Game Development Reference
In-Depth Information
11.3 Bézier Curves
Although we shall limit ourselves to studying the cubic variety, a Bézier (pro-
nounced BAY-ZEE-AY) curve can be defined for any polynomial degree n . Giv-
en
points 0 ,,, n
PP
, called the control points of the curve, the degree n
n
+
1
()
Bézier curve
B
t
is given by the parametric function
n
()
()
B
t
=
B
t
P ,
(11.16)
nk
,
k
k
=
0
()
t are the Bernstein polynomials defined by
where the blending functions
B
nk
n

nk
k
(
)
B
=
t
1
t
(11.17)

nk
,
k
with the binomial coefficient
n
n
!
 
=
.
(11.18)
 
(
)
k
kn k
!
!
 
The first and last control points, P and P , are interpolated by the curve, and the
interior control points 12
PP are approximated by the curve. The Bern-
stein polynomials can be generated by the recurrence relation
,
,
,
n
1
() (
)
B
t
=−
1
t B
+
tB
,
(11.19)
nk
,
nk
−−
1,
1
nk
1,
where
. As shown in Figure 11.3,
this recurrence resembles Pascal's triangle, but with the modification that each
value is the weighted average of the two closest values above it instead of the
sum.
B
=
1
, and
B
=
0
whenever
<
0
or k
>
n
k
0,0
nk
,
11.3.1 Cubic Bézier Curves
The cubic Bézier curve has four control points whose positions are blended to-
gether by evaluating Equation (11.16) for
n
=
3
:
3
()
()
B
t
=
B
t
P
3,
k
k
k
=
0
(
)
3
(
)
2
2
(
)
3
=−
1
t
P
+
3
t
1
t
P
+
3
t
1
t
P
+
t
P .
(11.20)
0
1
2
3
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