Implicitization of curves and surfaces using predicted support

Abstract

We reduce implicitization of rational parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For this, we may use any method for predicting the implicit support. We focus on methods that exploit input structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial. We offer a public-domain implementation of our methods, and study their numerical stability and efficiency on several classes of plane curves and surfaces, and discuss how it can be used for approximate implicitization in the setting of sparse elimination.