closed Bezier curve Influence of point position 33
34 Bezier curve... Bernstein polynomial B i,n (u) = C(n,i) u i (1-u) n-i Binomial coefficient: Combination Invariant under affine transformation Convex hull property: Straight interval Variation diminishing property Blending fn for a cubic Bezier curve 34
35 B-spline curves Piecewise Polynomials Approximating Splines B 0,1 B 1,1 B 2,1 B 3,1 B 4,1 B 5,1 B 6,1 B 0,2 B 1,2 B 2,2 B 3,2 B 4,2 B 5,2 B 0,3 B 1,3 B 2,3 B 3,3 B 4,3 B 0,4 B 1,4 B 2,4 B 3,4 NURBS B-Spline Synthetic Curves Analytical curves are insufficient for designing complex machinery parts and, therefore synthetic curves are used. Synthetic curves are commonly used when interpolation curves are needed and it is easy to modify these curves locally. 35
36 B-Spline Synthetic Curves CAD\CAM systems have got 3 types of synthetic curves such as Hermite cubic splines, Bezier and B-spline curves. Cubic splines are interpolating curves. Bezier and B-splines are approximating curves. On some cases B-splines can be interpolating. B-Spline Synthetic Curves B-spline curves are specified by giving set of coordinates, called control points, which indicates the general shape of the curve. B-splines can be either interpolating or approximating curves. Interpolation splines used for construction and to display the results of engineering. 36
37 Effect of curve order The higher the order of a curve, the stiffer the curve (less dramatic curvature changes) Maximum curve order dependent upon the number of control points order = one less than the number of control pts High order curves can exhibit irregularities B-Spline Synthetic Curves With (m1) control points, there are always (n=mp-1) basic functions. The basis functions are 1, at the end points of the curve defined as a and b. If there s no other definition, then a=0 and b=1. {P i } the set of the control points forms the control polygon from Figure. P 3 P 2 Control Polygon P 1 B-spline curve N 0,p (a)=1 N n,p (b)=1 P 4 37
38 B-splines Generalization of Bezier curve Bezier: p(u) = S P B i,n (u) B-spline: p(u) = S P N j,k (u) Basis fn or blending fn is different Local changes Degree of the curve is independent of # of control points Linear, quadratic, cubic B-spline 38
39 Influence of control point position Blending functions for linear B-spline 39
40 Quadratic B-spline blending fn (k=3) Non-Uniform Rational,B-splines Most modern CAD systems use the NURB curve representation scheme. NURB stands for Non-Uniform, Rational, B-spline. Uniformity deals with the spacing of control points. Rational functions include a weighting value at each control point for effect of control point. 40
41 Non-Uniform Rational,B-splines very popular due to their flexibility in curve generation. same mathematical form may be used to represent entire family of curves including: Bezier B-Spline conics NURBS & CAD For years major CAD software has been using NURBS for good surface definition. However, the exchange formats IGES and DXF are more limited - hence, the original model can become somewhat distorted in transfering from one program to another. 41
42 NURBS & CAD Add that to some inconsistency in how these formats are defined and consider a task like importing the NURBS sphere into a Coons Patch based program. With NURBS it is now possible to represent the geometry the same as the CAD packages represent it internally - so rather than using faulty exchange formats we have partnered with the major CAD vendors and now have a direct link that will pull a model from memory in the CAD program into one of our programs. Rational parametric curves Normal Bezier or B-spline cannot precisely represent conics and circles Homogeneous coordinate P h = (hx, hy, hz, h) NURBS (non-uniform rational B- splines) 84 42
43 Circular arc as a rational quadratic Bezier 85 43
We 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 informationComputer 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 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 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 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 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 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 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 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 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 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 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 real-life problem. algebraic expression - An expression with variables.
More informationComputer Aided Design (CAD)
16.810 Engineering Design and Rapid Prototyping Lecture 4 Computer Aided Design (CAD) Instructor(s) Prof. Olivier de Weck January 6, 2005 Plan for Today CAD Lecture (ca. 50 min) CAD History, Background
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 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 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 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 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 informationComputer Aided Design (CAD)
16.810 Engineering Design and Rapid Prototyping Lecture 3a Computer Aided Design (CAD) Instructor(s) Prof. Olivier de Weck January 16, 2007 Plan for Today CAD Lecture (ca. 50 min) CAD History, Background
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 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 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 informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationTWO-DIMENSIONAL TRANSFORMATION
CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
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 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 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 informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus 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 informationCATIA Wireframe & Surfaces TABLE OF CONTENTS
TABLE OF CONTENTS Introduction... 1 Wireframe & Surfaces... 2 Pull Down Menus... 3 Edit... 3 Insert... 4 Tools... 6 Generative Shape Design Workbench... 7 Bottom Toolbar... 9 Tools... 9 Analysis... 10
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 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 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 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 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 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 informationGrade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
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 informationSolidWorks Implementation Guides. Sketching Concepts
SolidWorks Implementation Guides Sketching Concepts Sketching in SolidWorks is the basis for creating features. Features are the basis for creating parts, which can be put together into assemblies. Sketch
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 informationRELEASED. Student Booklet. Precalculus. Fall 2014 NC Final Exam. Released Items
Released Items Public Schools of North arolina State oard of Education epartment of Public Instruction Raleigh, North arolina 27699-6314 Fall 2014 N Final Exam Precalculus Student ooklet opyright 2014
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 informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
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 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 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 informationAlgebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationCabri Geometry Application User Guide
Cabri Geometry Application User Guide Preview of Geometry... 2 Learning the Basics... 3 Managing File Operations... 12 Setting Application Preferences... 14 Selecting and Moving Objects... 17 Deleting
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 informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationRobust NURBS Surface Fitting from Unorganized 3D Point Clouds for Infrastructure As-Built Modeling
81 Robust NURBS Surface Fitting from Unorganized 3D Point Clouds for Infrastructure As-Built Modeling Andrey Dimitrov 1 and Mani Golparvar-Fard 2 1 Graduate Student, Depts of Civil Eng and Engineering
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 informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
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 information089 Mathematics (Elementary)
089 Mathematics (Elementary) MI-SG-FLD089-07 TABLE OF CONTENTS PART 1: General Information About the MTTC Program and Test Preparation OVERVIEW OF THE TESTING PROGRAM... 1-1 Contact Information Test Development
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University
More informationMathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationAdvanced Surface Modeling
This sample chapter is for review purposes only. Copyright The Goodheart-Willcox Co., Inc. ll rights reserved. Chapter dvanced Modeling Learning Objectives fter completing this chapter, you will be able
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 informationFurther Steps: Geometry Beyond High School. Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu
Further Steps: Geometry Beyond High School Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu Geometry the study of shapes, their properties, and the spaces containing
More informationIntroduction to CATIA V5
Introduction to CATIA V5 Release 16 (A Hands-On Tutorial Approach) Kirstie Plantenberg University of Detroit Mercy SDC PUBLICATIONS Schroff Development Corporation www.schroff.com www.schroff-europe.com
More informationCAD/CAM and the Exchange of Product Data
CAD/CAM and the Exchange of Product Data N.-J. Høimyr CERN, Geneva, Switzerland Abstract A 3D model defined in a CAD-system is used as a basis for design and product development. The concept of Computer
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 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 informationINVERSION AND PROBLEM OF TANGENT SPHERES
South Bohemia Mathematical Letters Volume 18, (2010), No. 1, 55-62. INVERSION AND PROBLEM OF TANGENT SPHERES Abstract. The locus of centers of circles tangent to two given circles in plane is known to
More informationUsing GeoGebra to create applets for visualization and exploration.
Handouts for ICTCM workshop on GeoGebra, March 2007 By Mike May, S.J. mikemaysj@gmail.com Using GeoGebra to create applets for visualization and exploration. Overview: I) We will start with a fast tour
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationPARAMETRIC MODELING. David Rosen. December 1997. By carefully laying-out datums and geometry, then constraining them with dimensions and constraints,
1 of 5 11/18/2004 6:24 PM PARAMETRIC MODELING David Rosen December 1997 The term parametric modeling denotes the use of parameters to control the dimensions and shape of CAD models. Think of a rubber CAD
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationVolumes of Revolution
Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students
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 informationSketcher. Preface What's New? Getting Started Basic Tasks Customizing Workbench Description Glossary Index
Sketcher Preface What's New? Getting Started Basic Tasks Customizing Workbench Description Glossary Index Dassault Systèmes 1994-99. All rights reserved. Preface CATIA Version 5 Sketcher application makes
More information1 Cubic Hermite Spline Interpolation
cs412: introduction to numerical analysis 10/26/10 Lecture 13: Cubic Hermite Spline Interpolation II Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Cubic Hermite
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 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 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 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 informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
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 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 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 informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
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, H-3515
More informationEL-9650/9600c/9450/9400 Handbook Vol. 1
Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations
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 informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationComputational Fluid Dynamics - CFD Francisco Palacios & Markus Widhalm fpalacios@gmail.com, markus.widhalm@dlr.de Basque Center for Applied
Computational Fluid Dynamics - CFD Francisco Palacios & Markus Widhalm fpalacios@gmail.com, markus.widhalm@dlr.de Basque Center for Applied Mathematics (BCAM). February 22nd to 26th, 2010 1 Grid generation
More informationAnalyzing Piecewise Functions
Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including
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 informationPro/ENGINEER Wildfire 4.0 Basic Design
Introduction Datum features are non-solid features used during the construction of other features. The most common datum features include planes, axes, coordinate systems, and curves. Datum features do
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 informationComputer Aided Systems
5 Computer Aided Systems Ivan Kuric Prof. Ivan Kuric, University of Zilina, Faculty of Mechanical Engineering, Department of Machining and Automation, Slovak republic, ivan.kuric@fstroj.utc.sk 1.1 Introduction
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
More information