The act of finding or constructing a model for a portion of a given polynomial or Bezier parametric surface from the whole original one is an encountered problem in surface modeling. A new method is described for constructing polynomial or Bezier piecewise model from an original one. It is based on the "Parametric Piecewise Model," abbreviated to PPM, of curve representation. The PPM representation is given by explicit expressions in terms of only control points or polynomial coefficients. The generated piecewise model behaves completely as a normal, polynomial or Bezier model in the same way as the original one for the piece of region considered. Also it has all characteristics, i. e, order and number of control points as the original one, and satisfies at the boundaries all order continuities. The PPM representation permits normalization, piecewise modeling, PPM reduction and systematic processes.

In this paper, a new explicit transformation method between Bezier and polynomial representation is proposed. An expression is given to approximate (n + 1) Bezier control points by another of (m + 1), and to perform simple and sufficiently good approximation without any additional transformation, such as Chebyshev polynomial. A criterion of reduction is then deduced in order to know if the given number of control points of a Bezier curve is reducible without error on the curve or not. Also an error estimation is given only in terms of control points. This method, unlike previous works, is more transparent because it is given in form of explicit expressions. Finally, we discuss some applications of this method to curve-fitting, order decreasing and increasing number of control points.