**Chapter 3 Interpolation and Polynomial Approximation**

**Parametric Interpolation/Approximation for Discrete
Nodes**

In this software, a polynomial or piecewise
polynomial parametric interpolation for is done by first introducing its associated
parametric sequence , with ,
and then constructing the interpolation polynomial or piecewise polynomial
individually for and ,
as in the previous section **Interpolation
for Monotonically Increasing Discrete Nodes**. Besides Lagrange, Hermite,
free and clamped cubic splines interpolation as in the previous section, Bezier
and B-spline interpolation/approximation are new here and are described as
below.

Given control points Bezier curve is defined as , where

and is Bernstein polynomial,

, where is the coefficient of binomial polynomial.

For interpolation in terms of Bezier curve,
given interpolation points control points are reversely determined and then the Bezier
curve is computed based on those control points, by de Casteljau algorithm
(illustrated in the **tracing point**
menu option), and displayed. For approximation in terms of Bezier curve, the
input points are seen as the control points and the
Bezier curve based on that is computed and displayed.

In the same way, given control points a B-spline of degree is defined as , where

where

,

and

is the basis polynomial of degree of B-spline which is nonzero only in . is called the knot vector, or knot sequence. In this software, the knot vector is determined in the way for uniform non-periodic B-spline:

when ,

with distinct values, equally spaced in , for .

When , the upper case is reduced to

Also when , , and the B-spline will be equivalent to the Bezier curve.

To illustrate knot vector, for example, given six control points , choosing B-spline of degree , the knot vector will be

and

involved with , associated with , are ;

involved with , associated with , are ;

involved with , associated with , are ;

involved with , associated with , are ;

involved with , associated with , are ;

involved with , associated with , are .

For interpolation in terms of a B-spline of
degree k, given interpolation points control points are reversely determined and then the B-spline
of degree is computed based on those control points, by
Cox de Boor algorithm (illustrated in the **tracing
point** menu option), and displayed. For approximation in terms of B-spline
of degree ,
the input points are seen as the control points and the
B-spline of degree based on that is computed and displayed.

References:

【1】
R. L. Burden and J. D. Faires, *Numerical Analysis*, PWS,

【2】
D. Kincaid and W. Cheney, *Numerical Analysis*, Brooks/Cole, Pacific Grove, 1996.

【3】
G. Farin, *Curves
and Surfaces for Computed Aided Geometric Design*, Academic Press, Boston,
1993.