Graphics Reference
In-Depth Information
You will discover (9.17) and (9.18) used in more advanced texts to describe
Bezier curves. Although they may initially appear intimidating, you should
now find them relatively easy to understand.
9.4 A recursive Bezier Formula
Note that (9.17) explicitly describes the polynomial terms needed to construct
the blending terms. With the use of
recursive functions
(a recursive function
is a function that calls itself), it is possible to arrive at another formulation
that leads towards an understanding of
B-splines
. To begin, we need to ex-
press
n
i
in terms of lower terms, and because the coe
cients of any row
in Pascal's triangle are the sum of the two coe
cients immediately above, we
can write
n
i
=
n
+
n
−
1
−
1
(9.20)
i
i
−
1
Therefore, we can write
B
i
(
t
)=
n
t
i
(1
t
)
n−i
+
n
t
i
(1
−
1
−
1
t
)
n−i
−
−
i
i
−
1
B
i
(
t
)=(1
t
)
B
n−
1
i
(
t
)+
tB
n−
1
−
i−
1
(
t
)
(9.21)
As with all recursive functions, some condition must terminate the process: in
this case it is when the degree is zero. Consequently,
B
0
(
t
)
1and
B
j
(
t
)
≡
≡
0
for
j<
0.
9.5 Bezier Curves Using Matrices
As we have already seen, matrices provide a very compact notation for alge-
braic formulae. So let's see how Bernstein polynomials lend themselves to this
form of notation. Recall (9.11), which defines the three terms associated with
a quadratic Bernstein polynomial. These can be expanded to
2
t
+
t
2
)(2
t
2
t
2
)(
t
2
)
(1
−
−
(9.22)
and can be written as the product of two matrices:
⎡
⎤
1
−
21
⎣
⎦
[
t
2
t
1]
·
−
220
100
(9.23)
This means that (9.13) can be expressed as
⎡
⎤
⎡
⎤
1
−
21
V
1
V
c
V
2
⎣
⎦
·
⎣
⎦
V
=[
t
2
t
1]
·
−
220
100
(9.24)
