Graphics Reference
In-Depth Information
p>
M
t
2
δ
2
.
(1.10)
From equation (1.10) it is clear that 2
is the error for a given approximation.
So, once we choose the error for an approximation,
then corresponding to
this
, we can search the data set and determine
δ
and hence the order of the
polynomial. For two dimensions, the extension is straightforward.
Example
1
1+
t
Approximate
f
(
t
)=
with a Bernstein polynomial for which
|
B
p
(
t
)
−
f
(
t
)
<
0
.
9.
We have,
|
|
M
t
B
kp
(
t
)
−
f
(
t
)
|
<
2
whenever
2
pδ
2
<
. Thus, we can write,
t
1
1
2
2
|
1+
t
−
1+
t
|
<
0
.
45 whenever
|
−
t
|
<
3
.Soweconsider
δ
=
3
. Also, from
equation (1.10),
M
t
2
δ
2
1
2(0
.
45)(2
/
3)
2
<
=2
.
5
.
Since,
p>
M
t
2
δ
2
we can choose,
p
= 3 (considering the nearest integer).
Hence,
3
i
f
(
i
3
3
)
t
i
(1
t
)
3
−i
B
3
(
t
)=
−
i
=0
t
)
3
+9
/
4
t
(1
t
)
2
+9
/
5
t
2
(1
=(1
−
−
−
t
)
,
is the required polynomial. Here,
f
(0) = 1,
f
(1
/
3) = 3
/
4,
f
(2
/
3) = 3
/
5and
f
(1) = 1
/
2).
1.3.2 Bezier-Bernstein Polynomial
The elementary properties of the Bernstein polynomial show that during ap-
proximation of a data set, having some ordered representative points
f
(
p
),
the approximating polynomial always remains confined within the convex hull
of the representative points of the data set. The polynomial interpolates the
end points of the ordered representative set of points. All other points are
approximated by the polynomial.
Bezier-Bernstein polynomial (BBP) of degree p is mathematically defined
as
p
P
(
t
)=
φ
ip
(
t
)
V
i
≤
t
≤
1
.
0
i
=0
The polynomial is based on the Bernstein basis or the blending function, given
by
φ
ip
(
t
)=
p
i
t
i
(1
t
)
p−i
,
−
i
∈
[0
,p
]
.
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