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v
1
i
=
B
2
i
(
t
i
)
−
(1
−
t
i
)
2
v
o
−
t
i
2
v
2
.
(3.6)
2
t
i
(1
−
t
i
)
n
−
2
1
v
1
i
be the average value of the second control points for (
n
-
Let
v
1
=
n
−
2
i
=1
2) p
o
lynomials an
d l
et the corresponding B-B
pol
ynomial with control points
v
o
,
v
1
,and
v
2
be
B
2
(
t
i
). The discrete form of
B
2
(
t
i
) can be written as
t
i
)
2
v
o
+2
t
i
(1
t
i
)
v
1
+
t
i
2
v
2
.
B
2
(
t
i
)=(1
−
−
(3.7)
From equations (3.5) and (3.7),
B
2
i
(
t
i
)
v
1
i
|
B
2
(
t
i
)
−
|
=
|
v
1
−
|×
2
t
i
(1
−
t
i
)
.
(3.8)
This equation denotes the absolute difference between the polynomial
B
2
(
t
i
)
and an arbitrary
i
th quadratic
B-B
polynomial
B
2
i
(
t
i
) at an instant
t
i
.The
maximum absolute difference of
B
2
(
t
i
)and
B
2
i
(
t
i
)is
B
2
i
v
1
i
|
B
2
−
|
max
=
|
v
1
−
|
max
×
[2
t
i
(1
−
t
i
)]
max
(3.9)
v
1
i
1
=
|
v
1
−
|
max
×
2
.
Note that
t
i
(1
−
t
i
) is always positive. Similarly,
B
2
i
v
2
i
|
B
2
−
|
min
=
|
v
1
−
|
min
×
[2
t
i
(1
−
t
i
)]
min
.
(3.10)
1
2
The expression
t
i
(1
−
t
i
) has maximum at
t
=
and the value falls sym-
1
metrically either side as
t
moves away from
2
. Since
t
i
∈
(0
,
1), the expression
2
t
i
(1
t
i
) is minimum for the possible minimum/maximum value of
t
i
.For
equally spaced data points, the minimum possible value of
t
i
is
−
1
(
n−
1)
and the
t
i
)]
min
=
2(
n
−
2)
maximum possible value of
t
i
is
n
−
2
n−
1
. In either case, [ 2
t
i
(1
−
(
n−
1)
2
.
With this,
(
n−
1)
2
2(
n
v
1
i
B
2
i
|
v
1
−
|
min
=
|
B
2
−
|
−
2)
min
(3.11)
1)
2
2(
n−
2)
(
n
−
=
min
and
B
2
i
v
1
i
|
v
1
−
|
max
=2
|
B
2
−
|
max
(3.12)
=2
max
B
2
i
B
2
i
where
|
max
=
max
are respectively
the minimum and maximum absolute errors in approximating a function
f
(
t
)
and
t
i
(1
|
B
2
−
|
=
min
and
|
B
2
−
min
1
−
t
i
) is maximum at
t
i
=
2
. It is straightforward to observe from
equation (3.11) and (3.12) that
v
1
i
v
1
i
v
1
i
|
v
1
−
|
min
≤|
v
1
−
|≤|
v
1
−
|
max
,
(3.13)
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