** Overview**

These notes present the direct definition of the B-Spline curve. This definition is given in two ways: first by an analytical definition using the normalized B-spline blending functions, and then through a geometric definition.

To get a pdf version of these notes look here.

** The B-Spline Curve - Analytical Definition**

A B-spline curve , is defined by

where

- the are the control points,
- is the order of the polynomial segments of the B-spline curve. Order means that the curve is made up of piecewise polynomial segments of degree ,
- the
are the ``normalized B-spline blending functions''.
They are described by the order
and by a non-decreasing sequence of
real numbers

normally called the ``knot sequence''. The functions are described as follows

and if ,

- and

We note that if, in equation (2), either of the terms on the right hand side of the equation are zero, or the subscripts are out of the range of the summation limits, then the associated fraction is not evaluated and the term becomes zero. This is to avoid a zero-over-zero evaluation problem. We also direct the readers attention to the ``closed-open'' interval in the equation (1).

The order is independent of the number of control points (). In the B-Spline curve, unlike the Bézier Curve, we have the flexibility of using many control points, and restricting the degree of the polymonial segments.

** The B-Spline Curve - Geometric Definition**

Given a set of Control Points , an order , and a set of knots , the B-Spline curve of order is defined to be

if |

where

and

It is useful to view the geometric construction as the following pyramid

Any in this pyramid is calculated as a convex combination of the two functions immediately to it's left.

2000-11-28