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
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Definition of the Blending Functions Utilizing Convolution
The uniform th order B-spline blending function is defined recursively by
The First Order Blending Function
The first order blending function is just the Haar scaling function
The support of this function is the interval .
The Second Order Blending Function
To calculate the second order blending function we must calculate
where in each case we have shaded the areas between the limits of integration 0 and .
So we have that
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
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
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.