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]:
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]
(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