Graphics Reference
In-Depth Information
This cubic curve is called the cubic Bézier curve based on the points p i . Notice how
the coefficients add up to 1, which tells us that the curve lies in the convex hull of the
points p i . This is one reason for our particular choice of a and b in equation (11.45).
We have a new way to design curves. Picking four points produces a curve that
starts at the first and ends at the last and has a tangent vector at the beginning and
end that is parallel to the lines between the first and last two points, respectively. We
manipulate the curve by simply moving one or more of these “control points.”
Next, we shall generalize this construction from four points to an arbitrary
number. We want to be able to define a curve by outlining its desired shape with some
points. We shall describe two approaches to defining general Bézier curves. They can
be defined by
(1) starting with the Bernstein polynomial approximation of continuous functions
and continuing with a “brute” force approach to derive various properties, or
(2) using a more geometric “multiaffine” approach.
We begin with the first approach. Section 11.5.2 will deal with the second.
Let f : [0,1] Æ R m be a continuous function. Define
Definition.
n
f i
n
Ê
Ë
ˆ
¯
 0
()( ) =
()
Ffu
Bu
,
(11.49)
n
in
,
i
=
where
n
i
() = Ê
Ë
ˆ
¯
ni
-
i
(
)
Bu
uu
1
-
.
(11.50)
in
,
The polynomial function F n (f)(u) is called the Bernstein polynomial approximation of
degree n to f.
Bernstein used these polynomials to give a constructive proof of the Weierstrass
approximation theorem, which showed that every continuous function could be
approximated by a polynomial. One can show that the Bernstein polynomials con-
verge uniformly to f, but they converge very slowly and so they are not normally used
for that in mathematics since there are better ways to approximate functions by poly-
nomials. On the other hand, they lead to a representation for curves that is good for
interactive curve design.
Definition. Given a sequence of points p i in R m , i = 0,1,..., n, define the Bézier curve
p(u), u Π[0,1], by
n
Â
() =
()
pu
B
u
p
.
(11.51)
in
,
i
i
=
0
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