1 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
2 TYPES OF CURVES AND THEIR MATHEMATICAL REPRESENTATIONS Wireframe Model ( 2D in 1960s for drafting, 3D in 1970s) Wireframe Entities analytic entities (points, lines, arcs and circles, fillets and chamfers, and conics synthetic entities (splines and Bezier curves) methods of defining points: P(x,y,z), P(r,,z), P(x+x,y+y,z+z),, end points of existing entity, center point, intersection of two entities. methods of defining lines: between two points, parallel to axis, parallel or perpendicular to a line, tangent to entity methods of defining arcs and circles: center and radius, three points, center and a point, a radius and tangent to a line passing through a point. methods of defining ellipses and parabolas: ellipse (center and axes lengths, four points, two conjugate diameters), parabola (vertex and focus, three points). methods of defining synthetic curves: cubic spline (a set of data points and end slopes), Bezier curves (a set of data points), B-spline curves (interpolate a set of data points with local control possible).
3 Curve Representation: Two types of representation are parametric and non-parametric representation. In parametric representation all variables (i.e., coordinates) are expressed in terms of common parameters. For example, a point can be expressed with respect to a parameter as P( u) [ x( u), y( u), z( u)], u u u min max Non-parametric representation is the conventional representation as P [ x, y( x), z( x)] Ex. Non-parametric form of a circle: x^2+y^2=r^2, parametric form: P(u)=[r cos 2u, r sin 2u], 0u1. This form can be used to find slopes at a certain angle for example.
4 PARAMETRIC REPRESENTATION OF ANALYTIC CURVES The following list shows most of the analytic curve that are used in CAD/CAM system for part design and modeling. Lines Circles Ellipses Parabolas Hyperbolas Conic Curves
5 LINE AND CIRCLE A line between two points P 1 and P 2 can be expressed with respect to a parameter. P P u( P 1 ) 1 2 P A circle for a center and the radius can be written as x x y y z z c c c R cosu R sin u, 0 u 2
6 ELLIPSE An ellipse with a center and major and minor axes of 2A and 2B can be expressed as. x y z x y z c c c Acosu B sin u, 0 u 2
9 EXAMPLE 1
10 EXAMPLE 2
12 EXAMPLE 1
13 CONICS The most general form of planar quadratic curves is conic curves or conic sections that include the previously covered curves; lines, circles, ellipses, parabolas, and hyperbolas. The general implicit nonparametric quadratic equation that describes the planar conic curve has five coefficients and naturally needs five conditions to complete it. The conic parametric equation can be described if five conditions are specified appropriately. One case is specifying five points on the curve. L1 0, L2 0, L3 0, L4 0 L1 L2 0, L3L4 0 L L 1 2 al3 L4 0
14 EXAMPLE 1
15 PARAMETRIC REPRESENTATION OF SYNTHETIC CURVES Hermite Cubic Splines Bezier Curves: cubic curve with four control points. B-Splines: general case of Bezier s curve (non-uniform) Rational Curves (algebraic ratio of two polynomials) NURB (non-uniform rational B-spline) curve combines all features of previous curves
16 HERMITE CUBIC SPLINES
22 BEZIER CURVES Another alternative to create curves is to use approximation techniques which produce curves that do not pass through the given data points that are rather used to control the shape of the curves. Approximation techniques arc more often preferred over interpolation techniques in curve design due to the added flexibility and the additional intuitive feel. Bezier curves and surfaces are credited to P. Bezier of the French car firm Regie Renault who developed (about 1962) and used them in his software system called UNISURF which has been used by designers to define the outer panels of several Renault cars.
26 B-SPLINES In contrast to Bezier curves, the theory of B-spline curves separates the degree of the resulting curve from the number of the given control points. The B-spline curve P(u) for the degree k defined by n + 1 control points n P( u) B ( u) P, 0 u u j0 j0 partition of unity n k k j B j ( u) 1 j max
27 otherwise otherwise u u u n j for otherwise u u u u B j j j j j, 0, 1,, 0, 1, ) ( the recursive property 1 ), ( ) ( ) ( k u B u u u u u B u u u u u B k j j k j k j k j j k j j k j open parametric knots otherwise k j n j k n k j u j, 1, 1 0, for 0 j n + k + 1
28 Knot Values number of knots = number of points + degree + 1 (n k = n+1+k+1 must be satisfied) The individual knot values are not meaningful by themselves; only the ratios of the difference between the knot values matter. To be able to use the parametric knots produced by the coordinates, the values must be in increasing order. Thus, if the curve intersects itself, this cannot be used. The number of duplicate values is limited to no more than the degree. Duplicate knot values make the b-spline curve less smooth. At the extreme, a full multiplicity knot in the middle of the knot list means there is a place on the b-spline curve that can be bent into a sharp kink.
29 Summary of B-spline Basis Function B j k a polynomial function of degree k nonnegative for all j and k (nonnegativity) non-zero only on [u j,u j+k+1 ] (local support) at most k+1 basis functions of degree k are non-zero on knot span [u j,u j+1 ], namely, B j-kk, B j-k+1k,, B j k sum of all degree k basis functions on knot span [u j,u j+1 ] is one (partition of unity) at a knot of multiplicity m, basis function B j k is C k-m continuous. a composite curve of degree k polynomials with joining knots in [u j,u j+k+1 ] n+1 control points, P 0,P 1,,P n n-2 basis functions (i.e., cubic polynomial curve segments, Q 3,Q 4,,Q n ) n-1 knot points, t 3,t 4,,t n+1 (any additional ones needed for computation are set to same values as the nearest) B j k is - defined over a knot interval [u j,u j+1 ] - defined by four of the control points, P j-3,,p j
30 Notable benefits of b-spline include: Independent degree: the degree is set by user and it has nothing to do with number of data points a single piecewise curve of a particular degree: there is no need to stitch together separate curves as in interpolation splines such as natural cubic spline
36 MCAD PROGRAMMING
40 UNIFORM QUADRATIC B-SPLINES
41 UNIFORM CUBIC B-SPLINES
42 RATIONAL CURVES
45 RATIONAL FORMS OF COMMON CURVES
48 NURBS Remember that the b-spline curve is weighted sum of its control points. n k P( u) P B ( u), 0 u u j0 j j max Since the weights depend on the knot vector only, it is useful to add another weight w j to every control point that can be set independently. The weights can be added as n w j B ( u) P( u) N ( u) P, N ( u), 0 u u j0 k j j k j n j0 w j k j B Here, N j k is NURBS basis functions. Thus, the NURBS curve of degree k is defined as weighted sum of control points with NURBS basis functions. k j ( u) max
49 Some of Valuable Properties of NURBS Curves and Surfaces invariant under affine as well as perspective transformations (affine invariance and projective invariance). single mathematical form for both analytical and free-form shapes. flexibility to design variety of shapes. less memory is used when storing shapes in comparison to other methods. numerical algorithms can evaluate them easily and quickly.
50 EXAMPLE Solution. For three control points n = 2 and for the second degree k=2. The uniform knots can be computed by the program in the B-spline example. The following shows result in MathCad with modified program for B jk. It now uses the knot vector as an input.
52 Shown below are various curves that vary with different weights. Note that the last shows violation of convex hull property with negative weight for midpoint.
53 CURVE MANIPULATIONS The cases that require manipulation of curves include: Displaying Evaluating Points on Curves Blending Segmentation Trimming Intersection Transformation
58 DESIGN APPLICATIONS Example 1. For the given state of stresses (a) determine the principal stresses and (b) state of stresses on a plane a-a. A Example 2. For a plane motion of a bar sliding down the step, find the locus of the point C that is located at one-third length from B. C B
59 TYPES OF SURFACES AND THEIR MATHEMATICAL REPRESENTATIONS Surface model is an extension of wireframe but has advantages: less ambiguous, provide realism for display with hidden lines, mesh, and shading. Surface Entities Plane surface Ruled (lofted) surface (surface created by two curves being blended) Surface of Revolution
60 Tabulated Cylinder (surface created by a curve and a vector) Bezier surface: only approximates the given data points permitting only global control B-spline: surface that can approximate and interpolate permitting local control Coons Patch: used to create a surface using curves that form closed boundaries in contrast to the above surfaces that use open boundaries or set of points. Fillet surface: B-spline surface that blends two surfaces together Free-form surface: formed by free-form curves that are extensions of Bezier, B-spline, and NURB curves.
67 BILINEAR SURFACE Bilinear surface is a linear interpolation of four corners in two different directions (u, v). P P u,0 u,1 ( u) P( u, v) ( u) (1 u) P (1 u) P (1 v) P 0,0 0,1 u,0 up up vp 1,0 1,1 u,1
68 RULED SURFACE
69 REVOLVED SURFACE
70 TABULATED CYLINDER
71 PARAMETRIC REPRESENTATION OF SYNTHETIC SURFACES
72 HERMITE BICUBIC SURFACES
76 BILINEAR SURFACE Bilinear surface is a linear interpolation of four corners in two different directions (u, v). P P u,0 u,1 ( u) P( u, v) ( u) (1 u) P (1 u) P (1 v) P 0,0 0,1 u,0 up up vp 1,0 1,1 u,1
77 COON S PATCH Bilinear surface generated by four corners has straight sides and produce quite flat surfaces. In contrast, Coon s surface uses four side curves. ), ( ), ( ), ( ) ( ) ( ) (1 ), ( ) ( ) ( ) (1 ), ( v u v u v u u v u v v u v u v u v u BT LR T B BT R L LR P P P P P P P P P The surface obtained as above does not produce the end curves. Thus, evaluating the surface on boundaries and forcing it to confirm to the boundary curves yields extra-terms that must be subtracted.
78 Evaluating the surface along edges one finds extra terms that must be subtraceted. P P LR BT ( u, v) ( u, v) P(0, v) (1 u) P (1 v) P P P P LR L L L B ( v) up ( u) vp (0, v) P BT ( v) (1 v) P ( v) ( u) (0, v) B R T (0) vp (0) T must ( v) (1 v) P0,0 vp0,1 PL ( v) On the left edge, the extra terms are identified as ( 1 v) P vp 0,0 0,1
79 Similarly, the other extra terms are found along the other edges and all surplus terms are surplus (1 v) P (1 v) P (1 u) P (1 u) P 0,0 1,0 0,0 0,1 vp vp up 0,1 1,1 up 1,0 1,1 P( u, v) P ( u, v) P ( u, v) surplus LR BT Advantages: Follows boundary curves Limitation: not able to control internal shape
80 EXAMPLE For given data along four edges, use cubic Bezier curves along the edges to create Coon s patch.
81 First find the curves along the edges. Their three components in (u,v) are L,R,B, and T refer to left, right, bottom, and top.
82 The plot is shown below. L,R,B, and T refer to left, right, bottom, and top.
83 The Coon s patch is then obtained as below and the plot is shown with data points.
84 BEZIER SURFACE
87 EXAMPLE [PROGRAMMING]
89 B-SPLINE AND NURBS SURFACES
90 EXAMPLE [PROGRAMMING] xp yp zp ,,, wt
96 OTHER SURFACES
97 SURFACE MANIPULATIONS As in the curve manipulations, the surfaces have to be manipulated for various reasons in the CAD designs. Some of the applications are
99 DESIGN AND ENGINEERING APPLICATIONS
100 SOLIDS AND THEIR REPRESENTATIONS Solid Model is based on informationally complete (or spatial addressability), valid, and unambiguous representation of objects and stores geometric data as well as topological information of associated objects. This representation permits automation and integration of tasks such as interference analysis, mass property calculation, finite element modeling, CAPP (computer-aided process planning), machine vision, and NC machining. It is very easy to define an object with a solid model than other two previous modeling techniques (curves and surfaces) because solid models do not need individual locations as with wireframe models.
101 GEOMETRY AND TOPOLOGY The above figure illustrates the difference between geometry and topology. The geometry that defines the object is the lengths of lines, areas of surfaces, the angles between the lines, and the radius and the center of the cylinder and the height. On the other hand, topology (sometimes called combinatorial structure), is the connectivity and associativity of the object entities. It has to do with the notion of neighborhood and determines the relational information between object entities. From a user point of view, geometry is visible and topology is considered to be nongraphical relational information that is stored in solid model databases and are not visible to users.
102 SOLID ENTITIES There are various basic building blocks, so called, primitives that can be combined in certain boolean operations to construct complex models. They include: block cylinder cone sphere wedge Torus In the previous figure, a block and a cylinder were combined with union (i.e., addition) and difference (i.e., subtraction). One more available Boolean operation is intersection.
103 Example Explain how to construct the solid model of the bearing support with primitives and boolean operations. EXAMPLE
104 SOLID REPRESENTATION Underlying fundamentals of solid modeling theory are geometry, topology, geometric closure, set theory, regularization of set operations, set membership classification, and neighborhood. Solid representation is based on the notion that a physical object divides an n-dimensional space, En, into two regions: interior and exterior separated by the boundaries. A region is a portion of space En and the boundary of a region is a closed surface. The set in the set theory is a collection of objects and the operations include complement, union, and intersection. Further, regularized set operations are used to avoid irregular object created by boolean operations. Regular set is a geometrically closed set. The set membership classification determines if some objects intersect with a given object.
105 Half-spaces are unbounded geometric entities; each one of them divides the representation space into two infinite portions, one filled with material and the other empty. By combining half-spaces in a building block fashion, various solids can be constructed. Thus, a solid model of an object can be defined as a point set S in threedimensional Euclidean space E3. If three sets for a region is denoted by Si (interior set), Se (exterior set), and Sb (boundary set), then the set for the solid model can be expressed as
106 The mathematical properties that the solid model should capture can be stated as: Rigidity Homogeneous three-dimensionality (ie. no dangling boundaries) Finiteness and finite describability Closure under rigid motion and regularized boolean operations Boundary determinism (ie. Boundary must contain the solid) The mathematical implication of the above properties suggests that valid solid models are bounded, closed, regular, and semi-analytic subsets of E3. These subsets are called r-sets (ie. regularized sets), which are curved polyhedra with well-behaved boundaries. Here, regular means no dangling portion and semi-analytic means no oscillation in value.
107 FUNDAMENTALS OF SOLID MODELING Various solid representation schemes require several underlying fundamentals of solid modeling theory. They include geometry, topology, geometric closure, set theory, regularization of set operations, set membership classification, and neighborhood. For detailed discussion of these topics, the readers are encouraged to refer to advanced topics on solid modeling. In the following some of most popular solid modeling techniques are discussed.
108 MODELING TECHNIQUES Boundary Representation (B-rep): A B-rep model is one of the two most popular and widely used schemes and is based on the topological notion that a physical object is bounded by a set of faces. The following shows various faceted B-rep solids.
109 Constructive Solid Geometry (CSG): This is based on the topological notion that a physical object can be divided into a set of primitives that can be combined in a certain order following a set of rules (i.e. Boolean operations) to form the object. The basic elements are block, cylinder, cone, sphere, wedge, and torus and building operations are Boolean operations. The following bearing support was constructed with various primitives in a certain sequence by CSG technique.
110 Sweep Representation: This is especially useful for two-and-half dimensional objects used most frequently for extruded solids and revolved solids. This is based on sweeping of a section along a path that may be linear, nonlinear, and hybrid operations. When the path is straight, it s a simple extrusion and when it is axisymmetric, it becomes the revolution. For nonlinear path, the sweeping is done along a nonlinear curve in space. Cutting tool path simulation is one good applications of this technique. The section may vary along the sweeping path. The following shows such an example with variable section.
111 Analytic Solid Modeling (ASM): Historically it is closely related to three dimensional isoparametric formulation of finite element analysis for 8- to 20-node hexahedral elements. This arose from the need to model complex objects for finite element analysis. ASM uses the parametric representation of an object in three-dimensional space that is a mapping of a cubical parametric domain (so called, master domain) into a solid described by the global coordinates (MCS).
112 Other Representations: primitive instancing: is based on notion of families of objects or family of parts. cell decomposition scheme: an object can be represented as the sum of cells. spatial enumeration scheme: a solid is represented by the sum of spatial cells that it occupies. The cells have fixed size. octree encoding scheme: is generalization of spatial enumeration scheme with variable cell size.
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
CHAPTER 6 SOLID MODELING 6.1 Application of Solid Models In mechanical engineering, a solid model is used for the following applications: 1. Graphics: generating drawings, surface and solid models 2. Design:
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
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 right-hand rule.
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
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
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
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
REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit
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:
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
Parametric Representation of Crves and Srfaces How does the compter compte crves and srfaces? MAE 455 Compter-Aided Design and Drafting Types of Crve Eqations Implicit Form Explicit Form Parametric Form
Comal Independent School District Pre- Pre-Calculus Scope and Sequence Third Quarter Assurances. The student will plot points in the Cartesian plane, use the distance formula to find the distance between
On-Line 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
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
CSCI 420 Computer Graphics Lecture 8 and Constructive Solid Geometry [Angel Ch. 10] Jernej Barbic University of Southern California Modeling Complex Shapes An equation for a sphere is possible, but how
Chapter 9 Working with Wireframe and Surface Design Learning Objectives After completing this chapter you will be able to: Create wireframe geometry. Create extruded surfaces. Create revolved surfaces.
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.
2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication
Voronoi Diagrams 4 A city builds a set of post offices, and now needs to determine which houses will be served by which office. It would be wasteful for a postman to go out of their way to make a delivery
Training Objective After watching the program and reviewing this printed material, the viewer will understand the basic concepts of computer-aided design, or CAD. The relationship between 2-dimensional
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
3D Design from Concept to Completion in AutoCAD David Cohn 4D Technologies AC6705-V Modeling in 3D can be easy. Learn how to use new and enhanced AutoCAD software functionality to apply your 2D AutoCAD
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
An introduction to 3D draughting & solid modelling using AutoCAD Faculty of Technology University of Plymouth Drake Circus Plymouth PL4 8AA These notes are to be used in conjunction with the AutoCAD software
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
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.
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
Introduction to SolidWorks Software Marine Advanced Technology Education Design Tools What is SolidWorks? SolidWorks is design automation software. In SolidWorks, you sketch ideas and experiment with different
Chapter 3 Drawing Sketches in the Sketcher Workbench-II Learning Objectives After completing this chapter, you will be able to: Draw ellipses. Draw splines. Connect two elements using an arc or a spline.
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 1-1: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 1-2: Algebraic Expressions
Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Welcome everybody. We continue the discussion on 2D
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
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
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
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
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect
(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much
Graphical objects geometrical data handling inside DWG file Author: Vidmantas Nenorta, Kaunas University of Technology, Lithuania, firstname.lastname@example.org Abstract The paper deals with required (necessary)
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 xz-plane, the intersection is + z 2 = 1 on a plane,
1 P a g e Mathematics Notes for Class 12 chapter 12. Linear Programming Linear Programming It is an important optimization (maximization or minimization) technique used in decision making is business and
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
5 Computer Aided Systems Ivan Kuric Prof. Ivan Kuric, University of Zilina, Faculty of Mechanical Engineering, Department of Machining and Automation, Slovak republic, email@example.com 1.1 Introduction
Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions
Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
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:
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
CS155b Computer Graphics Instructor: Giovanni Motta (firstname.lastname@example.org) Volen, Room #255. Phone: x62718 Class: Mon. and Wed. from 5 to 6:30pm Abelson #131 Teaching Assistants: Anthony Bucci (abucci@cs) John
1. Physical world and measurement 2. Kinematics 3. Laws of Motion 4. Work Energy Power 5. Motion of System of Particles & Rigid body 1. Physical world and measurement 2. Kinematics 3. Laws of Motion 4.
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
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,
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
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
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
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
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
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
REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working
Able Enrichment Centre - Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
Spring 2004 1 Course Description AE4375: advanced treatment for undergrads; focus on learning and aplying CAD to engineering; CAD modeling projects. AE6380: graduate course on CAD focusing on how tools
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,
COURSE OVERVIEW The geometry course is centered on the beliefs that The ability to construct a valid argument is the basis of logical communication, in both mathematics and the real-world. There is a need
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
Seminar Path planning using Voronoi diagrams and B-Splines Stefano Martina email@example.com 23 may 2016 This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International
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
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
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
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)
Surfacing using Pro/ENGINEER Wildfire 5.0 Overview Course Code TRN-2239 T Course Length 3 Days In this course, you will learn how to use various techniques to create complex surfaces with tangent and curvature
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
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