1.4.1 B-splines

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1.4.1 B-splines

An order

B-spline is formed by joining several pieces of polynomials of degree

with at most $C^{k-2}$ continuity at the breakpoints. A set of non-descending breaking points $t_0 \leq t_1 \leq \ldots \leq t_m$ defines a knot vector

$\displaystyle {\bf T} = (t_0, t_1, \ldots, t_{m})\;,$

(1.57)

which determines the parametrization of the basis functions.

Given a knot vector ${\bf T}$ , the associated B-spline basis functions, $N_{i,k}(t)$ , are defined as:

$\displaystyle N_{i,1}(t) = \left\{ \begin{array}{ll} 1 &\mbox{for $t_i \leq t < t_{i+1}$ } \\ 0 &\mbox{otherwise}\;, \end{array}\right.$

(1.58)

for

, and

$\displaystyle N_{i,k}(t) = \frac{t-t_i}{t_{i+k-1}-t_i}N_{i,k-1}(t)+ \frac{t_{i+k}-t}{t_{i+k}-t_{i+1}}N_{i+1,k-1}(t)\;,$

(1.59)

for

and $i = 0,1,\ldots,n$ . These equations have the following properties [175]:

Positivity: $N_{i,k}(t) > 0$ , for $t_i < t < t_{i+k}$ .
Local support: $N_{i,k}(t) = 0$ , for $t_0 \leq t \leq t_{i}$ , and $t_{i+k} \leq t \leq t_{n+k}$ .
Partition of unity: $\sum_{i=0}^n N_{i,k}(t) = 1$ , for $t \in [t_{0}, t_{m}]$ .
Recursion: Given by (1.59).
Continuity: $N_{i,k}(t)$ has $C^{k-2}$ continuity at each simple knot.

The concept of nodes or Greville abscissae [130,92], which are the averages of the knots, are important in B-spline approximations [130,452] and defined as follows:

$\displaystyle \xi_i = \frac{1}{k-1}(t_{i+1} + t_{i+2} + \cdots + t_{i+k-1})\;.$

(1.60)

The node $\xi_i$ generally lies near the parameter value which corresponds to a maximum of the basis function $N_{i,k}(t)$ [345,314].

The derivative of the B-spline basis function is given by [314]

$\displaystyle \frac{dN_{i,k}(t)}{dt} = \frac{k-1}{t_{i+k-1}-t_i}N_{i,k-1}(t)- \frac{k-1}{t_{i+k}-t_{i+1}}N_{i+1,k-1}(t)\;.$

(1.61)

Next: 1.4.2 B-spline curve Up: 1.4 B-spline curves and Previous: 1.4 B-spline curves and Contents Index

December 2009