Editing B-spline (section)

==Definition==
[[File:Cardinal quadratic B spline.svg|thumb|Cardinal quadratic B-spline with knot vector (0, 0, 0, 1, 2, 3, 3, 3) and control points (0, 0, 1, 0, 0), and its first derivative]]
[[File:Cardinal cubic B-spline2.svg|thumb|Cardinal cubic B-spline with knot vector (−2, −2, −2, −2, −1, 0, 1, 2, 2, 2, 2) and control points (0, 0, 0, 6, 0, 0, 0), and its first derivative]]
[[File:Cardinal quartic B-spline.svg|thumb|right|216px|Cardinal quartic B-spline with knot vector (0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5) and control points (0, 0, 0, 0, 1, 0, 0, 0, 0), and its first and second derivatives]]

A B-spline of order <math>p+1</math> is a collection of [[piecewise]] [[polynomial]] functions <math>B_{i,p}(t)</math> of degree <math>p</math> in a variable <math>t</math>.
The values of <math>t</math> where the pieces of polynomial meet are known as knots, denoted <math>t_0, t_1, t_2, \ldots, t_m</math> and sorted into nondecreasing order.

For a given sequence of knots, there is, up to a scaling factor, a unique spline <math>B_{i,p}(t)</math> satisfying
:<math>B_{i,p}(t) = \begin{cases}
 \text{non-zero} & \text{if }  t_i < t < t_{i+p+1}, \\
 0 & \text{if }  t < t_i \lor t_{i+p+1} \le t.
\end{cases}
</math>

If we add the additional constraint that 
:<math>\sum^{m-p-1}_{i=0} B_{i,p}(t) = 1</math> 
for all <math>t</math> between the knots <math>t_{p}</math> and <math>t_{m-p}</math>, then the scaling factor of <math>B_{i,p}(t)</math> becomes fixed. 
The knots in-between (and not including) <math>t_{p}</math> and <math>t_{m-p}</math> are called the internal knots.

B-splines can be constructed by means of the Cox–de Boor recursion formula. 
We start with the B-splines of degree <math>p=0</math>, i.e. piecewise constant polynomials.

:<math>B_{i,0}(t) := \begin{cases}
 1 & \text{if } t_i \leq t < t_{i+1}, \\
 0 & \text{otherwise}.
\end{cases}
</math>

The higher <math>(p+1)</math>-degree B-splines are defined by recursion
:<math>B_{i,p}(t) := \dfrac{t - t_i}{t_{i+p} - t_i} B_{i,p-1}(t) + \dfrac{t_{i+p+1} - t}{t_{i+p+1} - t_{i+1}} B_{i+1,p-1}(t).</math>