The Derivative Vectors of a B-Spline
This Demonstration shows up to the first three derivative vectors of a B-spline.
As described in , a B-spline curve of degree is defined by
where are the control points and the are the degree- B-spline basis functions (see [Related Links]) defined on the nondecreasing knot vector , where and .
The functions depend only on the knots given by the knot vector ; they are defined recursively by
Using the equation of the derivative of the basis function ,
The first and last terms evaluate to , which  takes to be 0. Thus
where . (1)
Now let be the knot vector obtained by dropping the first and last knots from ; that is,
( has knots) (2)
Use the property: The function computed at is computed at . Thus
where the are defined by equation (1) and the are computed at .
Hence is a B-spline curve of degree .
Since is a B-spline curve, we can apply equations (1), (2), and (3) recursively to obtain higher derivatives. Letting , we write
In this Demonstration, assume the initial control points are , , , , , , and the knot vector is .
The derivatives are scaled down by to see them better.
 L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Berlin: Springer–Verlag, 1997 pp. 93–94, 97–98.
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