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In-Depth Information
5.2 B-Spline Function
Cox [46] and de Boor [54] independently put forward a recursive definition for
numerical computation of normalized B-spline basis function. An (
m
−
1)
th
degree B-spline curve P(u) is defined as
n
P
(
u
)=
B
i,m
(
u
)
V
i
2
≤
m
≤
n
+1
,
(5.1)
i
=0
where
V
i
is the
i
th control point of the (
n
+1)th point control polygon vertices
and
B
i,m
are the B-spline blending functions, which are basically polynomials
of degree m-1.
B
i,m
are also called the B-spline basis functions. The order
m
canbechosenfrom2to
n
+ 1. The basis function
B
i,m
(
u
) is defined by the
recursion formula of Cox-de Boor.
t
i
)
t
i
+
m−
1
−
(
u
−
t
i
B
i,m−
1
(
u
)+
(
t
i
+
m
−
u
)
B
i,m
(
u
)=
t
i
+1
B
i
+1
,m−
1
(
u
)
,
(5.2)
t
i
+
m
−
where,
t
i
≤
u
≤
t
i
+
m
and
B
i,
1
(
u
)=
1
if t
i
≤
u
≤
t
i
+1
(5.3)
0
otherwise.
The
t
i
s in equation (5.2) are elements of a knot vector. From the equation
(5.2), it is clear that the basis function
B
i,m
(
u
) is non-zero in the interval
[
t
i
,t
i
+
m
]. For a cubic B-spline,
m
=4and
B
i,
4
is non-zero in the interval
[
t
i
,t
i
+4
]. The basis function spans the knots
t
i
,t
i
+1
,t
i
+2
,t
i
+3
,t
i
+4
. Note
that when knots are not repeated, B-spline is zero at the end-knots
t
i
and
t
i
+
m
, i.e.,
B
i,m
(
u
=
t
i
)=0
,
i,m
(
u
=
t
i
+
m
)=0
.
But in B-splines, we use repeated knots (i.e.,
t
i
=
t
i
+1
=
···
). Therefore,
B
i,m
0
0
can have the form
0
. Hence, we assume
0
= 0 to incorporate repeated knots.
1)th degree curve,
P
(
u
), in equation (5.1) the parameter
u
ranges from 0 to
n
To trace an (
m
−
m
+ 2. It can be shown that for any value of the
parameter
u
, the sum of the basis functions is
n
−
B
i,m
(
u
)=1
.
(5.4)
i
=0
Therefore, the B-spline curve lies within the convex hull defined by its control
polygon, which is a similar property exhibited by the B-B curve.
5.2.1 B-Spline Knot Structure for Uniform, Open Uniform, and
Nonuniform Basis
The equation (5.2) shows that we need to choose a set of knots ,
t
i
,which
relate the parameter
u
to the control points. This relation, together with the
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