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, Boston, 1993.

【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.