## B-Spline

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

 (1)

where is a nondecreasing Sequence with , and define control points , ..., . Define the degree as
 (2)

The knots'' , ..., are called Internal Knots.

Define the basis functions as

 (3) (4)

Then the curve defined by
 (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.