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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. - PowerPoint PPT Presentation
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Principles of Computer-Aided Design and Manufacturing
Second Edition 2004
ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche
University of Illinois-Chicago
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
21 cxcg(x)
1,2....Li,cxcyxgyr 2i1iiii
22i1i
L
1i
2i
L
1icxcyrR
0cxcyx2c
R2i1ii
L
1i1
0cxcy2c
R2i1i
L
1n2
2
1
2
1
2,22,1
1,21,1
z
z
c
c
aa
aa
We have two equations and two unknowns and the coefficient are given by :
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
points. data ofnumber total theis L where
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
2i
L
1i1,1 xa
i
L
1i2,11,2 xaa
LaL
1i2,2
0ix
ii
L
1i1 yxz
L
1ii2 yz
2,11,22,21,121,212,21 aaaa/zazac
2,11,22,21,112,221,12 aaaa/zazac
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
The solution to equation (4.6) is found by Cramer’s rule
CHAPTER 4 4.1 Line Fitting
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Example 4.1
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
CHAPTER 4 4.1 Line Fitting
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Solution:
19.4,6,1.2
813.1,1.2,91.0
22,21,2
12,11,1
zaa
zaa
19.4
813.1
61.2
1.291.0
2
1
c
c
0053.,98.1 21 cc
0.00531.98xxg
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
CHAPTER 4 4.1 Line Fitting
x
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
x
y
y=1.98x+.0053
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
CHAPTER 4 4.1 Line Fitting
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 4.2 Nonlinear Curve Fitting
4.2 NONLINEAR CURVE FITTING WITH A POWER
FUNCTION αβxxg βlogxαlogglog
xlogX
log(βoc
αc
log(g)G
2
1
21 cXcG where
(4.14)
(4.15)
(4.16)
(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
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.
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 0.25 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 1 .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
Table 4.4
CHAPTER 4 4.2 Nonlinear Curve Fitting
x x x
2 4 6 8 10 12 14 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x
g(x)
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
35.10,12,6233.0
6815.1,6233.0,9524.2
22,21,2
12,11,1
zaa
zaa
9405.1)log(,3917.0 21 cc
0.3197α 6.9622xβxxg
Figure 4.2 The curve fitted to the data
C2 = 0.8422
β =2.3215
CHAPTER 4 4.2 Nonlinear Curve Fitting
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
11
21 ... n
nn cxcxcxg
CHAPTER 4 4.3 Higher order Curve Fitting
Considering a set of data (xi, yi).
Let us try to interpolate the data with a polynomial of order n :
xi yi
x1 y1
x2 y2
…. …..
xL yL
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
LnnL
n
nn
y
y
y
y
c
c
c
c
x
x
xx
A.
,.
,
1..
....
1..
1.
2
1
1
2
1
2
111
** ycAoryAAcA tt ** 1
yAc
yAyandAAA tt **
11
21
11
32313
11
22212
11
12111
...
...........................................
...
...
...
nn
Ln
LL
nnn
nnn
nnn
cxcxcy
cxcxcy
cxcxcy
cxcxcyThe 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
In order to find the best fit, the error needs to be minimized :
11
21 ... n
nn cxcxcxg
Lixgyr iii ,...2,1,
L
iirR
1
21,...,2,1,0
njc
R
j
1,...2,1,1
11
1 1
22
nkyxcxL
ii
knij
n
j
L
i
kjni
L
ii
L
ii
ni
L
ii
ni
nL
ii
L
i
ni
L
i
ni
L
i
ni
L
i
ni
L
i
ni
L
i
ni
y
yx
yx
c
c
c
xx
xx
xxx
1
1
1
1
1
2
1
1
0
1
1
1
1
12
11
12
1
2
..
..
....
..
.
yAc
(4.18)
(4.19)
(4.21)(4.20)
(4.22)
(4.23)(4.24)
CHAPTER 4 4.3 Higher order Curve Fitting
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Example 4.3
A data set of a biomechanical experiment is provided in Table 4.5. Find a polynomial of order 12 that best fits the data.
CHAPTER 4 4.3 Higher order Curve Fitting
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Solution:
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2
4
6
8
10
12
14
X
Y
Figure 4.3 Plot of the quadratic polynomial fitted
CHAPTER 4 4.3 Higher order Curve Fitting
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
4.4 CHEBYSHEV POLYNOMIAL FIT
1. A Chebyshev polynomial is defined over the interval [-1,1].
2. The range of the independent variable must then be
3. The zeroth-order Chebyshev polynomial is
4. The first-order Chebyshev polynomial is
5. The second-order Chebyshev polynomial is
The definition of a Chebyshev polynomial is contained in the following rules:
.1...1 10 nxxx
.1)(0 xT
.)(1 xxT
.12)( 22 xxT
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
)()(2)( 21 xTxxTxT kkk
n
kkk xTaxf
0
).()(
Example 4.4
Figure 4.4 Free Body Analysis of a Vehicle on a Road
(4.29)
(4.30)
CHAPTER 4 4.4 Chebyshev Polynomial Fit
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
56.0448.0336.0224.0112.00]56.0,0[],[ ba
.12
ab
axI
,112
02
xx
I
,16.02.02.06.01 xI
xTaxTaxTaxTaxTaxTaxf o 55443322110 )()()()(
,1)( xTo
,)(1 xxT
,12 22 xxT
,342 3123 xxxTxTxxT
,188)( 244 xxxT
.52016)( 355 xxxxT
where
(4.31)
(4.32)
(4.33)
CHAPTER 4 4.4 Chebyshev Polynomial Fit
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
1883412 244
33
2210 xxaxxaxaxaaxf
xxxa 52016 355
0
176.1
902.1
902.1
176.1
0
111111
0788.0
845.0
843.0
693.0
936.0
568.0
28.0
92.0
6.0
2.0
1
1
84512.06928.0568.092.02.01
07584.08432.0936.028.06.01
111111
5
4
3
2
1
0
a
a
a
a
a
a
533.203490.001844.20 a
543210 533.20023.03490.00004.01844.20019.0)( TTTTTTxf
(4.34)
(4.35)
(4.36)
CHAPTER 4 4.4 Chebyshev Polynomial Fit
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
VALUE OF X IN THE FUNCTION Y=2*SIN X
RESULTS FROM APPROXIMAT
ION
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
TABLE 4.6
CHAPTER 4 4.4 Chebyshev Polynomial Fit
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
4.5 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
),0(0 fy ),(1 fy
].)1(2[12 Nfy N
.
.
.
m
jjj jbja
af
1
0 sincos2
)cos...2coscos(2 21
0 maaaa
f m )sin...2sinsin( 21 mbbb m
0
cos.2
djfa j .,....,2,1,0 mj
0
sin.2
djfb j .,....,2,1,0 mj
where is the Time Period.
j
(4.38)
(4.39)
(4.41)
(4.42)
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
12
0
2])([N
kkyfE
kkj jfN
a cos1 ,,.....,2,1,0 mj
kkj jfN
b sin)(1
,,.....,2,1,0 mj
where
)./( Nkkk
Figure 4.5 Mass M with Support Motion
(4.43)
(4.44)
(4.45)
3
23
4 2
f 3 3 0
0
0
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
)2sinsin()2coscos(2 2121
0 bbaaa
f
2sinsin 21 bbf
kkfN
b sin1
1
2sin*0
3
4sin*3
3
2sin*30sin*0
2
1
5.1
kkfN
b 2sin1
2
4sin*0
3
8sin*3
3
4sin*30sin*0
2
1
5.1 2sin5.1sin5.1 f
(4.46)
(4.47)
(4.48)
(4.49)
(4.50)
We apply Fourier series method to the data and use two-term Fourier series.
Because the function is odd all a’s are zeros.
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
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
3
X
Y
Figure 4.6 Graph for 2sin5.1sin5.1 f
f(y=2sin
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
2sinsin 21 bbf
kkfN
b sin1
1
29.154sin*8676.086.102sin*95.143.51sin*56.10sin*0(4
1
)360sin*058.308sin*56.115.257sin*95.172.205sin*868.0
75.1 kkf
Nb 2sin
12
58.308sin*8676.072.205sin*95.186.102sin*56.10sin*0(4
1
)720sin*016.617sin*56.13.514sin*95.144.411sin*868.0
sin75.1f
0
(4.52)
(4.53)(4.54)
2N=8
CHAPTER 4 4.5 Fourier Series
0 1 2 3 4 5 6 7-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
X
Y
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Figure 4.7 Graph for sin75.1f
732.1176.1416.0416.0176.1732.1989.1902.1486.1813.00
240216192168144120967248240
f 0813.0486.1902.1989.1
360336312288264
y=2sin
f(
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
kkfN
b sin1
1
2sinsin 21 bbf
120sin*732.196sin*989.172sin*902.148sin*486.124sin*813.00sin*0(8
1
240sin*732.1216sin*176.1192sin*416.0168sin*416.0144sin*176.1
)360sin*0336sin*813.0312sin*486.1288sin*902.1264sin*989.1
876.1
kkfN
b 2sin1
2
240sin*732.1192sin*989.1144sin*902.196sin*486.148sin*813.00sin*0(8
1
480sin*732.1432sin*176.1384sin*416.0336sin*416.0288sin*176.1 )720sin*0672sin*813.0624sin*486.1576sin*902.1528sin*989.1
0
,sin876.1 f
CHAPTER 4 4.5 Fourier Series
0 1 2 3 4 5 6 7-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
X
Y
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Figure 4.8 Graph for sin876.1f
f(
y=2sin
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
2sinsin 21 bbf
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
kkfN
b sin1
1
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
b2
CHAPTER 4 4.5 Fourier Series
0 1 2 3 4 5 6 7-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
X
Y
sin96.1fFigure 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
f(
y=2sin
CHAPTER 4 4.5 Fourier Series
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
CHAPTER 4 4.6 Cubic Splines
4.6 CUBIC SPLINES
4
1
1)(i
iixaxy (1<i<4)
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.
(4.55)
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
33,
22,1,,)( tatataatS iiioii
niforYXPSa iiiii .,.,.1),(0,
iii yxa 0,
niforYYXXt iiiii .,..,2121
211
.1,..,.10
,,,,,)(
1
33,3,
22,2,1,1,0,0,
niandttwhere
taataataaaatStStS
i
yixiyixiyixiyixiyixii
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)
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
UnknownsnaS
PtsControlnKnownsYXPaSTherefore
atataatd
tSdtSS
atatataatSS
i
iiiii
itiii
t
iii
itiiiiii
1,1
0,
1,0
23,2,1,
0
0,0
33,
22,1,0,
'
:),(
32)(
)0(''
)0(
3,3
3'"
3,2,2
2
23,2,1,
'
6)(
)(
62)(
)("
32)(
)(
ii
i
iii
i
iiii
i
atd
tSdtS
taatd
tSdtS
tataatd
tSdtS
33,
22,11 ')( tatatSStS iii
111
1111
')0(')('
)0()(
iiii
iiiii
StSttS
PStSttS
(4.60)
(4.61)
(4.62)
(4.63)
(4.64)
(4.65)
(4.66)
(4.67)
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
12
13,12,
13
13,2
12,1
''
'
iiiiii
iiiiiiii
StataS
StatatSS
)''(1
)(2
)2(13
121
131
3,
''1
112
12,
iii
iii
i
iii
iii
i
SSt
SSt
a
SSt
SSt
a
322
'2
22
'1
32
)212
2
'2
2
'1
22
12'11
(22)(3)( t
t
S
t
S
t
SSt
t
S
t
S
t
SSSStSi
Therefore, the spline function between P1 & P2 could simply be expressed as
(4.68)
(4.69)
(4.70)
(4.71)
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
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 S i(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?
32
1
'1
21
'1
31
12
1
'1
1
'
21
1' )(22)(3)( t
t
S
t
S
t
SSt
t
S
t
S
t
SStSStS
i
i
ii
ii
i
i
i
i
i
iiiii
taatS iii 3,2," 62
2," 20 ii aS
23,2,2" 62 taatS iii
0)( "12
" ii StS
(4.72)
(4.73)
(4.74)
(4.75)
(4.76)
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
)21()()/3(
)(2
12
2122
121
'21
'112
'2
niSStSSttt
StSttSt
iiiiiiii
iiiiiii
)()(3
)(33
3
(200
0)(200
0020
0002
212
12
11
23244
23
43
122323
22
32
'
'3
'2
'1
)1
4545
3434
2323
nnnnnnnn
nnnnn
SStSSttt
SStSSttt
SStSSttt
S
S
S
S
tttt
tttt
tttt
tttt
ti+2S’i
CHAPTER 4 4.7 Parametric Cubic Splines
i
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:
00"11" tSS
0)("" 1 nnn ttSS
212'2
'1 /)(5.15.0 tSSSS
)(/642 1''
1 nnnnn SStSS
b) Clamped Spline:
The boundary conditions for this spline are such that the first derivatives (slope) at t=0 and t=tn are specified.
(4.79)
(4.80)
(4.81)
(4.82)
CHAPTER 4 4.7 Parametric Cubic Splines
Adding Equations (4.81) and (4.82) to the n-2 equations given by Equation (4.78) we can solve for all the S’.
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
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).
Summary
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
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:
21
211 iiiii YYXXt
CHAPTER 4 4.7 Parametric Cubic Splines
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
3222313
122
21
''2')1
)(3
''2)0
StSttSti
SSt
SSi
122323
22
32
3SStSSt
tt
4333424 ''2')2 StSttSti 232434
23
43
3SStSSt
tt
)(3
'2')3 344
43 SSt
SSi
t32
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 siT CSC '
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
2100
031.1476.3707.00
0118.1298.4031.1
0012
TCwhere
4
34
232434
23
43
4
34
232434
23
43
122323
22
3212
2323
22
32
2
12
2
12
3
3
3
3
33
33
t
SS
SStSSttt
t
SS
SStSSttt
SStSSttt
SStSSttt
t
SS
t
SS
C
YY
yyyy
XX
xxxx
YYYYXXXX
YYXX
s
121.2121.2
672.1245.4
952.1637.4
683.2342.1
(4.86)
(4.85)
Last Eqn
t2t3
t42
t42
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
sTi CCS 1'
1,4
1,3
1,2
1,1
9745.06217.0
1720.08776.0
0996.07836.0
2917.12792.0
a
aa
a
= (4.87)
Since we have three splines we need to compute three coefficientsof ai,2 and ai,3.
To solve for Si’ we multiply equation (4.84) by [CT]-1
to get the ai,1 constants .
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
3,2,1'2'1
)(
31
12
1
12,
iforSStt
SSa ii
ii
iii
135.1361.0
067.1452.0
00
2,3
2,2
2,1
a
a
a
ii
i
ii
i
i SSt
SSt
a ''12
121
131
3,
536.0171.0
713.0263.0
317.0135.0
3,3
3,2
3,1
a
a
a
(4.88)
(4.90)
(4.89)
(4.91)
Si+1
Using equation (4.69) to find ai,2
Using equation (4.70) to find ai,3
(ti+1)3
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
32
3
322
321
536.0171.0135.1361.0172.0878.075.15.2
713.0263.0067.1452.01.0784.025.1
317.0135.000292.1279.011
tttS
tttS
tttS
1 1.5 2 2.5 3 3.5 1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
X
Y
Figure 4.10 Parametric cubic curve
(4.92)
S1
S2
S3
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
CHAPTER 4 4.8 Nonparametric Cubic Spline
4.8 NONPARAMETRIC CUBIC SPLINE
32 dxcxbxaS(x)
y)(xS ii
1i1i1i1i y)(xSxS
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)
(4.94)
(4.95)
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
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
3ii
2iiiiii xxd)x(xc)x(xba(x)S
2iiiii
'i )x(x3d)x(x2cbS
)x(x6d2cS iii"i
(4.98)
(4.99)
(4.100)
The non-parametric cubic spline can be expressed as:
Its first and second derivatives are
)(xS)(xS 1i'
1i1i'i
)(xS)(xS 1i"
1i1i"i
(4.96)
(4.97)
By considering the smoothness and continuity of the cubic splines the following conditions are derived:
CHAPTER 4 4.8 Nonparametric Cubic Spline
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
iiii ya)(xS
3ii
2iiiii1i1ii hdhchbaa)(xS
ii'i b)(xS
2iiiii1i1i
'1i1i
'i h3dh2cbb)(xS)(xS
iii1i1i"
1i1i"i h6d2c2c)(xS)(xS
(4.101)
(4.102)
(4.103)
(4.104)
(4.105)
CHAPTER 4 4.8 Nonparametric Cubic Spline
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
i1ii xxh
where
3
)c(2ch
h
aab 1iii
i
i1ii
3
)c(2ch
h
aab
as expressed be alsocan b
i1i1i
1i
1iii
i
1i
1ii
i
i1i1iiii1i1i1i h
aa
h
aa3chchh2ch
(4.106)
(4.107)
(4.108)
CHAPTER 4 4.8 Nonparametric Cubic Spline
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
2n
2n1n
1n
1nn
0
01
1
12
n
2
1
0
1n1n2n2n
322
2211
1100
h
aa
h
aa
h
aa
h
aa
3
c
c
c
c
hhh2h000
0hh2h00
0hhh2h0
00hhh2h
1.n1,...,iforAcH hi
3
)c(2ch
h
aab i1i1i
1i
1iii
1.n0,...,ifor3h
ccd
i
i1ii
(4.109)
(4.110)
(4.111)
(4.112)
CHAPTER 4 4.8 Nonparametric Cubic Spline
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
0)(xS")(xS" n0
0cc n0
4.9.2 Clamped Splines
)(xf')(xS' 00
)(xf')(xS' nn
S”(x0)
(4.113)
(4.114)
(4.115)
(4.116)
When substituted into equation (4.105) yields
CHAPTER 4 4.9 Boundary Conditions
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Example 4.6Find 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 (c0=c2=0) using equation ( 4.109 )
25.2
225.03 3
5.0
12
1
275.1301 15.0205.0
3 2
1
1
1
0
01
1
122111000
c
c
c
h
aa
h
aachchhch
i xi yi hi
0 1 1 0.5
1 1.5 2 1
n=2 2.5 1.75 -
CHAPTER 4 4.9 Boundary Conditions
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)
1.n0,...,iforh3
ccd
1.n0,...,ifor3
)c(2ch
h
aab
i
i1ii
1iii
i
i1ii
1.5h3
ccd
2.3753
)c(2ch
h
aab
0i
0
010
100
0
010
0.75h3
ccd
1.253
)c(2ch
h
aab
1i
1
121
211
1
121
CHAPTER 4 4.9 Boundary Conditions
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:
2.5x1.5 )x-0.75(x )x-2.25(x-)x-1.25(x2(x)S
1.5x1.0 )x-1.5(x-)x-2.375(x1(x)S
:bygiven are (4.98)equation from calculated splines theHence
3i
2ii2
3ii1
i xi hi yi=ai bi ci di
0 1 0.5 1 2.375 0 -1.5
1 1.5 1.0 2 1.25 -2.25 0.75
n=2 2.5 - 1.75 - 0 -
CHAPTER 4 4.9 Boundary Conditions
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
s1s2
CHAPTER 4 4.9 Boundary Conditions
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
iniin, t1t
i
n(t)J
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.
i)!(ni!
n!
i
n
where
*2)(n*1)(n*nn!
1)t(0(t)JStS in,i
n
1i
The curve points are defined by
where i=1 to n, and the Si contain the vector components of the various
points.
(4.117)
(4.118)
(4.119)
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
n
i
in, n
ini)(ni
i
n
n
iJ
The following example illustrates the Bezier curve method of curve fitting.
Example 4.7Define the Bezier Curve that passes through the following points:
5210 10 PP
1654 32 PP
Find the Bezier curve space that passes through these points.
Solution
33,3
23,2
23,1
3303,0
t(t)J
t)(13t(t)J
t)3(1(t)J
t)(1t)(1(1)t(t)J
3,323,223,113,00 JPJPJPJPS(t)
(4.120)
(4.121)
(4.122)
CHAPTER 4 4.10 Bezier Curves
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
100 S
529.29.015.0 S
The resulting S (t) function is then found as
733.3102.2)35.0( S
43)5.0( S
733.3904.365.0 S
529.2099.5)85.0( S
16)1( S
TABLE 4.8 Evaluation of the Bezier function J3,1(I=0,1,2,3) in
terms of the parameter t.
t J3,0 J3,1 J3,2 J3,3
0 1 0 0 0
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 0.375 0.125
0.65 0.043 0.239 0.444 0.275
0.85 0.0034 0.0574 0.325 0.614
1 0 0 0 1
CHAPTER 4 4.10 Bezier Curves
X
3.5 4 4.5 5 5.5 6 0
2
4
6
8
10
12
14
16
x
y
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
Figure 4.12 Bezier curve
CHAPTER 4 4.10 Bezier Curves
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 )10()(,
1
ttJStS ini
n
i
dt
tJStJStdS ini
n
iini
n
i)()( ,
1,
1
ini
in tti
n
dt
dtJ )1()(,
11 )1()1()1(
iniini tt
i
nntt
i
ni
)100.4()1()1()1()( 11
iini
iini Stt
i
nnStt
i
ni
dt
tdS
(4.123)
(4.124)
(4.125)
(4.126)
CHAPTER 4 4.11 Bezier Curves
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
j
nn
j
nj
1
1)1(
i
nn
i
nin
1)(
iijni
n
i
SStti
nn
dt
tdS
1
11
0
)1(1)(
iini
n
ij
jnjn
j
Stti
ninStt
j
nj
dt
tdS 11
01
11
0
)1()()1(1
)1()(
(4.128)
CHAPTER 4 4.11 Bezier Curves
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
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.
)()( 11,0
nkkii
n
ittttNStS
1
1,1
1
1,,
)()()()()(
iki
kiki
iki
kiiki tt
tNtt
tt
tNtttN
where
0
1)(1, tN i rest theall
1 ii ttt
(4.129)
(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
knin 2
(4.133) n ik 1
ki0 0
t
:knots periodicNon b)
(4.132) )k ni(0 k -i T
:knots Periodic a)
:knots of types twoare There
i
i
kn
ki
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
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.
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
0
1)(
0
1)(
0
1)(
0
1)(
0
1)(
1,4
1,3
1,2
1,1
1,0
tN
tN
tN
tN
tN
else
else
else
else
else
ttt
ttt
ttt
ttt
ttt
5
4
3
2
1
4
3
2
1
0
(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
(4.138) Stt)S2t(1St)(1S(t)
t(t)N
(4.137) t)2t(1(t)N
t)(1(t)N
(4.136) t
t)(t
tt
)Nt(t(t)N
and
t)(1
(4.135) N t)(1
tt
t)N(t
tt
)Nt(t(t)N
22
102
32,3
1,3
20,3
4
23
2,122,2
2,1
23
2,13
12
1,111,2
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.
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 4.13 Non-Uniform B-Spline Curve
4.13 NON-UNIFORM RATIONAL B-SPLINE CURVE (NURBS)
n
0iki,ii (t))N.x(hx.h
n
0iki,ii (t))N.y(hy.h
n
0iki,ii (t))N.z(hz.h
n
0iki,i (t)Nhh
n
0iki,i
n
0iki,ii
(t)Nh
(t)NShS(t)The equation for NURBS curve S(t) is given by:
(4.139)
(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 4.9Derive 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:
CHAPTER 4 4.13 Non-Uniform 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
t0 = 0, t1 = 0, t2 = 0, t3 = 1, t4 = 1 and t5 = 1 h0 = 1,
2
2
2
11 h 12 h
t)2t(1(t)N
t)(1(t)N
t(t)N
t(t)N
t1(t)N
0
1(t)N
where
(t)Nh(t)Nh(t)Nh
(t)NSh(t)NSh(t)NShS(t)
1,3
20,3
22,3
2,2
1,2
2,1
2,321,310,30
2,3221,3110,300
(4.141)
(4.142)
(4.143)
CHAPTER 4 4.13 Non-Uniform 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
with S0 = P0, S1 = P1 and S2 = P2 ; after substitution the NURBS equation is then found to be :
22
22
1tt)2t(12
2t)1.(1
t
0
1
0
1t)2t(1
0
1
1
2
2t)(1
0
0
1
1.
S(t)
(4.144)
CHAPTER 4 4.13 Non-Uniform 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 4.15 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
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 4.16 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
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
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
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
•Revolve-Create
Figure 4.20 Revolved surface
CHAPTER 4 4.19 Application Software
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
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
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
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
•Offset-Create
Figure 4.24 Offsetting of a surface
CHAPTER 4 4.19 Application Software
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
•Copy-Create
Figure 4.25 Copying of a surface by selection method
CHAPTER 4 4.19 Application Software