Graphics Reference
In-Depth Information
Let
τ
a
=
{
τ
1
,τ
2
,
···
τ
l
}
be the knots in addition to existing ones. If
m
=
n
+
l
τ
a
is the new knot sequence in nondecreasing
order, then
f
(
x
) can be written as a linear combination of the B-splines,
N
j,k
on
t
, with
j
=1
,
2
,
and
t
=
{
t
1
,t
2
,
···
t
m
+
k
}
=
τ
∪
···
m
, i.e.,
m
f
(
x
)=
d
j
N
j,k
(
x
)
,
(7.2)
j
=1
d
j
s are unknown coecients and need to be computed. There are several ways
to compute
d
j
s. We cite a few of them.
(1) One can choose
m
points, say,
ρ
1
,ρ
2
,
···
,ρ
m
, and solve the following in-
terpolation problem:
m
d
j
N
j,k
(
ρ
i
)=
f
(
ρ
i
)
,
i
=1
,
2
···
,m.
(7.3)
j
=1
This set of linear equations has a unique solution
d
1
,d
2
,
···
d
m
if
t
j
<ρ
j
<
t
j
+
k
,
j
=1
,
2
,
m
. The coecient matrix is totally positive, banded, and can
be inverted by Gaussian elimination without pivoting in
O
(
mk
3
) operations
[57].
(2) Another technique to compute
d
j
s is to use the quasi-interpolant of deBoor
and Fix [56]. If
···
k−
1
1
1)
k−
1
−r
Ψ
(
r
)
j
(
a
j
)
f
(
k−
1
−r
)
(
a
j
)
,
λ
i
f
=
(
−
(7.4)
(
k
−
1)!
r
=0
where
a
j
is any point on (
t
j
,t
j
+
k
)and
k−
1
Ψ
j
(
y
)=
(
y
−
t
j
+
r
)
,
(7.5)
r
=1
then
λ
j
N
i,k
=
δ
i,j
=1
,
i
=
j,
=0
,
i
=
j.
Therefore, applying
λ
j
on both sides of equation (7.2), we get
d
j
=
λ
j
f,
j
=1
,
2
,
···
m.
(7.6)
Computation of
d
j
gives advantages provided
f
is given in its piecewise poly-
nomial representation.
(3) One can also compute
d
j
recursively. This method is similar to the subdi-
vision scheme of Lane and Risenfeld [100] for the special case of Bezier curves
(k-tuple knots) and for uniform knots.
Let us assume
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