(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

for and . These equations have the following properties [175]:

*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 [314]