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Finally, using the chain rule and the fact that blossoms are symmetric functions we
get
[
]
d
-
1
d
-
1
¢
()
=
[
(
)
-
(
)
]
=
pu
dP u
1
, ,...,
u P u
0
, ,...,
u
d
bb
-
,
1
0
which is the case r = 1 of the theorem.
Theorem 11.5.2.6 generalizes the formula in equation (11.57).
Next, let us see where an analysis using blossoms leads when applied to the B-
spline functions N
i,k
(u) given the nondecreasing knot sequence
t
== =
t
...
t
,
t
,...,
t
,
t
=
t
= =
...
t
.
0
1
k
-
1
k
n
n
+
1
n
+
2
n
+
k
Let N
i,k
(u) denote the restriction of N
i,k
(u) to the interval I
j
= [t
j
,t
j+1
). The function
N
i,k
(u) is a polynomial of degree k - 1. Let n
i,k
be its blossom.
The functions n
i,r
(
u
) satisfy the recurrence relation
11.5.2.7
Theorem.
j
()
=
n
d
,
ij
i
,
1
ut
t
-
-
-
+-
r
1
1
i
j
j
(
)
=
(
)
nuu
,
,...,
u
nuu u
,
,...,
12
r
-
1
12
r
-
2
ir
,
ir
,
-
1
t
ir
i
t
-
-
u
ir
+
+
r
j
(
)
+
n
u
,
u
,...,
u
2
2
,
£
rrk
£
.
(11.94)
11
12
r
-
i
+-
,
r
t
t
ir
i
+
1
Proof.
The theorem is proved by induction. One shows that the relation defines
symmetric multiaffine maps and that the diagonals clearly agree with the N
i,r
. See
[Seid89].
11.5.2.8
Corollary.
If t
j
< t
j+1
and j-r+1 £
£ j, then
j
nt t
(
,
,...,
t
)
=
d
.
l
+
1
l
+
2
l
+
r
-
1
i
l
ir
,
Proof.
One uses the formulas in Theorem 11.5.2.7 and induction on r. See [Seid89].
Corollary 11.5.2.8 leads to an important algorithm that relates the control points
of a B-spline to associated blossoms. Let
n
Â
()
=
()
pu
N
u
p
(11.95)
ik
,
i
i
=
0
be a B-spline curve of order k. Since p(u) is a polynomial over each knot interval, let
p
j
(u) be the polynomial which defines p(u) over the interval I
j
= [t
j
,t
j+1
] and let P
j
be
its blossom.
11.5.2.9
Theorem.
If t
j
< t
j+1
and j-k+1 £
£ j, then the
de Boor point
p
is defined
by
