Basis Functions for Rational Continuity

Abstract

The parametric or geometric continuity of a rational polynomial curve h a s often been obtained by requiring the homogeneous polynomial curve associated with the rational curve to possess parametric or geometric continuity, respectively. Recently this approach has been shown overly restrictive. We m a k e use of the necessary and su cient conditions of rational parametric continuity for de ning basis functions for the homogeneous representation of a rational curve.These functions are represented in terms of shape parameters of rational continuity, which a r e i n troduced due to these exact conditions. The shape parameters may b e v aried globally, a ecting the entire curve, or modi ed locally thereby a ecting only a few segments. Moreover, the local parameters can be represented as continuous or discrete functions. Based on these properties, we i n troduce three classes of basis functions which can be used for the homogeneous representation of rational parametric curves.

Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves

“…For every degree of continuity, the rational continuity constraints contain a degree of freedom not present in the corresponding continuity constraints for projected curves. The construction of spline curves based on these rational continuity constraints is investigated in [35,361.…”