https://inria.hal.science/hal-02278661Emiris, IoannisIoannisEmirisDI NKUA - Department of Informatics and Telecomunications [Kapodistrian Univ] - NKUA - National and Kapodistrian University of AthensAROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - NKUA - National and Kapodistrian University of AthensKonaxis, ChristosChristosKonaxisDI NKUA - Department of Informatics and Telecomunications [Kapodistrian Univ] - NKUA - National and Kapodistrian University of AthensAROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - NKUA - National and Kapodistrian University of AthensLaroche, ClémentClémentLarocheDI NKUA - Department of Informatics and Telecomunications [Kapodistrian Univ] - NKUA - National and Kapodistrian University of AthensAROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - NKUA - National and Kapodistrian University of AthensImplicit representations of high-codimension varietiesHAL CCSD2019randomized algorithmconical hypersurfaceresultantMaple implementationImplicitizationspace curveChow form[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Laroche, Clément2019-09-05 10:44:282023-03-15 08:58:092019-09-05 11:22:14enJournal articleshttps://inria.hal.science/hal-02278661/document10.1016/j.cagd.2019.07.003application/pdf1Implicitization usually focuses on plane curves and (hyper)surfaces, in other words, varieties of codimension 1. In this paper we shift the focus on space curves and, more generally, on varieties of codimension larger than 1, and discuss approaches that are not sensitive to base points.Our first contribution is a direct generalization of an implicitization method based on interpolation matrices for objects of high codimension given parametrically or as point clouds. Our result shows the completeness of this approach which, furthermore, reduces geometric operations and predicates to linear algebra computations.Our second, and main contribution is an implicitization method of parametric space curves and varieties of codimension > 1, which exploits the theory of Chow forms to obtain the equations of conical (hyper)surfaces intersecting precisely at the given object. We design a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, which is the most common case, our algorithm returns 3 surfaces whose polynomials are of near-optimal degree; moreover, computation reduces to a Sylvester resultant. We illustrate our algorithm through a series of examples and compare our Maple code with other methods implemented in Maple. Our prototype is not faster but yields fewer equations and is more robust than Maple's implicitize. Although not optimized, it is comparable with Gröbner bases and matrix representations derived from syzygies, for degrees up to 6.