The normalized B-spline blending functions are defined recursively by
These functions are difficult to calculate directly for a general knot sequence. However, if the knot sequence is uniform, it is quite straightforward to calculate these functions - and they have some suprising properties.
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Calculating the Blending Functions using a Uniform Knot Sequence
Assume that is a uniform knot sequence, i.e., . This will simplify the calculation of the blending functions, as .
Blending Functions for k = 1
if , then by using equation (1), we can write the normalized blending functions as
These functions have support (the region where the curve is nonzero) in an interval, with having support on . They are also clearly shifted versions of each other - e.g., is just shifted one unit to the right. In fact, we can write
Blending Functions for k = 2
If then can be written as a weighted sum of and by equation (2). This gives
These functions are commonly referred to as ``hat'' functions and are used as blending functions in many linear interpolation problems.
Similarly, we can calculate to be
Finally, we have that
These nonzero portion of these curves each cover the intervals spanned by three knots - e.g., spans the interval . The curves are piecewise linear, made up of two linear segments joined continuously.
Sinve the curves are shifted versions of each other, we can write
Blending Functions for k = 3
For the case , we again use equation (2) to obtain
The nonzero portion of these two curves each span the interval between four consecutive knots - e.g., the nonzero portion of spans the interval . Again, can be seen visually to be a shifted version of . (This fact can also be seen analytically by substituting for in the equation for .) We can write
Blending Functions of Higher Orders
It is not too difficult to conclude that the blending functions will be piecewise cubic functions. The support of will be the interval and each of the blending functions will be shifted versions of each other, allowing us to write
In the case of the uniform knot sequence, the blending functions are fairly easy to calculate, are shifted versions of each other, and have support over a simple interval determined by the knots. These characteristics are unique to the uniform blending functions.
For other special characteristics see the notes that describe writing the uniform blending functions as a convolution, and the notes that describe the two-scale relation for uniform B-splines