Graphics Reference
In-Depth Information
The points
p
i
are called the
Bézier
or
control points
for p(u) and the polygonal curve
defined by them is said to form the
Bézier
or
characteristic
or
control polygon
for p(u).
The functions B
i,n
(u) are called the
Bézier basis functions
.
Note.
P. Bézier, who invented the Bézier curves, originally used coefficient functions
in (11.51) that were slightly different from but related to the Bernstein polynomials.
See [Fari97]. One does not use his original functions anymore because the Bernstein
polynomials are easier to use.
Bézier curves have a number of nice properties. The first of these is that they lie
in the convex hull of their characteristic polygon. This follows from the following two
facts and Theorem 1.7.2 in [AgoM05]:
(1) B
i,n
(u) ≥ 0.
n
Â
()
=
(2)
Bu
1
.
in
,
i
=
0
Equation (2) holds because the B
i,n
(u) are just the terms that one gets in the binomial
expansion of the right hand side of the equation
n
uu.
n
(
(
)
+
)
11
==-
1
It is also easy to check that the Bézier curve starts at the first control point and ends
on the last, that is, p(0) =
p
0
and p(1) =
p
n
.
Bézier curves are also
symmetric
. What this means is that if we list the control
points of the curve in reverse order we get the same curve, although it will be tra-
versed in the opposite direction. This follows from the easily checked fact that
()
=
(
)
BuB
1
-
u
,
(11.52)
in
,
n in
-
,
so that
n
n
Â
Â
()
=
(
)
Bu
p
B
1
-
u
p
.
in
,
i
in
,
n i
-
i
=
0
i
=
0
Next, we show that Bézier curves have a simple recursive definition by rewriting
the formula that defines them. Using the identity for binomial coefficients
n
i
n
i
-
1
n
i
-
-
1
1
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
+
Ê
Ë
ˆ
¯
=
,
(11.53)
we get
