1. 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
Author: Prof. Farid. Amirouche
University of Illinois-Chicago

2. CHAPTER 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
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)
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Author: Prof. Farid. Amirouche, University of Illinois-Chicago

4. The solution to equation (4.6) is found by Cramer’s rule
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

5. CHAPTER 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
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6. 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 3
0.3
0.57
0.5993 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
TABLE 4.3
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

7. Figure 4.1 The line fitted to the data
CHAPTER Line Fitting
Figure 4.1 The line fitted to the data
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8. 4.2 NONLINEAR CURVE FITTING WITH A POWER FUNCTION
CHAPTER Nonlinear Curve Fitting
4.2 NONLINEAR CURVE FITTING WITH A POWER FUNCTION
(4.14)
(4.15)
(4.16)
where
(4.17)
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Author: Prof. Farid. Amirouche, University of Illinois-Chicago

9. 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
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10. C2 = 0.8422 β =2.3215 CHAPTER 4 4.2 Nonlinear Curve Fitting
Figure 4.2 The curve fitted to the data
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11. 4.3 CURVE FITTING WITH A HIGHER-ORDER POLYNOMIAL
CHAPTER 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)
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Author: Prof. Farid. Amirouche, University of Illinois-Chicago

12. 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.
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13. Solution: CHAPTER 4 4.3 Higher order Curve Fitting
Figure 4.3 Plot of the quadratic polynomial fitted
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14. 4.4 CHEBYSHEV POLYNOMIAL FIT
CHAPTER 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
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Author: Prof. Farid. Amirouche, University of Illinois-Chicago

15. 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
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16. CHAPTER 4 4.4 Chebyshev Polynomial Fit
(4.31)
(4.32)
where
(4.33)
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17. The approximating function becomes
CHAPTER Chebyshev Polynomial Fit
The approximating function becomes
(4.34)
(4.35)
(4.36)
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Author: Prof. Farid. Amirouche, University of Illinois-Chicago

18. 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
1.9991
2.1
1.7264
2.6
1.0310
3.1
0.0832
3.6
1.5274
4.1
2.1458
4.6
1.0104
5.1
5.6
6.1
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19. 4.5 FOURIER SERIES OF DISCRETE SYSTEMS
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

20. 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
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21. CHAPTER 4 4.5 Fourier Series
(4.43)
(4.44)
(4.45)
where
Figure 4.5 Mass M with Support Motion
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22. 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)
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23. CHAPTER 4 4.5 Fourier Series
f(q)
y=2sinq
Figure 4.6 Graph for
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24. 2N=8 CHAPTER 4 4.5 Fourier Series (4.52) (4.53) (4.54)
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25. CHAPTER 4 4.5 Fourier Series
y=2sinq
f(q)
Figure 4.7 Graph for
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26. CHAPTER 4 4.5 Fourier Series
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27. CHAPTER 4 4.5 Fourier Series
y=2sinq
f(q)
Figure 4.8 Graph for
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28. CHAPTER 4 4.5 Fourier Series
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29. CHAPTER 4 4.5 Fourier Series
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30. CHAPTER 4 4.5 Fourier Seriesb2
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31. CHAPTER 4 4.5 Fourier Series
y=2sinq
f(q)
Figure 4.9 Graph for
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32. 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.
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33. 4.7 PARAMETRIC CUBIC SPLINES
CHAPTER 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)
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Author: Prof. Farid. Amirouche, University of Illinois-Chicago

34. CHAPTER 4 4.7 Parametric Cubic Splines
(4.60)
(4.61)
(4.62)
(4.63)
(4.64)
(4.65)
(4.66)
(4.67)
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35. 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)
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36. 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)
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37. CHAPTER 4 4.7 Parametric Cubic Splines
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38. Boundary Conditions a) Natural Spline: b) Clamped Spline:
CHAPTER 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.
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39. 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).
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40. We first compute the cord length
CHAPTER 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
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41. (4.83) (4.84) Equation (4.78) in notational form is
CHAPTER 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)
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42. CHAPTER 4 4.7 Parametric Cubic Splines
where
(4.85)
(4.86)
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43. Since we have three splines we need to compute three co-efficients
CHAPTER 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.
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44. 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)
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45. CHAPTER 4 4.7 Parametric Cubic Splines
(4.92) S3 S2 S1
Figure 4.10 Parametric cubic curve
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46. 4.8 NONPARAMETRIC CUBIC SPLINE
CHAPTER 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)
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47. 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)
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48. CHAPTER 4 4.8 Nonparametric Cubic Spline
(4.101)
(4.102)
(4.103)
(4.104)
(4.105)
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49. CHAPTER 4 4.8 Nonparametric Cubic Spline
where
(4.106)
(4.107)
(4.108)
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50. CHAPTER 4 4.8 Nonparametric Cubic Spline
(4.109)
(4.110)
(4.111)
(4.112)
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51. 4.9 BOUNDARY CONDITIONS 4.9.1 Natural Splines (4.113) (4.114)
CHAPTER 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)
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52. 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 ( )
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53. Step 3: Solve for bi and di from equation ( 4.106)
CHAPTER Boundary Conditions
Step 3: Solve for bi and di from equation ( 4.106)
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54. The results are compiled in the following table:
CHAPTER 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
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55. Figure 4.11: Nonparametric cubic spline function
CHAPTER Boundary Conditions s2 s1
Figure 4.11: Nonparametric cubic spline function
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56. CHAPTER 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.
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57. 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)
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58. The resulting S (t) function is then found as
CHAPTER 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
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59. CHAPTER 4 4.10 Bezier Curves Figure 4.12 Bezier curve
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60. 4.11 DIFFERENTIATION OF BEZIER CURVE EQUATION
CHAPTER Bezier Curves
4.11 DIFFERENTIATION OF BEZIER CURVE EQUATION
(4.123)
(4.124)
(4.125)
(4.126)
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61. CHAPTER 4 4.11 Bezier Curves (4.128)
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62. 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)
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63. CHAPTER 4 4.12 B-Spline Curve
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64. 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
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65. (4.134) CHAPTER 4 4.12 B-Spline Curve
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66. 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.
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67. 4.13 NON-UNIFORM RATIONAL B-SPLINE CURVE (NURBS)
CHAPTER 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)
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68. 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:
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69. 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

70. 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

71. Figure 4.14 Plane surface formed by intersecting lines
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

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

73. Figure 4.16 Ruled surface formed by 2 Curves
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

74. Figure 4.17 Rectangular surface formed by 4 curves
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

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

76. Different Ways to Create a Surface
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago

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

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

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

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

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

82. Figure 4.25 Copying of a surface by selection method
CHAPTER 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
Author: Prof. Farid. Amirouche, University of Illinois-Chicago