Details

For definitions and details of the B-spline curve and B-spline basis function, see Related Links.

Assume that and are given, where . We seek a -degree nonrational curve

, ,

satisfying

(1) , ,

(2) the remaining are approximated in the least-squares sense; that is,

is a minimum with respect to the variables.

Here, and are precomputed parameter values.

Let

, .

Then set

.

So is a scalar-valued function of the variables . Applying the standard technique of linear least-squares fitting to minimize, set the derivatives with respect to the points, , to zero. The of these derivatives is

,

which implies that

.

It follows that

. (1)

Equation (1) is one linear equation in the unknowns . Letting yields the system of equations in unknowns,

,

where is the matrix of scalars

, (2)

is the vector of points

, (3)

and

.

In order to set up equations (2) and (3), a knot vector with parameters is required.

The can be computed using the following method (with , in each case):

(1) Equally spaced

, ,

(2) Chord length

let , then

, ,

(3) Centripetal

let , then

, ,

The placement of the knots should reflect the distribution of the . If , denote by the largest integer less than or equal to . We need a total of knots; there are internal knots and internal knot spans.

Let

.

Then define the internal knots by

, ,

, . (4)

Equation (4) guarantees that every knot span contains at least one , and under this condition the matrix is positive definite and well-conditioned.

Reference

[1] L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Berlin: Springer-Verlag, 1997 pp. 410–413.