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

Overview

A B-spline blending function $N_{i,k}(t)$ 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.

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The Support of the Function

Given an order $k$, and a knot sequence $\left\{ t_0, t_1, t_2, ..., t_{n+k} \right\}$, the normalized B-spline blending function $N_{i,k} (t)$ is positive if and only if $t \in [t_i , t_{i+k})$.

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

$$\begin{array}{ccccccc} & & & & & & N_{i,1} \\ & & & & & ... ...\ & & & & & & \\ & & & & & & N_{i+k-1,1} \\ \end{array}$$

The definition of the normalized blending function $N_{i,k}$ as a weighted sum of $N_{i,k-1}(t)$ and $N_{i+1,k-1}(t)$. Thus for any of the $N$ 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 $N_{j,1}$ that contribute to $N_{i,k}$ are those with $i \leq j \leq i+k-1$, and these function are collectively nonzero when $t \in [t_i , t_{i+k})$.

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 $k+1$ knots if the curve is or order $k$.