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Limit ed. All right s reserved
11
P
3
P
2
P
1
P
Table 1: Some of the parameters that are commonly used in the system of units
System of Units I nput Par ameter s
Output Par ameter s
Lengt h
, ζ
For ce
, ƒ
Mass
, η
Density
, ρ
Displacement
, δ
For ce
, ƒ
Str ess
, υ
1 m
N kg
kgm
-3
m N
Pa 2
mm N
mg mgmm
-3
mm N
MPa 3
ft Ib
f
slug slugft
-3
ft Ib
f
psf 4
in Ib
f
Ib
f
.s
2
i n
-1
Ib
f
.s
2
i n
-4
in Ib
f
psi
3.0 M
ETHOD OF
D
ESCRIPTIONS
3.1 Bézier-Like Cubic Curve as a Function
Bézier -like cubic cur ve also know n as the cubic alter native cur ve is a new basis function in the field of Computer Aided Geometr ic Design CAGD. Bézier -like basis function simplifies the pr ocess of contr olling
the cur ve since it has only tw o shape par ameter s,
λ
and
λ
1
to contr ol or change the shape of the cur ve such in Figur e 4, compar ed to the cubic Bézier cur ve for w hich the contr ol points need to be adjusted
Farin, 1997. Bézier -like cubic cur ve is for mulated by applying the for m of Her mite inter polation given by Ali, 1994; Ali et al., 1996; Ahmad, 2009.
1 ,
3 3
2 2
1 1
t P
t P
t P
t P
t t
z
1 w ith,
1 2
1 ,
1 1
, 2
1 1
1 2
3 2
1 2
2 1
2
t t
t t
t t
t t
t t
t t
2
w her e 3
2 1
, ,
, P
P P
P
ar e the contr ol points, and 3
2 1
, ,
,
t t
t
t
ar e Bézier -like cubic basis functions.
Hence, Bézier -like cubic cur ve can be w r itten as 3
1 2
2 2
1 1
2 2
1 2
1 1
1 2
1 1
P t
t P
t t
tP t
P t
t t
z
3
Figure 4: Distribution of Bézier-like cubic curves with the different values of
λ and
λ
1
.
In the next subsection, w e w ill explain the contr ol point s identification in Bézier -like cubic cur ve based on the cir cle to cir cle template Figur e 2
3.2 Third Case of Circle Templates as an S-Shaped Transition Curve
The cubic Bézier cur ve had been pr edominantly used in cur ve design, as show n in studies by Habib and Sakai 2003 and Walton and Meek 1999. The similar ity betw een this function and Bézier -like cubic
cur ve is that both functions have four contr ol points. By r efer r ing to Habib and Sakai 2003, the contr ol points ar e stated as
© 2012 GETview
Limit ed. All right s reserved
12
} cos
, sin
{ },
sin ,
{cos }
cos ,
sin {
}, sin
, {cos
3 2
1 1
3 1
k P
P r
c P
h P
P r
c P
4 w her e, c
and c
1
are the centre points of circles ψ and ψ
1
, r and r
1
are the radii of circles ψ
and
ψ
1
, h and k ar e the length or nor m of
1
, P P
and
2 3
, P P
w hile
α
is an angles cir cles measur ed anticlockw ise. These contr ol points ar e unique only in the case of the S-shaped tr ansition cur ve. The
impr ovements ar e made in Eq. 4 to incr ease the degr ee of fr eedom and its applicability such
} cos
, sin
{ },
sin ,
{cos }
cos ,
sin {
}, sin
, {cos
3 2
1 1
3 1
k P
P r
c P
h P
P r
c P
5
with the angles in the circle {ψ , ψ
1
} denoted as
α
and
β
w hich ar e measur ed anticlockw ise. The other par ameter s r emain the same as in Eq. 4. The value of h and k w ill be calculated by using the
cur vatur e continuity G
2
continuity show n as
1
1 1
, 1
r t
r t
6 By applying these theor ies, an S-shaped cur ve is demonstr ated in Figur e 5.
Figure 5: An S-shaped transition curve produced by using G
2
Bézier-like cubic curve.
3.3 Fifth
