B-spline basis functions. More...

#include <bspline.hpp>

Collaboration diagram for BSpline:

Public Member Functions
	BSpline (Natural p, Natural n, const std::vector< Real > &knots)

Real	operator() (Natural i, Real x) const

Private Member Functions
Real	N (Natural i, Natural p, Real x) const

Private Attributes
Natural	p_

Natural	n_

std::vector< Real >	knots_

Detailed Description

B-spline basis functions.

Follows treatment and notation from:

Weisstein, Eric W. "B-Spline." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/B-Spline.html

\( (p+1) \)-th order B-spline (or p degree polynomial) basis functions \( N_{i,p}(x), i = 0,1,2 \ldots n \), with \( n+1 \) control points, or equivalently, an associated knot vector of size \( p+n+2 \) defined at the increasingly sorted points \( (x_0, x_1 \ldots x_{n+p+1}) \). A linear B-spline has \( p=1 \), quadratic B-spline has \( p=2 \), a cubic B-spline has \( p=3 \), etc.

The B-spline basis functions are defined recursively as follows:

\[ \begin{array}{rcl} N_{i,0}(x) &=& 1 \textrm{\ if\ } x_{i} \leq x < x_{i+1} \\ &=& 0 \textrm{\ otherwise} \\ N_{i,p}(x) &=& N_{i,p-1}(x) \frac{(x - x_{i})}{ (x_{i+p-1} - x_{i})} + N_{i+1,p-1}(x) \frac{(x_{i+p} - x)}{(x_{i+p} - x_{i+1})} \end{array} \]

Definition at line 58 of file bspline.hpp.