7.6 Nonparametric Curves

A functional, or nonparametric, integral B zier curve is defined by

It is easily represented in terms of a parametric B zier curve: its B zier points are given by b _i = [ i/ n, b _i] ^T. This works since the x coordinates satisfy

(7.25)

The situation in the rational case is somewhat more complicated. Let

(7.26)

be a rational function of x. Can we write it as a parametric rational B zier curve, and if so, what are its B zier points b _i and the corresponding weights v _i? Written in parametric form, (7.26) takes the form

Using the identity and degree elevation, we obtain

where

and

In the integral case, the abscissae i/ n of the B zier points of a functional B zier curve do not depend on the function; now they depend on the weights of the function under consideration. Thus a rational function y = a( x)/ b( x) with both a and b polynomials of degree n has a rational parametric representation of degree n + 1.

The abscissae iw _{i- 1}/ v _i are in the range [0, 1]. But note that they do not necessarily have to be increasing! For example, if the original weight sequence { w _i} is 1, 10, 1, 10, then the abscissae sequence { iw _{i- 1}/ v _i} is 0, 1/31, 10/11,...