WOLFRAM NOTEBOOK

Calculating and Plotting B-Spline Basis Functions

​
p
2
case
plot
formula
schematic diagram
N
i,p
N
0,2
N
1,2
N
2,2
N
3,2
N
4,2
N
5,2
Let
U={
u
0
,
u
1
,…,
u
m
}
be a nondecreasing sequence of real numbers, that is,
u
i
≤
u
i+1
,
i=0,…,m-1
. The
u
i
are called knots and
U
is the knot vector. The
th
i
B-spline basis function of degree
p
(or order
p+1
), denoted by
N
i,p
(u)
, is defined as
N
i,0
(u)=
1
if
u
i
≤u<
u
i+1
0
otherwise
and for
i=1,…,m-(p+1)
, as
N
i,p
(u)=
u-
u
i
u
i+p
-
u
i
N
i,p-1
(u)+
u
i+p+1
-u
u
i+p+1
-
u
i+1
N
i+1,p-1
(u)
.
This Demonstration assumes the knots are
{0,0,1,1,2,3,4,4,5}
in order to calculate and plot the
N
i,p
(u)
.

Details

For drawing the schematic diagram of the algorithm, see[2].

References

[1] L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Berlin: Springer-Verlag, 1997 pp. 50–51.
[2] ShutaoTang. "Drawing the Schematic Diagram of Algorithm." Mathematica Stack Exchange. (Oct 18, 2014) mathematica.stackexchange.com/questions/63471/drawing-the-schematic-diagram-of-algorithm.
[3] Michael E2. Answer to "How to Deal with the Condition
u
i
=
u
i+1
in B-Spline Basis Function?" Mathematica Stack Exchange. (Oct 15, 2014) mathematica.stackexchange.com/questions/63192/how-to-deal-with-the-condition-u-i-u-i1-in-b-spline-basis-function.
[4] Mr.Wizard. Answer to "A Problem about Sorting Final Result." Mathematica Stack Exchange. (Oct 18, 2014) mathematica.stackexchange.com/questions/63468/a-problem-about-sorting-final-result.
​

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