On-Line Geometric Modeling Notes
THE SUPPORT OF A
NORMALIZED B-SPLINE BLENDING FUNCTION

Overview

A B-spline blending function has compact support. This means that the function is zero outside of some interval. In these notes, We find this interval explicitly in terms of the knot sequence.

The Support of the Function

Given an order , and a knot sequence , the normalized B-spline blending function is positive if and only if .

We can show that this is true by considering the following pyramid structure.

The definition of the normalized blending function as a weighted sum of and . Thus for any of the functions in the pyramid, it is a weighted sum of the two items immediately to its right. If we follow the pyramid to its right edge, we see that the only blending functions that contribute to are those with , and these function are collectively nonzero when .

Summary

A B-spline blending function has compact support. The support of this function depends on the knot sequence and always covers an interval of containing several knots - containing knots if the curve is or order .

Ken Joy
2000-11-28