Editing De Boor's algorithm (section)

== Introduction ==

A general introduction to B-splines is given in the [[B-spline|main article]]. Here we discuss de Boor's algorithm, an efficient and numerically stable scheme to evaluate a spline curve <math> \mathbf{S}(x) </math> at position <math> x </math>. The curve is built from a sum of B-spline functions <math> B_{i,p}(x) </math> multiplied with potentially vector-valued constants <math> \mathbf{c}_i </math>, called control points,
<math display="block"> \mathbf{S}(x) = \sum_i \mathbf{c}_i B_{i,p}(x). </math>
B-splines of order <math> p + 1 </math> are connected piece-wise polynomial functions of degree <math> p </math> defined over a grid of knots <math> {t_0, \dots, t_i, \dots, t_m} </math> (we always use zero-based indices in the following). De Boor's algorithm uses [[Big O notation|O]](p<sup>2</sup>) + [[Big O notation|O]](p) operations to evaluate the spline curve. Note: the [[B-spline|main article about B-splines]] and the classic publications<ref name="de_boor_paper"></ref> use a different notation: the B-spline is indexed as <math> B_{i,n}(x) </math> with <math>n = p + 1</math>.