Chapter 4 Description of Curves and Surfaces

1
University of Illinois-Chicago
Chapter 4 Description of Curves and Surfaces
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
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Amirouche University of Illinois-Chicago
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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
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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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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
Cramers rule
(4.12)
(4.13)
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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
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CHAPTER 4

4.1 Line Fitting
Solution
TABLE 4.3
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CHAPTER 4

4.1 Line Fitting
Figure 4.1 The line fitted to the data
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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)
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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
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CHAPTER 4
4.2
Nonlinear Curve Fitting
C2 0.8422 ß 2.3215
Figure 4.2 The curve fitted to the data
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CHAPTER 4
4.3 Higher
order Curve Fitting
4.3 CURVE FITTING WITH A HIGHER-ORDER POLYNOMIAL
(4.18)
(4.19)
(4.21)
(4.20)
(4.22)
(4.23)
(4.24)
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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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CHAPTER 4
4.3 Higher
order Curve Fitting
Solution
Figure 4.3 Plot of the quadratic polynomial
fitted
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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

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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
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CHAPTER 4
4.4 Chebyshev
Polynomial Fit
(4.31)
(4.32)
where
(4.33)
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CHAPTER 4
4.4 Chebyshev
Polynomial Fit
The approximating function becomes
(4.34)
(4.35)
(4.36)
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CHAPTER 4
4.4 Chebyshev
Polynomial Fit
TABLE 4.6
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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

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CHAPTER 4

4.5 Fourier Series
(4.38)
. . .
(4.39)
(4.41)
(4.42)
where is the Time Period.
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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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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 as are zeros.
(4.48)
(4.49)
(4.50)
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CHAPTER 4

4.5 Fourier Series
f(q)
y2sinq
Figure 4.6 Graph for
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CHAPTER 4

4.5 Fourier Series
2N8
(4.52)
(4.53)
(4.54)
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CHAPTER 4

4.5 Fourier Series
y2sinq
f(q)
Figure 4.7 Graph for
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CHAPTER 4

4.5 Fourier Series
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CHAPTER 4

4.5 Fourier Series
y2sinq
f(q)
Figure 4.8 Graph for
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CHAPTER 4

4.5 Fourier Series
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CHAPTER 4

4.5 Fourier Series
           
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b2
CHAPTER 4

4.5 Fourier Series
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CHAPTER 4

4.5 Fourier Series
y2sinq
f(q)
Figure 4.9 Graph for
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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.
(1ltilt4)
(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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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 i1,,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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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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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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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), Si1(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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CHAPTER 4
4.7
Parametric Cubic Splines
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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 t0 and
ttn are specified.
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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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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
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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)
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CHAPTER 4
4.7
Parametric Cubic Splines
where
(4.85)
(4.86)
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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.
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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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CHAPTER 4
4.7
Parametric Cubic Splines
(4.92)
S3
S2
S1
Figure 4.10 Parametric cubic curve
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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,xi1 we can write
(4.94)
(4.95)
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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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CHAPTER 4
4.8 Nonparametric
Cubic Spline
(4.101)
(4.102)
(4.103)
(4.104)
(4.105)
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CHAPTER 4
4.8 Nonparametric
Cubic Spline
where
(4.106)
(4.107)
(4.108)
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CHAPTER 4
4.8 Nonparametric
Cubic Spline
(4.109)
(4.110)
(4.111)
(4.112)
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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)
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CHAPTER 4
4.9
Boundary Conditions
Example 4.6 Find the nonparametric cubic spline
(natural spline) for the points shown in the
Table below.
Solution
Step 1 Control points. Intervals, and ai
Step 2 Solve for c1 Natural Spline (c0c20)
using equation ( 4.109 )
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CHAPTER 4
4.9
Boundary Conditions
Step 3 Solve for bi and di from equation ( 4.106)
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CHAPTER 4
4.9
Boundary Conditions
The results are compiled in the following table
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CHAPTER 4
4.9
Boundary Conditions
s2
s1
Figure 4.11 Nonparametric cubic spline function
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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 i1 to n, and the Si contain the vector
components of the various points.
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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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CHAPTER 4

4.10 Bezier Curves
The resulting S (t) function is then found as
TABLE 4.8 Evaluation of the Bezier function
J3,1(I0,1,2,3) in terms of the parameter t.
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CHAPTER 4

4.10 Bezier Curves
Figure 4.12 Bezier curve
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CHAPTER 4

4.11 Bezier Curves
4.11 DIFFERENTIATION OF BEZIER CURVE EQUATION
(4.123)
(4.124)
(4.125)
(4.126)
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CHAPTER 4

4.11 Bezier Curves
(4.128)
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CHAPTER 4

4.12 B-Spline Curve
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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CHAPTER 4

4.12 B-Spline Curve
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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 (nk1) 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.
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CHAPTER 4

4.12 B-Spline Curve
(4.134)
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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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4.13 Non-Uniform
B-Spline Curve
4.13 NON-UNIFORM RATIONAL B-SPLINE CURVE (NURBS)
(4.139)
(4.140)
The equation for NURBS curve S(t) is given by
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4.13 Non-Uniform
B-Spline Curve
Example 4.9 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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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
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
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
71
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
72
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
73
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
74
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
75
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
76
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
77
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
78
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
79
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
80
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
81
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
82
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