Description of Curves and Surfaces 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

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. i x y 1 xi yi 2 xi+1 yi+1 3 xi+2 yi+2 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

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

The solution to equation (4.6) is found by Cramer’s rule 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

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 i xi yi x2i xiyi 1 0.1 0.22 0.01 0.022 2 0.2 0.39 0.04 0.078 3 0.3 0.57 0.09 0.171 4 0.4 0.81 0.16 0.324 5 0.5 1.02 0.25 0.51 6 0.6 1.18 .36 0.708 Total 2.1 4.19 0.91 1.813 a21 z2 a11 z1 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Solution: TABLE 4.3 CHAPTER 4 4.1 Line Fitting i xi yi g=c1x+c2 Deviation (error) 1 0.1 0.22 0.2033 0.0167 2 0.2 0.39 0.4013 -0.0113 3 0.3 0.57 0.5993 -0.0293 4 0.4 0.81 0.7973 0.0127 5 0.5 1.02 0.9953 0.0247 6 0.6 1.18 1.1933 -0.0753 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

Figure 4.1 The line fitted to the data 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

4.2 NONLINEAR CURVE FITTING WITH A POWER FUNCTION 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

Example 4.2 CHAPTER 4 4.2 Nonlinear Curve Fitting 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 i 1 2 3 4 5 6 7 8 9 10 11 12 Total x 0.1 0.25 0.39 0.60 1.03 1.32 1.78 2.13 2.45 3.07 3.98 4.64 y 3.21 3.81 4.09 5.21 7.97 8.32 8.88 9.27 9.97 10.8 11.34 13.08 X=log(x) -1 -.602 -.408 -.22 .0128 .1205 0.328 .389 0.487 .60 0.666 .6233 Y=log(y) 0.506 .580 0.611 0.716 0.9014 0.920 0.948 .967 .998 1.033 1.054 1.116 10.35 X2 .3624 .1664 .0484 0.0001 0.014 .0625 .1075 0.151 .2371 .36 .443 2.9524 XY -.506 -.349 -.249 -.1575 .0115 .1108 .237 .3171 .388 .5030 0.6324 .7432 1.6815 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

C2 = 0.8422 β =2.3215 CHAPTER 4 4.2 Nonlinear Curve Fitting 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

4.3 CURVE FITTING WITH A HIGHER-ORDER POLYNOMIAL CHAPTER 4 4.3 Higher order Curve Fitting 4.3 CURVE FITTING WITH A HIGHER-ORDER POLYNOMIAL (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

CHAPTER 4 4.3 Higher order Curve Fitting where (4.25) (4.26) (4.27) 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

Solution: CHAPTER 4 4.3 Higher order Curve Fitting 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

4.4 CHEBYSHEV POLYNOMIAL FIT 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

Example 4.4 CHAPTER 4 4.4 Chebyshev Polynomial Fit (4.29) (4.30) 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

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

The approximating function becomes 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

CHAPTER 4 4.4 Chebyshev Polynomial Fit TABLE 4.6 VALUE OF X IN THE FUNCTION Y=2*SIN X RESULTS FROM APPROXIMATION DESIRED RESULTS 0.1 1.6071 0.1997 0.6 4.1959 1.1293 1.1 2.0987 1.7824 1.6 -0.6599 1.9991 2.1 -2.0871 1.7264 2.6 -1.7240 1.0310 3.1 -0.1475 0.0832 3.6 1.5274 -0.8850 4.1 2.1458 -1.6366 4.6 1.0104 -1.9874 5.1 -1.6205 -1.8516 5.6 -4.0349 -1.2625 6.1 -2.5692 -0.3643 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

4.5 FOURIER SERIES OF DISCRETE SYSTEMS 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

CHAPTER 4 4.5 Fourier Series (4.38) . (4.39) (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

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

CHAPTER 4 4.5 Fourier Series We apply Fourier series method to the data and use two-term Fourier series. (4.46) Because the function is odd all a’s are zeros. (4.47) (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

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

2N=8 CHAPTER 4 4.5 Fourier Series (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

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

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

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

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

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

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

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

4.6 CUBIC SPLINES CHAPTER 4 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

4.7 PARAMETRIC CUBIC SPLINES 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

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

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

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

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

Boundary Conditions a) Natural Spline: b) Clamped Spline: 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

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

We first compute the cord length 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

(4.83) (4.84) Equation (4.78) in notational form is CHAPTER 4 4.7 Parametric Cubic Splines (4.83) 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

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

Since we have three splines we need to compute three co-efficients CHAPTER 4 4.7 Parametric Cubic Splines To solve for Si’ we multiply equation (4.84) by [CT]-1 to get the ai,1 constants . = (4.87) Since we have three splines we need to compute three co-efficients 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

CHAPTER 4 4.7 Parametric Cubic Splines Using equation (4.69) to find ai,2 (4.88) (4.89) Using equation (4.70) to find ai,3 (4.90) (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

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

4.8 NONPARAMETRIC CUBIC SPLINE 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

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

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

CHAPTER 4 4.8 Nonparametric Cubic Spline where (4.106) (4.107) (4.108) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.8 Nonparametric Cubic Spline (4.109) (4.110) (4.111) (4.112) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

4.9 BOUNDARY CONDITIONS 4.9.1 Natural Splines (4.113) (4.114) CHAPTER 4 4.9 Boundary Conditions 4.9 BOUNDARY CONDITIONS 4.9.1 Natural Splines (4.113) When substituted into equation (4.105) yields (4.114) 4.9.2 Clamped Splines (4.115) (4.116) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.9 Boundary Conditions Example 4.6 Find the nonparametric cubic spline (natural spline) for the points shown in the Table below. i xi yi hi 1 0.5 1.5 2 n=2 2.5 1.75 - Solution: Step 1: Control points. Intervals, and ai Step 2: Solve for c1: Natural Spline (c0=c2=0) using equation ( 4.109 ) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Step 3: Solve for bi and di from equation ( 4.106) CHAPTER 4 4.9 Boundary Conditions Step 3: Solve for bi and di from equation ( 4.106) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

The results are compiled in the following table: CHAPTER 4 4.9 Boundary Conditions The results are compiled in the following table: i xi hi yi=ai bi ci di 1 0.5 2.375 -1.5 1.5 1.0 2 1.25 -2.25 0.75 n=2 2.5 - 1.75 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.11: Nonparametric cubic spline function CHAPTER 4 4.9 Boundary Conditions s2 s1 Figure 4.11: Nonparametric cubic spline function Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.10 Bezier Curves 4.10 BEZIER CURVES The shapes of Bezier curves are defined by the position of the points, and the curves may not intersect all the given points except for the endpoints. (4.117) where (4.118) The curve points are defined by (4.119) where i=1 to n, and the Si contain the vector components of the various points. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Example 4.7 Solution CHAPTER 4 4.10 Bezier Curves (4.120) The following example illustrates the Bezier curve method of curve fitting. Example 4.7 Define the Bezier Curve that passes through the following points: Find the Bezier curve space that passes through these points. Solution (4.121) (4.122) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

The resulting S (t) function is then found as CHAPTER 4 4.10 Bezier Curves The resulting S (t) function is then found as t J3,0 J3,1 J3,2 J3,3 1 0.15 0.614 0.325 0.0574 0.0034 0.35 0.275 0.444 0.239 0.043 0.5 0.125 0.375 0.65 0.85 TABLE 4.8 Evaluation of the Bezier function J3,1(I=0,1,2,3) in terms of the parameter t. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.10 Bezier Curves Figure 4.12 Bezier curve Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

4.11 DIFFERENTIATION OF BEZIER CURVE EQUATION CHAPTER 4 4.11 Bezier Curves 4.11 DIFFERENTIATION OF BEZIER CURVE EQUATION (4.123) (4.124) (4.125) (4.126) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.11 Bezier Curves (4.128) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.12 B-Spline Curve B-Splines were introduced to overcome some weaknesses in the Bezier curve. It seems that the number of control points affect the degree of the curve. Furthermore the properties of the blending functions used in the Bezier curve do not allow for an easier way to modify the shape of the curve locally. (4.129) where (4.130) (4.131) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.12 B-Spline Curve Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.12 B-Spline Curve Example 4.8   Define the B-spline curve of order 3 for non-periodic uniform knots. The control points for the curve are given by P0, P1 and P2 Solution: We obtain the (n+k+1) knot values as follows:  t0 = 0, t1 = 0, t2 = 0, t3 = 1, t4 = 1 and t5 = 1   (Note that n = 2 and k = 3) Order 1. Let us compute all possible functions. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

(4.134) CHAPTER 4 4.12 B-Spline Curve Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.12 B-Spline Curve We obtain order 2 Ni,2 function as follows: In a similar fashion, we obtain the Ni,3(t) functions for order 3. Where S0, S1 and S2 correspond to control points P0,P1 and P2, respectively. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

4.13 NON-UNIFORM RATIONAL B-SPLINE CURVE (NURBS) CHAPTER 4 4.13 Non-Uniform B-Spline Curve 4.13 NON-UNIFORM RATIONAL B-SPLINE CURVE (NURBS) (4.139) The equation for NURBS curve S(t) is given by: (4.140) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Example 4.9 Solution: CHAPTER 4 4.13 Non-Uniform B-Spline Curve Derive a NURBS representation of a quarter circle of radius 1. Let the arc be defined in the (x, y) plane. Determine the corresponding coordinates of the control points, and the knot values. Solution: Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.13 Non-Uniform B-Spline Curve t0 = 0, t1 = 0, t2 = 0, t3 = 1, t4 = 1 and t5 = 1 h0 = 1, (4.141) (4.142) (4.143) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 4 4.13 Non-Uniform B-Spline Curve with S0 = P0, S1 = P1 and S2 = P2 ; after substitution the NURBS equation is then found to be : (4.144) Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.14 Plane surface formed by intersecting lines CHAPTER 4 4.15 Plane Surface 4.15 PLANE SURFACE Figure 4.14 Plane surface formed by intersecting lines Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.15 Plane surface formed by intersecting curves CHAPTER 4 4.15 Plane Surface Figure 4.15 Plane surface formed by intersecting curves Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.16 Ruled surface formed by 2 Curves CHAPTER 4 4.16 Ruled Surface 4.16 RULED SURFACE Figure 4.16 Ruled surface formed by 2 Curves Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.17 Rectangular surface formed by 4 curves CHAPTER 4 4.17 Rectangular Surface 4.17 RECTANGULAR SURFACE Figure 4.17 Rectangular surface formed by 4 curves Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.18 Revolved Surface CHAPTER 4 4.18 Surface of Revolution 4.18 SURFACE OF REVOLUTION Figure 4.18 Revolved Surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Different Ways to Create a Surface CHAPTER 4 4.19 Application Software 4.19 APPLICATION SOFTWARE Different Ways to Create a Surface Extrude-Create Figure 4.19 Plane surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.20 Revolved surface CHAPTER 4 4.19 Application Software Revolve-Create Figure 4.20 Revolved surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Sweep-Create Figure 4.21 Sweep surface CHAPTER 4 4.19 Application Software Sweep-Create Figure 4.21 Sweep surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Blend-Create Figure 4.22 Blend surface CHAPTER 4 4.19 Application Software Blend-Create Figure 4.22 Blend surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Flat-Create Figure 4.23 Flat surface CHAPTER 4 4.19 Application Software Flat-Create Figure 4.23 Flat surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.24 Offsetting of a surface CHAPTER 4 4.19 Application Software Offset-Create Figure 4.24 Offsetting of a surface Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago

Figure 4.25 Copying of a surface by selection method CHAPTER 4 4.19 Application Software Copy-Create Figure 4.25 Copying of a surface by selection method Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago