Graphics Reference
In-Depth Information
Figure 11.9.
Graphing a Bézier curve geometrically.
Control points
p
i
, i = 0,1,...,n in
R
m
for a Bzier curve p(u)
u Π[0,1]
Inputs:
Output:
p(u)
Set
p
i
0
=
p
i
.
Step 1:
Step 2:
For r = 1,2,...,n and i = 0,1,...,n-r compute
p
r
= (1-u)
p
r-1
r-1
.
+ u
p
i+1
When one has finished,
p
0
n
= p (u) .
Algorithm 11.4.1.
The de Casteljau algorithm.
points
C
and
D
are one third of the way from
A
to
B
and from
B
to
C
. The point p(1/3)
is one third of the way from
D
to
E
.
The geometric construction just described translates into the de Casteljau evalu-
ation algorithm shown in Algorithm 11.4.1.
11.4.1
Theorem.
Algorithm 11.4.1 computes p(u).
Proof.
Clear.
Algorithm 11.4.1 can be visualized via the following triangular array:
pp
...
...
p
p
01
n
-
1
n
1
1
1
p
p
p
n
-
2
n
-
1
...
...
n
-
1
n
-
1
pp
p
0
1
n
(11.55)
0
