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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 continuity at the breakpoints. A set of non-descending breaking points defines a knot vector
 (1.57)

which determines the parametrization of the basis functions.

Given a knot vector , the associated B-spline basis functions, , are defined as:

 (1.58)

for , and
 (1.59)

for and . These equations have the following properties :
• Positivity: , for .
• Local support: , for , and .
• Partition of unity: , for .
• Recursion: Given by (1.59).
• Continuity: has 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:

 (1.60)

The node generally lies near the parameter value which corresponds to a maximum of the basis function [345,314].

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

 (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