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

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 [175]:

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 [314]

    (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