B-Spline

A B-spline is 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)

where , 2, ..., . 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; illustrated above) and uniform B-spline (internal knots are equally spaced). A B-spline with no internal knots is a Bézier curve.

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.

-splines are implemented in the Wolfram Language as BSplineCurve[pts].