Graphics Reference
In-Depth Information
Control points
b
i
, i = 0,1, º ,d in
R
m
Fixed
distinct
real numbers s and t
u Œ
R
Inputs:
Output:
p(u) as defined by equation (11.92)
Let u = a
0
s + a
1
t , where a
0
+ a
1
= 1 .
Set
b
i
0
=
b
i
.
Step 1:
Step 2:
For r = 1,2, º ,d and i = 0,1, º ,d-r compute
b
r
= a
0
b
r-1
r-1
+ a
1
b
i
+
1
.
When one has finished,
b
d
= p(u) .
Algorithm 11.5.2.1.
The de Casteljau algorithm.
11.5.2.6
Theorem.
The rth derivative of the Bézier curve p(u) defined by equation
(11.92) is given by
r
d
du
d
dr
!
()
=
rdr
-
()
pu
D
b
0
u
.
(
)
r
-
!
Proof.
See [Fari97]. To show the reader some more sample computations with blos-
soms, we will work through the main steps for the case r = 1. Let P(u
1
,...,u
d
) be the
blossom of p(u). To simplify the notation we assume that [s,t] = [0,1]. Then
Pu
(
,...,
u
+
h
,...,
u
)
=- +
(
1
(
u
h Pu
)
) (
,...,
u
, ,
0
u
,...,
u
)
1
i
d
i
1
i
-
1
i
+
1
d
++
(
uhPu
) (
,...,
u
, ,
1
u
,...,
u
)
i
1
i
-
1
i
+
1
d
=-
(
1
uPu
)
(
,...,
u
, ,
0
u
,...,
u
)
i
1
i
-
1
i
+
1
d
+
uPu
(
,...,
u
, ,
1
1
u
,...,
u
)
i
1
i
-
1
i
+
1
d
[
]
+
hPu
(
,...,
u
, ,
u
,...,
u
)
-
Pu
(
,...,
u
, ,
0
u
,...,
u
)
1
i
-
1
i
+
1
d
1
i
-
1
i
+
1
d
=
Pu
(
,...,
u
,
uu
,
,...,
u
)
1
i
-
1
i
i
+
1
d
[
]
+
hPu
(
,...,
u
, ,
1
u
,...,
u
)
-
Pu
(
,...,
u
, ,
0
u
,...,
u
)
.
1
i
-
1
i
+
1
d
1
i
-
1
i
+
1
d
This formula shows that
∂
∂
P
u
Pu
(
,...,
u
+
h
,...,
u
)
-
Pu
(
,...,
u
,...,
u
)
1
i
d
1
i
d
=
lim
h
h
Æ
0
i
=
Pu
(
,..., ,...,
1
u
)
-
Pu
(
,..., ,...,
0
u
)
.
1
d
1
d
