Graphics Reference
In-Depth Information
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
x
Fig. 9.8.
Quadratic Bezier curve between (1, 1) and (4, 3), with (3, 4) as the control
vertex.
9.3.3 Cubic Bernstein Polynomials
One of the problems with quadratic curves is that they are so simple. If we
wanted to construct a complex curve with several peaks and valleys, we would
have to join together a large number of such curves. A
cubic curve
,onthe
other hand, naturally supports one peak and one valley, which simplifies the
construction of more complex curves.
When
n
= 3 in (9.7), we obtain the following terms:
t
)
t
)
3
=(1
t
)
3
+3
t
(1
t
)
2
+3
t
2
(1
t
)+
t
3
((1
−
−
−
−
(9.12)
which can be used as a cubic interpolant, as
t
)
3
+
V
c1
3
t
(1
t
)
2
+
V
c2
3
t
2
(1
t
)+
V
2
t
3
V
=
V
1
(1
−
−
−
(9.13)
Once more the terms sum to unity, and the convex hull and slope properties
also hold. Figure 9.9 shows the graphs of the four polynomial terms.
This time we have two control values,
V
c1
and
V
c2
. These can be set to any
value, independent of the values chosen for
V
1
and
V
2
. To illustrate this, let's
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. 9.9.
The cubic Bernstein polynomial curves.
