Lecture 2.2 Cubic Splines
|
|
- Clifton Nichols
- 5 years ago
- Views:
Transcription
1 Lecture. Cubic Splines
2 Cubic Spline The equation for a single parametric cubic spline segment is given by 4 i t Bit t t t i (..) where t and t are the parameter values at the beginning and end of the segment. (t) is the position vector of any point on the cubic spline segment. Let and be the position vectors at the end points of the spline segment (see Fig...). Fig.. Schematic of a spline segment
3 (t) = [x(t) y(t) z(t)] is a vector valued function. The three components of (t) are the Cartesian coordinates of the position vector. Each component has a similar formulation to (t), i.e., 4 i x t Bixt t t t i 4 i y t Biyt t t t i 4 i z t Bizt t t t i The coefficients B i are determined by specifying four boundary conditions for the spline segment. Expanding Eq. (..) gives 3 t B Bt B3t B4t t t t (..) 3
4 Also let and, derivatives with respect to t, be the tangent vectors at the end points of the spline segment, refer Fig.... Differentiating Eq. (..) yields 4 ' [ ' ' ' ] i i i t x t y t z t B i t t t t (..3) Expanding Eq. (..3) gives t B B t 3B t t t t ' 3 4 (..4) Assuming, without loss of generality, that t = 0, and applying the four boundary conditions, (0) = (..5a) (t ) = (..5b) (0) = (..5c) (t ) = (..5d) 4
5 Using the four Eqns. (..5), the four unknown coefficients B i, i =,,3,4 can be evaluated. Specifically for i =,, 3, 4 (0) = B = 4 ' ' 0 i i i t 0 i t B B (..6a) (..6b) 4 i 3 i 3 4 i t t t B t B B t B t B t (..6c) 4 i i i t t t i t B B B t B t (..6d) 5
6 Solving for B 3 and B 4 yields and B 3 ' ' 3 t t t B ' ' 4 3 t t t (..7a) (..7b) B, B, B 3 and B 4 determine the cubic spline completely. This completes the computation of the coefficients for one coordinate (x or y or z) of a spline segment. Notice that the value of the parameter t = t at the end of the segment occurs in the results. Since each of the end position and tangent vectors has three components, the parametric equation and hence the shape of the cubic space curve depends on twelve vector components and the parameter value t at the ends of the 6 segment.
7 Substituting (..6) and (..7) into (..) gives the required cubic spline: 3 t t t t ' ' ' ' ' 3 3 t t t t t t Equation (..8) represents a single cubic spline segment. (..8) However, to represent a complete curve, multiple segments are joined together. Two adjacent segments are shown in Figs... and..3, one with points, and 3 and the other with generalized points k, k+ and k+. Referring to Fig..., assume that the position vectors,, 3, the tangent vectors,, 3 and the parameter values t, t 3 are known. Eq. (..8), applied to each of the two segments yields their shapes. 7
8 Fig.. Multiple segments However, it is unlikely that the tangent vector at the internal joint between the two segments is known. The internal tangent vector can be determined by imposing a continuity condition of the derivation at the internal joint. Recall that a piecewise spline of degree K has continuity of order (K-) at the internal joints. Thus, a cubic spline has second order continuity at the internal joints. This means that the second derivative (t) is continuous across the joints; i.e., the curvature is also continuous across the joint. 8
9 Differentiating Eq. (..) twice gives 4 i 3 t i i Bit t t t i 3 (..9) Noting that for the first cubic spline segment the parameter range is 0 t t,evaluating Eq. (..9) at the end of the segment where t = t gives 6B4t B3 (..9a) For the second cubic spline segment the parameter range is t t t 3. Evaluating Eq. (..9) at the beginning of this second segment, where t = 0, yields B 3 (..9a) 9
10 Equating (..9a) and (..9b) and then using (..6a, b) and (..7a) gives 3 6t ' ' ' ' 3 t t t t t t 3 ' ' 3 3 t3 t3 t3 Here the left hand side of the equation represents the curvature at the end of the first segment and the right hand side is the curvature at the beginning of the second segment. Multiplying by t t 3 and simplifying gives 3 t t t t t t ' ' ' tt 3 (..0) 0
11 Equation (..0) can be solved for, the unknown tangent vector at the internal joint. Again notice that the end values of the parameter t, i.e., t and t 3, occur in the resulting equation. These results can be generalized for n data points to give n piecewise cubic spline segments with position, slope and curvature, i.e., C continuity at the internal joints. Now, using the notation shown in Fig...3 the generalized equation for any two adjacent cubic spline segments k (t) and k+ (t) can be written as follows: 3 t t t ' ' ' k k k k k k k tk tk tk ' ' k k k k 3 t 3 k k k (..) t t t
12 for the first segment, and 3 t t t ' ' ' k k k k k k k tk tk tk ' ' k k k k 3 t 3 tk tk tk (..) for the second segment. Recall that the parameter range begins at zero for each segment, for the first segment 0 t t k+ and for the second 0 t t k+. For any two adjacent spline segments, the second derivatives at the common internal joint are equated. In other words, letting k (t k ) = k+ (0), yields the generalized result, equivalent to Eq. (..0).
13 It is given by t t t t ' ' ' k k k k k k k t k 3 t k t t k n k k k k k k (..3) Equation (..3) is used for determining the tangent vector at the internal joint between any two spline segments k and k+. Applying Eq. (..3) recursively over all the spline segments yields (n ) equations for the determination of tangent vectors k, k n. 3
14 In matrix form, the result is t t t t t t t t 0... ' ' ' 3 ' n t t t t t t t t.. t n 3 t 3 tt 3 tt n t t 3 3 t t t n n n... t n n n n n n n (..4) or [M*][ ] = [R] 4
15 Since there are only n equations for the n tangent vectors, [M*] is not square and thus cannot be inverted to obtain the solution for [ ]; i.e., the problem is indeterminate. By assuming that the end tangent vectors and n are known, the problem becomes determinant. The matrix formulation is now [M][ ] = [R] (..5) Here, [M] is a square matrix and invertible. Notice also that [M] is tri-diagonal, which reduces the computational work required to invert it. Further, [M] is diagonally dominant. Hence it is nonsingular, and inversion yields a unique solution. 5
16 The equation (..5) can be expanded as t t t t t t t t 0... ' ' ' 3 ' n t t t t 0.. ' t t t t n n n n t n ' n 3 t 3 tt 3 tt n t t t t... 3 t t n n n n n n 6
17 The solution for [ ] is thus [ ] = [M] - [R] (..6) Once the k s are known, the coefficients (B) for each spline segment can be determined. Generalizing Eqs. (..6) (..) gives B B k k 3 ' ' k k k k 3k tk tk tk B 3 B ' ' k k k k 4k 3 tk tk tk k ' k 7
18 Since k and k are vector valued, B i are also vector valued. That is, if the k and k have x, y, z components then the B i also have the corresponding x, y, z components. In the matrix form, the equations for B for any spline segment k are: B B B B B k k 3k 4k ' 3 3 k k k k k k t t t t t t t t 3 3 k k k k ' k k (..7) 8
19 To generate a piecewise cubic spline through n given position vectors k, k n, with end tangent vectors and n, Eq. (..6) is used to determine the internal tangent vectors k, k n. Then for each piecewise cubic spline segment the end position and tangent vectors for that segment are used to determine the B ik s, i 4 for that segment using Eq. (..7). Finally the generalization of Eq. (..) gives 4 i k ik k i t B t 0 t t, k n (..8) Now, Eq. (..8) can be used to determine points on the spline segment. 9
20 In matrix form Eq. (..8) becomes 3 B k k t t t t 0 t t B Substituting Eq. (..7) and rearranging yields B B k 3k 4k k (..9) T F T F T F T F T k k 3 4 ' k ' k 0 T 0 k n (..0) Eq. (..0) is written in matrix form as k (T) = [F][G] (..) k 0
21 F is called a matrix of blending or weighing functions and G is a matrix of geometric conditions. T t t k F T T T k k 3 3 F T T 3T 3 F T T T T t 3k k F T T T T t 4k k (..a) (..b) (..c) (..d) The form of Eq. (..0) frequently appears in curve and surface descriptions.
22 Generation of Cascade Geometries Figure..3 shows an example of a turbomachinery cascade. All the boundaries of a typical block consisting of two straight line segments L and L and three curves C, C and C 3. In Module it has been shown that the flow through turbomachinery passages is generally studied through cascade models. Two-dimensional cascades of stator and rotor rows at the midspan of turbine stages are introduced here as physical models relevant to the problem of turbine rotor-stator interactions. The blade profiles determine the turbomachinery passages with high turning of flow through them. For the rotor blades the physical models need to be developed for use in relative motion, capable of giving accurate and computationally efficient numerical solutions.
23 Fig...3 Boundaries of a typical computational block 3
24 The model represented in Fig...4 considers that the stator and rotor with equal pitches. Therefore the computational domain, shown with solid lines in Fig...4, consists of single passages of stator and rotor with periodic boundaries. However the periodic boundaries get extended over multiple passages when the stator and rotor pitches are unequal. For example, as the stator to rotor pitch ratio is.5 in the model presented in Fig...5, two stator and three rotor passages are taken for computation. In Fig...4 the periodic pairs of the boundary are curves {C and C } for the stator and {C 3 and C 4 } for the rotor. Similarly the periodic pairs in Fig...5 are {C and C }, {C 3 and C 4 } and {C 5 and C 6 } for the first-stage stationary row, first-stage rotating row and second-stage stationary row to enable consideration of a multi-stage problem.
25 Fig...4 The physical model for the analysis of a turbine stage (Equal stator and rotor pitches) 5
26 Fig...5 The physical model for the analysis of a turbine stage (Unequal stator and rotor pitches) 6
27 Generation of Blade rofiles Using Splines It is shown that line segments of arbitrary shaped curve can be generated by specifying a set of several coordinate points. However, for obtaining smooth and accurate arbitrary curve such as the blade profile, the coordinates of a large number of points on the curve are required. They are found from a limited number of known points on the curve by using spline fitting techniques. The nodes are then selected from the large number of coordinates obtained by spline fitting.
28 As the spacing between nodes on the boundary curve varies along the curve, the curve is first divided into segments, say six, as shown in Fig...6 (a). The number and distribution of nodes placed on each boundary segment varies depending on the grid density distribution within the domain. For the blade profile shown in Fig...6, it is required to have fine grids near the leading and trailing edges. Therefore, the distance between two adjacent nodes at the leading edge and trailing edge regions should be reducing towards the respective forward and rear stagnation points. 8
29 Fig...6 Boundary discretization of an airfoil (a) Subdivision of an airfoil 9
30 For providing such a distribution of nodes, a simple and generalised method is used by adopting the following exponential stretching function. i L e S N i (..3) Here δ i is the distance along the curve of the i th point from the starting point of the curve, L is the length of the curve, N is the total number of divisions required and S is the stretching factor. The purpose of the stretching factor is to vary the distance between two adjacent nodes. A negative value for S leads to close nodes near the starting point of the curve and vice versa. Large values of S leads to increase in stretching, very small values of S (say, S < 0.00) results in almost uniform division of 30 the curve. N e S
31 Figure..6 (b) shows the pattern of discretization of the segment-i of the curve obtained using a stretching factor S = and N = 0. Figure..6 (c) shows the distribution of points on segment-iv of the curve obtained using a positive value of for S and with N = 0. It can be seen that close points are obtained near the end of the curve. The procedure described for boundary discretization and generation of large number of data points on the boundary by spline fitting is again used for the purpose of grid generation of an aerofoil. Figure..6(d) shows the discretization of the entire aerofoil. 3
32 Fig...6 Boundary discretization of an airfoil (b) Discretization of segment-i (c) Discretization of segment-iv 3
33 Fig...6 Boundary discretization of an airfoil (d) The fully discretized airfoil 33
34 Summary of Lecture. Cubic spline is one of the popular technique used for generating space curves. The procedure for generating spline curves is illustrated. END OF LECTURE. 34
Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur
Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur Email: jrkumar@iitk.ac.in Curve representation 1. Wireframe models There are three types
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More information2D Spline Curves. CS 4620 Lecture 13
2D Spline Curves CS 4620 Lecture 13 2008 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes [Boeing] that is, without discontinuities So far we can make things with corners
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationInteractive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More informationLecture 9: Introduction to Spline Curves
Lecture 9: Introduction to Spline Curves Splines are used in graphics to represent smooth curves and surfaces. They use a small set of control points (knots) and a function that generates a curve through
More informationRepresenting Curves Part II. Foley & Van Dam, Chapter 11
Representing Curves Part II Foley & Van Dam, Chapter 11 Representing Curves Polynomial Splines Bezier Curves Cardinal Splines Uniform, non rational B-Splines Drawing Curves Applications of Bezier splines
More informationIntro to Curves Week 4, Lecture 7
CS 430/536 Computer Graphics I Intro to Curves Week 4, Lecture 7 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationCurves and Surfaces 1
Curves and Surfaces 1 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2 The Teapot 3 Representing
More informationBezier Curves, B-Splines, NURBS
Bezier Curves, B-Splines, NURBS Example Application: Font Design and Display Curved objects are everywhere There is always need for: mathematical fidelity high precision artistic freedom and flexibility
More informationCurves and Surfaces Computer Graphics I Lecture 9
15-462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] February 19, 2002 Frank Pfenning Carnegie
More informationCOMPUTER AIDED ENGINEERING DESIGN (BFF2612)
COMPUTER AIDED ENGINEERING DESIGN (BFF2612) BASIC MATHEMATICAL CONCEPTS IN CAED by Dr. Mohd Nizar Mhd Razali Faculty of Manufacturing Engineering mnizar@ump.edu.my COORDINATE SYSTEM y+ y+ z+ z+ x+ RIGHT
More informationCurves and Surfaces Computer Graphics I Lecture 10
15-462 Computer Graphics I Lecture 10 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] September 30, 2003 Doug James Carnegie
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationFour equations are necessary to evaluate these coefficients. Eqn
1.2 Splines 11 A spline function is a piecewise defined function with certain smoothness conditions [Cheney]. A wide variety of functions is potentially possible; polynomial functions are almost exclusively
More informationLECTURE #6. Geometric Modelling for Engineering Applications. Geometric modeling for engineering applications
LECTURE #6 Geometric modeling for engineering applications Geometric Modelling for Engineering Applications Introduction to modeling Geometric modeling Curve representation Hermite curve Bezier curve B-spline
More information2D Spline Curves. CS 4620 Lecture 18
2D Spline Curves CS 4620 Lecture 18 2014 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes that is, without discontinuities So far we can make things with corners (lines,
More informationIntro to Curves Week 1, Lecture 2
CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University Outline Math review Introduction to 2D curves
More informationCS 536 Computer Graphics Intro to Curves Week 1, Lecture 2
CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 Outline Math review Introduction to 2D curves
More informationCSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016
CSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016 Announcements Project 3 due tomorrow Midterm 2 next
More informationCSE 167: Introduction to Computer Graphics Lecture 12: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2013
CSE 167: Introduction to Computer Graphics Lecture 12: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2013 Announcements Homework assignment 5 due tomorrow, Nov
More information(Refer Slide Time: 00:02:24 min)
CAD / CAM Prof. Dr. P. V. Madhusudhan Rao Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture No. # 9 Parametric Surfaces II So these days, we are discussing the subject
More informationCentral issues in modelling
Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction includes: manual modelling; fitting to
More informationSplines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationCurves. Computer Graphics CSE 167 Lecture 11
Curves Computer Graphics CSE 167 Lecture 11 CSE 167: Computer graphics Polynomial Curves Polynomial functions Bézier Curves Drawing Bézier curves Piecewise Bézier curves Based on slides courtesy of Jurgen
More informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More informationChapter 18. Geometric Operations
Chapter 18 Geometric Operations To this point, the image processing operations have computed the gray value (digital count) of the output image pixel based on the gray values of one or more input pixels;
More informationGraphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationCOMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationTO DUY ANH SHIP CALCULATION
TO DUY ANH SHIP CALCULATION Ship Calculattion (1)-Space Cuvers 3D-curves play an important role in the engineering, design and manufature in Shipbuilding. Prior of the development of mathematical and computer
More informationModule 3 Mesh Generation
Module 3 Mesh Generation 1 Lecture 3.1 Introduction 2 Mesh Generation Strategy Mesh generation is an important pre-processing step in CFD of turbomachinery, quite analogous to the development of solid
More informationCSE 167: Introduction to Computer Graphics Lecture #13: Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2017
CSE 167: Introduction to Computer Graphics Lecture #13: Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2017 Announcements Project 4 due Monday Nov 27 at 2pm Next Tuesday:
More informationComputergrafik. Matthias Zwicker Universität Bern Herbst 2016
Computergrafik Matthias Zwicker Universität Bern Herbst 2016 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling 2 Piecewise Bézier curves Each
More informationYear 6 Mathematics Overview
Year 6 Mathematics Overview Term Strand National Curriculum 2014 Objectives Focus Sequence Autumn 1 Number and Place Value read, write, order and compare numbers up to 10 000 000 and determine the value
More informationComputergrafik. Matthias Zwicker. Herbst 2010
Computergrafik Matthias Zwicker Universität Bern Herbst 2010 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling Piecewise Bézier curves Each segment
More informationCS 450 Numerical Analysis. Chapter 7: Interpolation
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationAdvanced Turbomachinery Methods. Brad Hutchinson ANSYS, Inc. Industry Marketing
Advanced Turbomachinery Methods 1 Brad Hutchinson ANSYS, Inc. Industry Marketing Presentation Overview 1. Turbomachinery challenges 2. ANSYS TurboSystem 3. 2 Blade row fluid dynamics solution methods Available
More informationThe Free-form Surface Modelling System
1. Introduction The Free-form Surface Modelling System Smooth curves and surfaces must be generated in many computer graphics applications. Many real-world objects are inherently smooth (fig.1), and much
More information1 Trajectories. Class Notes, Trajectory Planning, COMS4733. Figure 1: Robot control system.
Class Notes, Trajectory Planning, COMS4733 Figure 1: Robot control system. 1 Trajectories Trajectories are characterized by a path which is a space curve of the end effector. We can parameterize this curve
More informationIntroduction to Computer Graphics
Introduction to Computer Graphics 2016 Spring National Cheng Kung University Instructors: Min-Chun Hu 胡敏君 Shih-Chin Weng 翁士欽 ( 西基電腦動畫 ) Data Representation Curves and Surfaces Limitations of Polygons Inherently
More informationNew Swannington Primary School 2014 Year 6
Number Number and Place Value Number Addition and subtraction, Multiplication and division Number fractions inc decimals & % Ratio & Proportion Algebra read, write, order and compare numbers up to 0 000
More informationParametric Curves. University of Texas at Austin CS384G - Computer Graphics
Parametric Curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Parametric Representations 3 basic representation strategies: Explicit: y = mx + b Implicit: ax + by + c
More informationDriven Cavity Example
BMAppendixI.qxd 11/14/12 6:55 PM Page I-1 I CFD Driven Cavity Example I.1 Problem One of the classic benchmarks in CFD is the driven cavity problem. Consider steady, incompressible, viscous flow in a square
More informationUnit 3 Higher topic list
This is a comprehensive list of the topics to be studied for the Edexcel unit 3 modular exam. Beside the topics listed are the relevant tasks on www.mymaths.co.uk that students can use to practice. Logon
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 36
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 36 In last class, we have derived element equations for two d elasticity problems
More informationB-spline Curves. Smoother than other curve forms
Curves and Surfaces B-spline Curves These curves are approximating rather than interpolating curves. The curves come close to, but may not actually pass through, the control points. Usually used as multiple,
More informationFor each question, indicate whether the statement is true or false by circling T or F, respectively.
True/False For each question, indicate whether the statement is true or false by circling T or F, respectively. 1. (T/F) Rasterization occurs before vertex transformation in the graphics pipeline. 2. (T/F)
More informationNumber and Place Value
Number and Place Value Reading and writing numbers Ordering and comparing numbers Place value Representing and estimating numbers Rounding numbers Counting Finding other numbers Solving problems Roman
More informationNumber- Algebra. Problem solving Statistics Investigations
Place Value Addition, Subtraction, Multiplication and Division Fractions Position and Direction Decimals Percentages Algebra Converting units Perimeter, Area and Volume Ratio Properties of Shapes Problem
More informationFOUNDATION HIGHER. F Autumn 1, Yr 9 Autumn 2, Yr 9 Spring 1, Yr 9 Spring 2, Yr 9 Summer 1, Yr 9 Summer 2, Yr 9
Year: 9 GCSE Mathematics FOUNDATION F Autumn 1, Yr 9 Autumn 2, Yr 9 Spring 1, Yr 9 Spring 2, Yr 9 Summer 1, Yr 9 Summer 2, Yr 9 HIGHER Integers and place value Decimals Indices, powers and roots Factors,multiples
More informationParametric Curves. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Parametric Curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Parametric Representations 3 basic representation strategies: Explicit: y = mx + b Implicit: ax + by + c
More informationFast marching methods
1 Fast marching methods Lecture 3 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 Metric discretization 2 Approach I:
More informationGeometric Modeling of Curves
Curves Locus of a point moving with one degree of freedom Locus of a one-dimensional parameter family of point Mathematically defined using: Explicit equations Implicit equations Parametric equations (Hermite,
More informationMath background. 2D Geometric Transformations. Implicit representations. Explicit representations. Read: CS 4620 Lecture 6
Math background 2D Geometric Transformations CS 4620 Lecture 6 Read: Chapter 2: Miscellaneous Math Chapter 5: Linear Algebra Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector
More informationNatural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
More informationLecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10
Lecture 25: Bezier Subdivision And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 1. Divide and Conquer If we are going to build useful
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More information7 Fractions. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability
7 Fractions GRADE 7 FRACTIONS continue to develop proficiency by using fractions in mental strategies and in selecting and justifying use; develop proficiency in adding and subtracting simple fractions;
More informationChapter 7 Practical Considerations in Modeling. Chapter 7 Practical Considerations in Modeling
CIVL 7/8117 1/43 Chapter 7 Learning Objectives To present concepts that should be considered when modeling for a situation by the finite element method, such as aspect ratio, symmetry, natural subdivisions,
More informationNumber Mulitplication and Number and Place Value Addition and Subtraction Division
Number Mulitplication and Number and Place Value Addition and Subtraction Division read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to
More informationPO 2. Identify irrational numbers. SE/TE: 4-8: Exploring Square Roots and Irrational Numbers, TECH: itext; PH Presentation Pro CD-ROM;
Arizona Mathematics Standards Articulated by Grade Level Strands 1-5, Performance Objectives (Grade 8) STRAND 1: NUMBER SENSE AND OPERATIONS Concept 1: Number Sense Locate rational numbers on a number
More informationAQA GCSE Maths - Higher Self-Assessment Checklist
AQA GCSE Maths - Higher Self-Assessment Checklist Number 1 Use place value when calculating with decimals. 1 Order positive and negative integers and decimals using the symbols =,, , and. 1 Round to
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2015 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationCS770/870 Spring 2017 Curve Generation
CS770/870 Spring 2017 Curve Generation Primary resources used in preparing these notes: 1. Foley, van Dam, Feiner, Hughes, Phillips, Introduction to Computer Graphics, Addison-Wesley, 1993. 2. Angel, Interactive
More informationGeometric Transformations
Geometric Transformations CS 4620 Lecture 9 2017 Steve Marschner 1 A little quick math background Notation for sets, functions, mappings Linear and affine transformations Matrices Matrix-vector multiplication
More informationRational Bezier Curves
Rational Bezier Curves Use of homogeneous coordinates Rational spline curve: define a curve in one higher dimension space, project it down on the homogenizing variable Mathematical formulation: n P(u)
More informationECE 600, Dr. Farag, Summer 09
ECE 6 Summer29 Course Supplements. Lecture 4 Curves and Surfaces Aly A. Farag University of Louisville Acknowledgements: Help with these slides were provided by Shireen Elhabian A smile is a curve that
More information8 Piecewise Polynomial Interpolation
Applied Math Notes by R. J. LeVeque 8 Piecewise Polynomial Interpolation 8. Pitfalls of high order interpolation Suppose we know the value of a function at several points on an interval and we wish to
More informationLinear Equation Systems Iterative Methods
Linear Equation Systems Iterative Methods Content Iterative Methods Jacobi Iterative Method Gauss Seidel Iterative Method Iterative Methods Iterative methods are those that produce a sequence of successive
More informationform. We will see that the parametric form is the most common representation of the curve which is used in most of these cases.
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 36 Curve Representation Welcome everybody to the lectures on computer graphics.
More informationInformation Coding / Computer Graphics, ISY, LiTH. Splines
28(69) Splines Originally a drafting tool to create a smooth curve In computer graphics: a curve built from sections, each described by a 2nd or 3rd degree polynomial. Very common in non-real-time graphics,
More informationShape Representation Basic problem We make pictures of things How do we describe those things? Many of those things are shapes Other things include
Shape Representation Basic problem We make pictures of things How do we describe those things? Many of those things are shapes Other things include motion, behavior Graphics is a form of simulation and
More informationIntro to Modeling Modeling in 3D
Intro to Modeling Modeling in 3D Polygon sets can approximate more complex shapes as discretized surfaces 2 1 2 3 Curve surfaces in 3D Sphere, ellipsoids, etc Curved Surfaces Modeling in 3D ) ( 2 2 2 2
More informationY6 MATHEMATICS TERMLY PATHWAY NUMBER MEASURE GEOMETRY STATISTICS
Autumn Number & Place value read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to a required degree of accuracy use negative numbers in
More information3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More informationFunctions and Transformations
Using Parametric Representations to Make Connections Richard Parr T 3 Regional, Stephenville, Texas November 7, 009 Rice University School Mathematics Project rparr@rice.edu If you look up parametric equations
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 24
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 24 So in today s class, we will look at quadrilateral elements; and we will
More information8. BASIC TURBO MODEL WITH UNSTRUCTURED MESH
8. BASIC TURBO MODEL WITH UNSTRUCTURED MESH This tutorial employs a simple turbine blade configuration to illustrate the basic turbo modeling functionality available in GAMBIT. It illustrates the steps
More informationB-Splines and NURBS Week 5, Lecture 9
CS 430/585 Computer Graphics I B-Splines an NURBS Week 5, Lecture 9 Davi Breen, William Regli an Maxim Peysakhov Geometric an Intelligent Computing Laboratory Department of Computer Science Drexel University
More informationBeams. Lesson Objectives:
Beams Lesson Objectives: 1) Derive the member local stiffness values for two-dimensional beam members. 2) Assemble the local stiffness matrix into global coordinates. 3) Assemble the structural stiffness
More informationCurves, Surfaces and Recursive Subdivision
Department of Computer Sciences Graphics Fall 25 (Lecture ) Curves, Surfaces and Recursive Subdivision Conics: Curves and Quadrics: Surfaces Implicit form arametric form Rational Bézier Forms Recursive
More informationShape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011
CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell 2/15/2011 1 Motivation Geometry processing: understand geometric characteristics,
More informationBirkdale High School - Higher Scheme of Work
Birkdale High School - Higher Scheme of Work Module 1 - Integers and Decimals Understand and order integers (assumed) Use brackets and hierarchy of operations (BODMAS) Add, subtract, multiply and divide
More informationAlmost Curvature Continuous Fitting of B-Spline Surfaces
Journal for Geometry and Graphics Volume 2 (1998), No. 1, 33 43 Almost Curvature Continuous Fitting of B-Spline Surfaces Márta Szilvási-Nagy Department of Geometry, Mathematical Institute, Technical University
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #12: Curves and Surfaces Prof. James O Brien University of California, Berkeley V2007-F-12-1.0 Today General curve and surface representations Splines and other polynomial
More informationGL9: Engineering Communications. GL9: CAD techniques. Curves Surfaces Solids Techniques
436-105 Engineering Communications GL9:1 GL9: CAD techniques Curves Surfaces Solids Techniques Parametric curves GL9:2 x = a 1 + b 1 u + c 1 u 2 + d 1 u 3 + y = a 2 + b 2 u + c 2 u 2 + d 2 u 3 + z = a
More informationYear 6 programme of study
Year 6 programme of study Number number and place value read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to a required degree of accuracy
More informationUse decimal notation for tenths, hundredths and thousandths
Maths Long Term Plan Year AUTUMN Number and place value () Read, write, order and compare numbers up to 0 000 000 and determine the value of each digit Use decimal notation for tenths, hundredths and thousandths
More informationMathematics Year 9-11 Skills and Knowledge Checklist. Name: Class: Set : 1 Date Year 9 MEG :
Personal targets to help me achieve my grade : AFL Sheet Number 1 : Standard Form, Decimals, Fractions and Percentages Standard Form I can write a number as a product of it s prime factors I can use the
More informationAlaska Mathematics Standards Vocabulary Word List Grade 7
1 estimate proportion proportional relationship rate ratio rational coefficient rational number scale Ratios and Proportional Relationships To find a number close to an exact amount; an estimate tells
More informationMontana Instructional Alignment HPS Critical Competencies Mathematics Honors Pre-Calculus
Content Standards Content Standard 1 - Number Sense and Operations Content Standard 2 - Data Analysis A student, applying reasoning and problem solving, will use number sense and operations to represent
More informationExpected Standards for Year 6: Mathematics Curriculum (taken from ncetm progression maps)
Expected Standards for Year 6: Mathematics Curriculum (taken from ncetm progression maps) Place Value Addition and Subtraction Multiplication and Division Fractions Ratio and Proportion Measurement Geometry
More informationMA 323 Geometric Modelling Course Notes: Day 21 Three Dimensional Bezier Curves, Projections and Rational Bezier Curves
MA 323 Geometric Modelling Course Notes: Day 21 Three Dimensional Bezier Curves, Projections and Rational Bezier Curves David L. Finn Over the next few days, we will be looking at extensions of Bezier
More informationYear 6 Maths Long Term Plan
Week & Focus 1 Number and Place Value Unit 1 2 Subtraction Value Unit 1 3 Subtraction Unit 3 4 Subtraction Unit 5 5 Unit 2 6 Division Unit 4 7 Fractions Unit 2 Autumn Term Objectives read, write, order
More informationSection 4.3: Derivatives and the Shapes of Curves
1 Section 4.: Derivatives and the Shapes of Curves Practice HW from Stewart Textbook (not to hand in) p. 86 # 1,, 7, 9, 11, 19, 1,, 5 odd The Mean Value Theorem If f is a continuous function on the closed
More information