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Equation (5.9) provides the basis functions over the
m
subintervals. It can be
decomposed as follows.
⎨
1
2
u
2
0
≤
u<
1
1
u
)+
2
(
u
B
0
,
3
(
u
)=
2
u
(2
−
−
1)(3
−
u
)1
≤
u<
2
(5.10)
⎩
1
u
)
2
2
(3
−
2
≤
u<
3
.
Computing
B
0
,
3
(
u
), we can get the other periodic basis functions by subtract-
ingashiftofoneunitfrom
u
.Thus,
⎧
⎨
1
1)
2
2
(
u
−
1
≤
u<
2
1
u
)+
2
(
u
B
1
,
3
(
u
)=
2
(
u
−
1)(3
−
−
2)(4
−
u
)2
≤
u<
3
(5.11)
⎩
1
u
)
2
2
(4
−
3
≤
u<
4
⎧
⎨
1
2)
2
2
(
u
−
2
≤
u<
3
1
u
)+
2
(
u
B
2
,
3
(
u
)=
2
(
u
−
2)(4
−
−
3)(5
−
u
)3
≤
u<
4
(5.12)
⎩
1
u
)
2
2
(5
−
4
≤
u<
5
⎧
⎨
1
3)
2
2
(
u
−
3
≤
u<
4
1
u
)+
2
(
u
B
3
,
3
(
u
)=
2
(
u
−
3)(5
−
−
4)(6
−
u
)4
≤
u<
5
(5.13)
⎩
1
u
)
2
2
(6
−
5
≤
u<
6
.
With all the basis functions in hand, we can draw the uniform periodic
quadratic B-spline curve.
5.4 B-Spline Curves on Unit Interval
We now want to examine the periodic B-spline curves on a unit interval, in-
stead of considering different intervals because for the periodic B-splines, the
blending functions in different intervals are translates of one another. There-
fore, we need to reparameterize the B-spline parameter on the unit interval.
We have already seen that the influence of a given blending function is limited
to
m
intervals. Hence, considering these facts, we can write a periodic B-spline
curve on the unit interval as
m−
1
P
j
(
s
)=
N
i
+1
,m
(
s
)
V
j
+
i
≤
j
≤
n
−
m
+1
1
(5.14)
i
=0
and,
0
≤
s<
1
.
In equation (5.14),
s
is the reparameterized form of the parameter
u
and
N
i,m
(
s
) is the reparameterized blending function corresponding to the blend-
ing function
B
i,m
(
u
);
j
gives the number of curve segments and
n
is one less
than the number of vertices of the control polygon. Equation (5.14) can be
extended as
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