On-Line Geometric Modeling Notes
THE UNIFORM B-SPLINE BLENDING FUNCTION

Overview

The uniform B-splines are based upon a knot sequence that has uniform spacing. This implies that the uniform B-spline blending functions are all translates of a single blending function where

This single blending function can be defined by convolution of blending functions of lower degree. This is the topic of these notes.

Definition of the Blending Functions Utilizing Convolution

The uniform th order B-spline blending function is defined recursively by

and

That is, the th order blending function is defined by convolving the st order blending function with the first order blending function. This convolution can be seen to be the integral

The First Order Blending Function

The first order blending function is just the Haar scaling function

and is shown by the graph

The support of this function is the interval .

The Second Order Blending Function

To calculate the second order blending function we must calculate

The function is nonzero only when . Thus, we can get nonzero values in the integrand for any where . The integral splits naturally into the two cases shown below - for and .

where in each case we have shaded the areas between the limits of integration 0 and .

So we have that

which is illustrated by

It is clear that the support of is the interval

The Third Order Blending Function

To calculate the third order blending function, we must calculate

The function is nonzero only when , so we can get nonzero values in the integrand for any where .

This is straightforward to calculate once the reader sees that there are three cases, each depending on . These three cases are illustrated below as

In each case the section of the curve that lies between the integration bounds of 0 and has been shaded.

So now we can calculate the integral by

This curve is a piecewise quadratic - i.e. it has quadratic pieces that are smoothly joined together. The curve is drawn as

It is clear that the support of is the interval

Summary

The uniform B-spline is somewhat unique as all blending functions are given as a translate of only one function. We have shown here that this single blending function can be calculated in an interesting way using convolution.

Bibliography

1
BARTELS, R., BEATTY, J., AND BARSKY, B.
An Introduction to Splines for Use in Computer Graphics and Geometric Modeling.
Morgan Kaufmann Publishers, Palo Alto, CA, 1987.

2
UEDA, M., AND LODHA, S.
Wavelets: An elementary introduction and examples.
Technical Report UCSC-CRL-94-47, Jan. 1994.

Ken Joy
2000-11-28