Changing Representation of Curves and Surfaces: Exact and Approximate Methods

In this thesis we explore recent methods for computing the Newton polytope of the implicit equation and study their applicability to the representation change from the parametric form to implicit. Computing a (super)set of the monomials appearing in the implicit equation allows us to determine the interpolation space. Following this phase we implement interpolation by exact or numeric linear algebra (applying singular value decomposition). We evaluate the monomials at the points, most suitable for the task, thus building a numeric matrix, ideally of corank 1, whose kernel vector contains the coefficients of the implicit equation. We propose techniques for handling the case of higher corank. This yields an efficient, output-sensitive algorithm for computing the implicit equation. The method can be applied to polynomial or rational parameterizations of planar curves or (hyper)surfaces of any dimension including parameterizations with base points. Moreover, this technique can be used for problems such as the computation of the discriminant of a multivariate polynomial or the resultant of a system of multivariate polynomials.