B-Spline

$\begin{figure}\begin{center}\BoxedEPSF{BSpline.epsf scaled 700}\end{center}\end{figure}$

A generalization of the Bézier Curve. Let a vector known as the Knot Vector be defined

$\begin{displaymath} {\bf T}=\{t_0, t_1, \ldots, t_m\}, \end{displaymath}$

(1)

where ${\bf T}$ is a nondecreasing Sequence with $t_i\in[0,1]$ , and define control points ${\bf P}_0$ , ..., ${\bf P}_n$ . Define the degree as

$\begin{displaymath} p\equiv m-n-1. \end{displaymath}$

(2)

The ``knots'' $t_{p+1}$ , ..., $t_{m-p-1}$ are called Internal Knots.

Define the basis functions as

$\displaystyle N_{i,0}(t)$	$\textstyle =$	$\displaystyle \left\{\begin{array}{ll} 1 & \mbox{if $t_i\leq t<t_{i+1}$\ and $t_i<t_{t+1}$}\\ 0 & \mbox{otherwise}\end{array}\right.$	(3)
$\displaystyle N_{i,p}(t)$	$\textstyle =$	$\displaystyle {t-t_i\over t_{i+p}-t_i}N_{i,p-1}(t)+{t_{i+p+1}-t\over t_{i+p+1}-t_{i+1}}N_{i+1,p-1}(t).$
			(4)

Then the curve defined by

$\begin{displaymath} {\bf C}(t)=\sum_{i=0}^n {\bf P}_iN_{i,p}(t) \end{displaymath}$

(5)

is a B-spline. Specific types include the nonperiodic B-spline (first

knots equal 0 and last

equal to 1) and uniform B-spline (Internal Knots are equally spaced). A B-Spline with no Internal Knots is a Bézier Curve.

The degree of a B-spline is independent of the number of control points, so a low order can always be maintained for purposes of numerical stability. Also, a curve is times differentiable at a point where duplicate knot values occur. The knot values determine the extent of the control of the control points.

See also Bézier Curve, NURBS Curve