Graphics Reference
In-Depth Information
Figure 11.8.
A Bézier curve and its data.
to the curve at these points by graphically picking any points
p
1
and
p
2
along these
lines. Then
¢
()
=
¢
()
=
p
p
0
1
a
b
pp
pp
,
,
01
(11.45)
23
for some a and b. Now let us turn this construction around. Rather than starting with
the curve p(u), let us start with the points
p
i
and ask what curve p(u) these points and
equations (11.45) define. We could of course let a and b be any fixed positive real
numbers, but for reasons that will become apparent shortly, we fix a=b=3.
By definition, the Hermite matrix
B
h
for the cubic curve p(u) is just
M
hb
B
b
, where
10 00
00 01
33 00
00 30
p
p
p
p
Ê
ˆ
Ê
ˆ
0
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
1
M
=
and
B
=
.
hb
b
-
2
Ë
¯
Ë
¯
-
3
The matrix
B
b
defines the geometric data for the Bézier curve. It follows that
()
=
pu
UM B
=
UM M
B
=
UM B
hh
h hbb
bb
=
FB
,
(11.46)
bb
where
F
b
=
UM
b
and
-
1331
3630
3300
1000
-
Ê
ˆ
Á
Á
Á
˜
˜
˜
-
MMM
=
=
(11.47)
b
h
hb
-
Ë
¯
Definition.
The elements of the matrix
B
b
, namely, the points
p
i
, are called the
Bézier
coefficients
of the curve p(u). The matrix
M
b
is called the
Bézier matrix
.
Multiplying the matrices in equation (11.44) leads to the following formula for
p(u):
3
2
()
=-
(
)
(
)
2
(
)
3
pu
1
u
p
+-
3
u
1
u
p
+
3
u
1
-
u
p
+
u
p
.
(11.48)
0
1
2
3
