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.
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