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Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8

University of Illinois-Chicago. Chapter 4 Description of Curves and Surfaces. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche University of Illinois-Chicago.

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Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8

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  1. University of Illinois-Chicago Chapter 4 Description of Curves and Surfaces Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche University of Illinois-Chicago

  2. CHAPTER 4 4.1 Line Fitting 4.1 LINE FITTING • Suppose we desire to fit a linear function to the data set, as illustrated in Table 4.1. Table 4.1 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  3. CHAPTER 4 4.1 Line Fitting (4.1) (4.2) We have two equations and two unknowns and the coefficient are given by : (4.3) (4.4) (4.5) (4.6) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  4. CHAPTER 4 4.1 Line Fitting (4.7) (4.8) (4.9) (4.10) (4.11) The solution to equation (4.6) is found by Cramer’s rule (4.12) (4.13) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  5. CHAPTER 4 4.1 Line Fitting Example 4.1 Determine the regression line for the data in Table 4.2 by solving Equation (4.6). After the regression line is obtained, examine the deviation error of the line from the data.  Table 4.2 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  6. CHAPTER 4 4.1 Line Fitting Solution: TABLE 4.3 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  7. CHAPTER 4 4.1 Line Fitting Figure 4.1 The line fitted to the data Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  8. CHAPTER 4 4.2 Nonlinear Curve Fitting 4.2 NONLINEAR CURVE FITTING WITH A POWER FUNCTION (4.14) (4.15) (4.16) where (4.17) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  9. CHAPTER 4 4.2 Nonlinear Curve Fitting Example 4.2 A following data set is used to demonstrate how curve fitting of a power function can be carried out making use of the regression line technique. Consider Table 4.4, when x, y represent experimental data between force (lbs) and displacement (mm). We need to find a mathematical function to describe the data and it is perceived that a power function is most suitable. Table 4.4 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  10. CHAPTER 4 4.2 Nonlinear Curve Fitting C2 = 0.8422 β =2.3215 Figure 4.2 The curve fitted to the data Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  11. CHAPTER 4 4.3 Higher order Curve Fitting 4.3 CURVE FITTING WITH A HIGHER-ORDER POLYNOMIAL Considering a set of data (xi, yi). Let us try to interpolate the data with a polynomial of order n : Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  12. CHAPTER 4 4.3 Higher order Curve Fitting The system can be written : In a matrix form : Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  13. CHAPTER 4 4.3 Higher order Curve Fitting In order to find the best fit, the error needs to be minimized : (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  14. CHAPTER 4 4.3 Higher order Curve Fitting Example 4.3 A data set of a biomechanical experiment is provided in Table 4.5. Find a polynomial of order 12 that best fits the data. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  15. CHAPTER 4 4.3 Higher order Curve Fitting Solution: Figure 4.3 Plot of the quadratic polynomial fitted Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  16. CHAPTER 4 4.4 Chebyshev Polynomial Fit 4.4 CHEBYSHEV POLYNOMIAL FIT The definition of a Chebyshev polynomial is contained in the following rules: • A Chebyshev polynomial is defined over the interval [-1,1]. • The range of the independent variable must then be • The zeroth-order Chebyshev polynomial is • The first-order Chebyshev polynomial is 5. The second-order Chebyshev polynomial is Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  17. CHAPTER 4 4.4 Chebyshev Polynomial Fit (4.29) (4.30) Example 4.4 Figure 4.4 Free Body Analysis of a Vehicle on a Road Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  18. CHAPTER 4 4.4 Chebyshev Polynomial Fit (4.31) (4.32) where (4.33) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  19. CHAPTER 4 4.4 Chebyshev Polynomial Fit The approximating function becomes (4.34) (4.35) (4.36) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  20. CHAPTER 4 4.4 Chebyshev Polynomial Fit TABLE 4.6 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  21. CHAPTER 4 4.5 Fourier Series 4.5 FOURIER SERIES OF DISCRETE SYSTEMS • By performing a variable transformation, we can transform the physical interval by using a new independent variable  that has the range from some given interval . We, then subdivide this interval into 2N equally spaced parts by using . The function is then known at the points . There are 2N known values of the function through which the series will be fitted. Then we have Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  22. CHAPTER 4 4.5 Fourier Series (4.38) . . . (4.39) j (4.41) . (4.42) . where is the Time Period. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  23. CHAPTER 4 4.5 Fourier Series (4.43) (4.44) (4.45) where Figure 4.5 Mass M with Support Motion Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  24. CHAPTER 4 4.5 Fourier Series We apply Fourier series method to the data and use two-term Fourier series. (4.46) (4.47) Because the function is odd all a’s are zeros. (4.48) (4.49) (4.50) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  25. CHAPTER 4 4.5 Fourier Series f(q) y=2sinq Figure 4.6 Graph for Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  26. CHAPTER 4 4.5 Fourier Series 2N=8 (4.52) (4.53) (4.54) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  27. CHAPTER 4 4.5 Fourier Series y=2sinq f(q) Figure 4.7 Graph for Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  28. CHAPTER 4 4.5 Fourier Series Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  29. CHAPTER 4 4.5 Fourier Series y=2sinq f(q) Figure 4.8 Graph for Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  30. CHAPTER 4 4.5 Fourier Series Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  31. CHAPTER 4 4.5 Fourier Series Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  32. CHAPTER 4 4.5 Fourier Series b2 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  33. CHAPTER 4 4.5 Fourier Series y=2sinq f(q) Figure 4.9 Graph for Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  34. CHAPTER 4 4.6 Cubic Splines 4.6 CUBIC SPLINES A spline is a smooth curve that can be generated by computer to go through a set of data points. The mathematical spline derives from its physical counterpart - the thin elastic beam. Because the beam is supported at specified points (we call them knots), it can be shown that its deflection (assumed small) is characterized by a polynomial of order three, hence a cubic spline. It is not a mere coincidence that the principle of explaining the deflection of beams under different loads results into a function of a third order. (1<i<4) (4.55) The benefits of using cubic splines are as follows: 11. They reduce computational requirements and numerical instabilities that arise from higher-order curves. 2. They have the lowest degree space curve that allows inflection points. 33. They have the ability to twist in space. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  35. CHAPTER 4 4.7 Parametric Cubic Splines 4.7 PARAMETRIC CUBIC SPLINES Consider a set of data points described in the x-y plane by (xi yi) with i=1,…,n. Our objective is to pass a parametric cubic spline between all these points. A parametric cubic spline is a curve that is represented as a function of one or more parameters. (4.56) (4.57) (4.58) (4.59) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  36. CHAPTER 4 4.7 Parametric Cubic Splines (4.60) (4.61) (4.62) (4.63) (4.64) (4.65) (4.66) (4.67) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  37. CHAPTER 4 4.7 Parametric Cubic Splines (4.68) (4.69) (4.70) Therefore, the spline function between P1 & P2 could simply be expressed as (4.71) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  38. CHAPTER 4 4.7 Parametric Cubic Splines IIn the context of computer graphics and general-purpose algorithm development, we need to ask the following questions:  11. How can we generate a solution for and for all cubic functions Si(t), Si+1(t), . . . Sn(t)?  22. How do we select t, t1, and t2 for a given set of data points?  3. How do we assure continuity between the splines at knots P1, P2,. . . , Pn? (4.72) (4.73) (4.75) (4.74) (4.76) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  39. CHAPTER 4 4.7 Parametric Cubic Splines i ti+2S’i Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  40. CHAPTER 4 4.7 Parametric Cubic Splines Boundary Conditions a) Natural Spline: (4.79) (4.80) (4.81) (4.82) Adding Equations (4.81) and (4.82) to the n-2 equations given by Equation (4.78) we can solve for all the S’. b) Clamped Spline: The boundary conditions for this spline are such that the first derivatives (slope) at t=0 and t=tn are specified. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  41. CHAPTER 4 4.7 Parametric Cubic Splines Summary TThe parametric cubic spline between any two points is constructed as follows: 11. Find the maximum cord length and determine t1, t2, . . . ,tn. 22. Use Equation (4.78) together with the corresponding boundary conditions to solve for the , , . . .. , . 33. Solve for the coefficients that make up the parametric cubic splines using equations (4.62), (4.69) and (4.70). Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  42. CHAPTER 4 4.7 Parametric Cubic Splines Example 4.4 For following data set (1,1), (1.5,2), (2.5,1.75) & (3.0,3.25). Find the parametric cubic spline assuming a relaxed condition at both ends of the data. Solution: We first compute the cord length Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  43. CHAPTER 4 4.7 Parametric Cubic Splines t32 (4.83) t42 The above equations are found using boundary conditions given by equations (4.81), (4.82) and (4.77). Equation (4.78) in notational form is (4.84) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  44. CHAPTER 4 4.7 Parametric Cubic Splines where (4.85) (4.86) t2t3 Last Eqn t42 t42 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  45. CHAPTER 4 4.7 Parametric Cubic Splines To solve for Si’ we multiply equation (4.84) by [CT]-1to get the ai,1 constants . = (4.87) Since we have three splines we need to compute three coefficients of ai,2 and ai,3. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  46. CHAPTER 4 4.7 Parametric Cubic Splines Using equation (4.69) to find ai,2 Si+1 (4.88) (4.89) Using equation (4.70) to find ai,3 (4.90) (ti+1)3 (4.91) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  47. CHAPTER 4 4.7 Parametric Cubic Splines (4.92) S3 S2 S1 Figure 4.10 Parametric cubic curve Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  48. CHAPTER 4 4.8 Nonparametric Cubic Spline 4.8 NONPARAMETRIC CUBIC SPLINE A nonparametric cubic spline is defined as a curve having a function of only one parameter. Non-parametric cubic splines allow a direct variable relationship between the parameter value x and the value of the cubic spline function to be determined. (4.93) Cubic spline S(x) is composed of (n-1) cubic segment splines. Each point has an x and y value. For the interval [xi,xi+1] we can write (4.94) (4.95) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  49. CHAPTER 4 4.8 Nonparametric Cubic Spline By considering the smoothness and continuity of the cubic splines the following conditions are derived: (4.96) (4.97) The non-parametric cubic spline can be expressed as: (4.98) Its first and second derivatives are (4.99) (4.100) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

  50. CHAPTER 4 4.8 Nonparametric Cubic Spline (4.101) (4.102) (4.103) (4.104) (4.105) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

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