Editing De Boor's algorithm (section)

== Local support ==

B-splines have local support, meaning that the polynomials are positive only in a compact domain and zero elsewhere. The Cox-de Boor recursion formula<ref>C. de Boor, p. 90</ref> shows this:
<math display="block">B_{i,0}(x) :=
\begin{cases}
1 & \text{if } \quad t_i \leq x < t_{i+1} \\
0 & \text{otherwise}
\end{cases}
</math>
<math display="block">B_{i,p}(x) := \frac{x - t_i}{t_{i+p} - t_i} B_{i,p-1}(x) + \frac{t_{i+p+1} - x}{t_{i+p+1} - t_{i+1}} B_{i+1,p-1}(x). </math>

Let the index <math> k </math> define the knot interval that contains the position, <math> x \in [t_{k},t_{k+1}) </math>. We can see in the recursion formula that only B-splines with <math> i = k-p, \dots, k </math> are non-zero for this knot interval. Thus, the sum is reduced to:
<math display="block"> \mathbf{S}(x) = \sum_{i=k-p}^{k} \mathbf{c}_i B_{i,p}(x). </math>

It follows from <math> i \geq 0 </math> that <math> k \geq p </math>. Similarly, we see in the recursion that the highest queried knot location is at index <math> k + 1 + p </math>. This means that any knot interval <math> [t_k,t_{k+1}) </math> which is actually used must have at least <math> p </math> additional knots before and after. In a computer program, this is typically achieved by repeating the first and last used knot location <math> p </math> times. For example, for <math> p = 3 </math> and real knot locations <math> (0, 1, 2) </math>, one would pad the knot vector to <math> (0,0,0,0,1,2,2,2,2) </math>.