# Is cardinal $B$-spline of order $n$ really piecewise Bezier order $n$ curve?

Is cardinal $B$-spline of order $n$ really piecewise Bezier curve $n$? I think I saw this in some lecture notes, but I can't recall where.

## 1 Answer

Any b-spline curve is just a piecewise polynomial, i.e. a sequence of polynomial segments, joined end-to-end. And, in fact, in deBoor's well-known book on splines, he talks a lot about the piecewise polynomial representation of splines.

Each of the polynomial pieces can be represented as a Bézier curve (because every polynomial can be represented using the Bernstein basis).

So, yes, every parametric polynomial spline curve is just a sequence of Bézier curves strung together.

To compute the control points of the Bezier curves from the control points of the b-spline, you can use Boehm's algorithm, as outlined in this answer: Convert a B-Spline into Bezier curves

The book by Carl deBoor is an excellent account of both theory and practice. The title is "A Practical Guide to Splines". I'd give you a link, but my ability to insert links seems to be broken. It's a well-known book, and should be easy to find at Amazon or your local university library.

Another good one is "Bezier and B-Spline Techniques" by Hartmut Prautzsch and Wolfgang Boehm.

If you want something really theoretical, try "Spline Functions: Basic Theory" by Larry Schumaker.

• Thanks! Could you recommend a good introduction to $B$-splines? I liked the "practical" approach by Gerald Farin (Curves and Surfaces for Computer-AIded Geometric Design) but now I'd like something with theoretical slant. – blazs Sep 15 '15 at 7:56