hal-02278661
https://inria.hal.science/hal-02278661
https://inria.hal.science/hal-02278661/document
https://inria.hal.science/hal-02278661/file/ImplicitRepresentationsOfHighCodimensionVarieties.pdf
doi:10.1016/j.cagd.2019.07.003
[INRIA] INRIA - Institut National de Recherche en Informatique et en Automatique
[INRIA-SOPHIA] INRIA Sophia Antipolis - Méditerranée
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[INRIASO] INRIA-SOPHIA
[INRIA_TEST] INRIA - Institut National de Recherche en Informatique et en Automatique
[TESTALAIN1] TESTALAIN1
[INRIA2] INRIA 2
[UNIV-COTEDAZUR] Université Côte d'Azur
[INRIAARTDOI] INRIAARTDOI
Implicit representations of high-codimension varieties
Emiris, Ioannis,
Konaxis, Christos
Laroche, Clément
[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]
ART
randomized algorithm
conical hypersurface
resultant
Maple implementation
Implicitization
space curve
Chow form
Implicitization 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.
2019-10
2019-09-04
en
Computer Aided Geometric Design
Elsevier