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.
On-Line Geometric Modeling Notes
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.
For a postscript version of these notes look here.
Definition of the Blending Functions Utilizing Convolution
The uniform kth order B-spline blending function 
is defined
recursively by
and
That is, the kth order blending function is defined by convolving the k-1st
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 [0,1].
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 t
where 0 < t < 2.
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 1.
So we have that
which is illustrated by
It is clear that the support
of 
is the interval [0,2]
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 t
where 0 < t < 3.
This is straightforward to calculate once the reader sees that there are three cases, each depending on t. These three cases are illustrated below as
In each case the section of the curve
that lies between the
integration bounds of 0 and 1 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 [0,3]
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.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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