Degree Reduction Approximations

The majority of CAD systems provide some form of parametric curve and surface representation but the precise form of this representation varies considerably. Some systems use simple parametric polynomial curves and surfaces, others provide some form of rational parametric representation. Even amongst those systems using parametric polynomials they differ in the maximum degree allowed and the form of representation, which may be explicit polynomial base, B-spline or Bézier. Provided the degrees are the same, exact conversions between these different representations are possible as detailed in Chapter 4 but approximation problems arise when transferring data from a rational or other non-polynomial based system to a polynomial system. The generalised Chebyshev technique described in this chapter addresses the degree reduction problem, the orthogonal polynomial methods described in Chapter 6 can be used for all other types of approximation problem.