Tatjana Kalinka

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial. In this work, we study how this interpolation matrix can be used to reduce some key geometric predicates on the hyper-surface to simple numerical operations on the matrix, namely membership and sidedness for given query points. We illustrate our results with examples based on our Maple implementation.

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.

ABSTRACT Cylindrical Algebraic Decomposition (CAD, first introduced in [Col75]) of Euclidean space has become an important tool in mathematics and allows for practical quantifier elimination (QE) over the reals. Much research has gone into improving the projection ...