Suppose we are given a set of points

,

, and we want to interpolate these points with a

th-degree non-rational B-spline curve. If we assign a parameter value,

, to each

, and select an appropriate knot vector

, we can set up the

system of linear equations

The control points,

, are the

unknowns. Let

be the number of coordinates in the

(typically 2, 3, or 4). Eq. (1) has one coefficient matrix, with

right hand sides and, correspondingly,

solution sets for the r coordinates of the

.

The problem of choosing the

and

remains, and their choice affects the shape and parameterization of the curve. We assume that the parameter lies in the range

Three common methods of choosing the

are:

let

, then

let

, then

Lastly, the following method was used to generate the knots. Namely,

To understand the process of this algorithm, let us give a example.

Let

, and assume that we want to interpolate the

with a cubic curve. We use Eqs. (3) and (5) to compute the

and

, and then set up the system of linear equations, Eq. (1). The separate chord lengths are

and the total chord length is

. Thus

So

hence

The system of linear equations is

[1] L. Piegl and W. Tiller,

*The NURBS Book*, 2nd ed., Berlin: Springer-Verlag, 1997 pp. 364-369.