B´
ezier Curves
James Keesling
1
Definition of a B´
ezier Curve
A B´
ezier curve is a parametric curve that is used widely in computer graphics.
They
were originally used as an aid in design in the automobile industry. The B´
ezier curves are
based on the Bernstein polynomials. The Bernstein polynomial of degree
n
is given by the
following formula.
B
n
(
t
) =
n
X
i
=0
n
i
t
i
·
(1

t
)
n

i
In themselves the Bernstein polynomials are not very interesting.
They are just the
binomial expansion of (
t
+ (1

t
))
n
≡
1. However, for B´
ezier curves we will restrict
t
to
the interval [0
,
1].
Then the individual terms in the expansion will be nonnegative and
will sum to one.
We will use them to produce a convex combination of points in
R
k
.
Let
{
P
0
, P
1
, . . . , P
n
}
be a set of
n
+ 1 points in
R
k
. The B´
ezier curve with control points
{
P
0
, P
1
, . . . , P
n
}
is given by the following formula.
B
(
t
) =
n
X
i
=0
n
i
t
i
·
(1

t
)
n

i
·
P
i
The curve begins at the point
P
0
and ends at point
P
n
, but will not generally go through
the other control points.
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 Summer '08
 Kyung
 Statistics, Englishlanguage films, Bez, Bernstein polynomial, B´zier curve

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