Abstract. Implicitization and parameterization are two fundamental problems in computational algebraic geometry, with immediate practical applications to computer-aided geometric design (CAGD) where one typically requires thousands of conversions between implicit and parametric forms of equations of curves and surfaces in real time. Thus it is important to have good algorithms at hand for both problems.

Many efficient and robust algorithms for implicitization and parameterization have been developed in Symbolic Computation using a wide variety of techniques such as resultants, Gröbner bases, characteristic sets, perturbations, multidimensional Newton formulae, symmetric functions, elimination, moving frames, linear systems, integral bases, polynomial factorization and adjoint functions. These algorithms provide constructive answers to the implicitization and parameterization problems and have been implemented in commercial and freely available Computer Algebra Systems as well as in stand-alone programs. Some of the available algorithms are designed for special categories of curves and surfaces only.

We give an overview of the available algorithms and report on a new implicitization algorithm as well as some future work.