 # The Derivative Vectors of a B-Spline Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows up to the first three derivative vectors of a B-spline.

Contributed by: Shutao Tang (June 2015)
(Northwestern Polytechnical University, Xi'an City, China)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

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 , (3)

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 .

Then (4)

with ( )

and .

In this Demonstration, assume the initial control points are , , , , , , and the knot vector is .

The derivatives are scaled down by to see them better.

References

 L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Berlin: Springer–Verlag, 1997 pp. 93–94, 97–98.

 施法中. 计算机辅助几何设计与非均匀有理B样条 CAGD&NURBS[M]．北京：高等教育出版社. 1994 pp. 244–247.

## Permanent Citation

Shutao Tang

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send