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.