Lecture 2.2 Cubic Splines

Size: px
Start display at page:

Download "Lecture 2.2 Cubic Splines"

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 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 information

An introduction to interpolation and splines

An 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 information

Parameterization. Michael S. Floater. November 10, 2011

Parameterization. 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 information

2D Spline Curves. CS 4620 Lecture 13

2D 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 information

Parameterization of triangular meshes

Parameterization 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 information

Interactive 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 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 information

The 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 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 information

Lecture 9: Introduction to Spline Curves

Lecture 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 information

Representing Curves Part II. Foley & Van Dam, Chapter 11

Representing 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 information

Intro to Curves Week 4, Lecture 7

Intro 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 information

Curves and Surfaces 1

Curves 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 information

Bezier Curves, B-Splines, NURBS

Bezier 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 information

Curves and Surfaces Computer Graphics I Lecture 9

Curves 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 information

COMPUTER AIDED ENGINEERING DESIGN (BFF2612)

COMPUTER 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 information

Curves and Surfaces Computer Graphics I Lecture 10

Curves 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 information

Sung-Eui Yoon ( 윤성의 )

Sung-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 information

Four equations are necessary to evaluate these coefficients. Eqn

Four 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 information

LECTURE #6. Geometric Modelling for Engineering Applications. Geometric modeling for engineering applications

LECTURE #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 information

2D Spline Curves. CS 4620 Lecture 18

2D 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 information

Intro to Curves Week 1, Lecture 2

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 Outline Math review Introduction to 2D curves

More information

CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2

CS 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 information

CSE 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 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 information

CSE 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 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)

(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 information

Central issues in modelling

Central 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 information

Splines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes

Splines. 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 information

Fall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.

Fall 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 information

Curves. Computer Graphics CSE 167 Lecture 11

Curves. 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 information

Computer Graphics Curves and Surfaces. Matthias Teschner

Computer 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 information

1.2 Numerical Solutions of Flow Problems

1.2 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 information

Chapter 18. Geometric Operations

Chapter 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 information

Graphics and Interaction Transformation geometry and homogeneous coordinates

Graphics 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 information

COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates

COMP30019 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 information

TO DUY ANH SHIP CALCULATION

TO 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 information

Module 3 Mesh Generation

Module 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 information

CSE 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 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 information

Computergrafik. Matthias Zwicker Universität Bern Herbst 2016

Computergrafik. 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 information

Year 6 Mathematics Overview

Year 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 information

Computergrafik. Matthias Zwicker. Herbst 2010

Computergrafik. 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 information

CS 450 Numerical Analysis. Chapter 7: Interpolation

CS 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 information

Advanced Turbomachinery Methods. Brad Hutchinson ANSYS, Inc. Industry Marketing

Advanced 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 information

The Free-form Surface Modelling System

The 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 information

1 Trajectories. Class Notes, Trajectory Planning, COMS4733. Figure 1: Robot control system.

1 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 information

Introduction to Computer Graphics

Introduction 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 information

New Swannington Primary School 2014 Year 6

New 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 information

Parametric Curves. University of Texas at Austin CS384G - Computer Graphics

Parametric 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 information

Driven Cavity Example

Driven 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 information

Unit 3 Higher topic list

Unit 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 information

Finite 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 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 information

B-spline Curves. Smoother than other curve forms

B-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 information

For each question, indicate whether the statement is true or false by circling T or F, respectively.

For 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 information

Number and Place Value

Number 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 information

Number- Algebra. Problem solving Statistics Investigations

Number- 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 information

FOUNDATION 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

FOUNDATION 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 information

Parametric 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 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 information

Fast marching methods

Fast 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 information

Geometric Modeling of Curves

Geometric 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 information

Math background. 2D Geometric Transformations. Implicit representations. Explicit representations. Read: CS 4620 Lecture 6

Math 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 information

Natural Quartic Spline

Natural 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 information

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

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 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 information

3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013

3D 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 information

7 Fractions. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability

7 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 information

Chapter 7 Practical Considerations in Modeling. Chapter 7 Practical Considerations in Modeling

Chapter 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 information

Number Mulitplication and Number and Place Value Addition and Subtraction Division

Number 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 information

PO 2. Identify irrational numbers. SE/TE: 4-8: Exploring Square Roots and Irrational Numbers, TECH: itext; PH Presentation Pro CD-ROM;

PO 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 information

AQA GCSE Maths - Higher Self-Assessment Checklist

AQA 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 information

Chapter 15 Introduction to Linear Programming

Chapter 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 information

CS770/870 Spring 2017 Curve Generation

CS770/870 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 information

Geometric Transformations

Geometric 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 information

Rational Bezier Curves

Rational 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 information

ECE 600, Dr. Farag, Summer 09

ECE 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 information

8 Piecewise Polynomial Interpolation

8 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 information

Linear Equation Systems Iterative Methods

Linear 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 information

form. We will see that the parametric form is the most common representation of the curve which is used in most of these cases.

form. 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 information

Information Coding / Computer Graphics, ISY, LiTH. Splines

Information 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 information

Shape 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 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 information

Intro to Modeling Modeling in 3D

Intro 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 information

Y6 MATHEMATICS TERMLY PATHWAY NUMBER MEASURE GEOMETRY STATISTICS

Y6 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 information

3D Modeling Parametric Curves & Surfaces

3D 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 information

Functions and Transformations

Functions 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 information

Until now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple

Until 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 information

Finite 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 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 information

8. BASIC TURBO MODEL WITH UNSTRUCTURED MESH

8. 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 information

B-Splines and NURBS Week 5, Lecture 9

B-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 information

Beams. Lesson Objectives:

Beams. 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 information

Curves, Surfaces and Recursive Subdivision

Curves, 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 information

Shape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011

Shape 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 information

Birkdale High School - Higher Scheme of Work

Birkdale 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 information

Almost Curvature Continuous Fitting of B-Spline Surfaces

Almost 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 information

CS-184: Computer Graphics

CS-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 information

GL9: Engineering Communications. GL9: CAD techniques. Curves Surfaces Solids Techniques

GL9: 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 information

Year 6 programme of study

Year 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 information

Use decimal notation for tenths, hundredths and thousandths

Use 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 information

Mathematics Year 9-11 Skills and Knowledge Checklist. Name: Class: Set : 1 Date Year 9 MEG :

Mathematics 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 information

Alaska Mathematics Standards Vocabulary Word List Grade 7

Alaska 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 information

Montana Instructional Alignment HPS Critical Competencies Mathematics Honors Pre-Calculus

Montana 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 information

Expected Standards for Year 6: Mathematics Curriculum (taken from ncetm progression maps)

Expected 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 information

MA 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 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 information

Year 6 Maths Long Term Plan

Year 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 information

Section 4.3: Derivatives and the Shapes of Curves

Section 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