Graphics Reference
In-Depth Information
Table 9.2.
Expansion of the terms t and (1
− t
)
i
n
0
1
2
3
4
1
t
(1
−
t)
2
t
2
t(1
−
t)
(1
−
t)
2
3
t
3
t
2
(1
−
t)
t(1
−
t)
2
(1
−
t)
3
4
t
4
t
3
(1
−
t)
t
2
(1
−
t)
2
t(1
−
t)
3
(1
−
t)
4
Table 9.3.
The Bernstein polynomial terms
i
n
0
1
2
3
4
1
1t
1(1
− t
)
2
1t
2
2t(1
−
t)
1(1
−
t)
2
3
1t
3
3t
2
(1
−
t)
3t(1
−
t)
2
1(1
−
t)
3
4
1t
4
4t
3
(1
−
t)
6t
2
(1
−
t)
2
4t(1
−
t)
3
1(1
−
t)
4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1 0.2 0.3 0.4 0.5
t
0.6 0.7 0.8 0.9
1
Fig. 9.3.
The graphs of the quadratic Bernstein polynomials.
t
)
2
graph starts at 1 and decays to zero, whereas the
t
2
graph starts at zero and
rises to 1. The 2
t
(1
Figure 9.3 shows the graphs of the three polynomial terms of (9.8). The (1
−
t
) graph starts at zero, reaches a maximum of 0.5 and
returns to zero. Thus the central polynomial term has no influence at the
end-points where
t
=0and
t
=1.
We can use these three terms to interpolate between a pair of values as
follows:
−
V
=
V
1
(1
− t
)
2
+2
t
(1
− t
)+
V
2
t
2
(9.9)
If
V
1
= 1 and
V
2
= 3 we obtain the curve shown in Figure 9.4. But there
is nothing preventing us from multiplying the middle term 2
t
(1
−
t
)byany
arbitrary number
V
c
:
t
)
2
+
V
c
2
t
(1
t
)+
V
2
t
2
V
=
V
1
(1
−
−
(9.10)
