Analysis and construction of rational curve parametrizations with non-ordinary singularities (Q668991)

Rational algebraic curves, i.e., curves which can be represented as the (Zariski-closure of the) image of a pair of rational functions, play an important role in computer aided geometric design. Bézier curves are examples of rational curves. Indeed exactly the irreducible curves of genus 0 admit a rational parametrizaton. Often we want to transform the defining algebraic equation of a curve into a rational representation, and vice versa. Such transformations (parametrization and implicitization) are readily available in the literature. But genus computation and blow-ups around a non-ordinary singularity are notoriously complex procedures. What this paper achieves is a new approach to the characterization of non-ordinary singularities and their neighboring points by means of linear equations involving a given parametrization. Based on such a characterization one can design rational curves with predescribed singularities (ordinary or not) and prescribed multiplicities.