82 research outputs found
Reverse engineering of CAD models via clustering and approximate implicitization
In applications like computer aided design, geometric models are often
represented numerically as polynomial splines or NURBS, even when they
originate from primitive geometry. For purposes such as redesign and
isogeometric analysis, it is of interest to extract information about the
underlying geometry through reverse engineering. In this work we develop a
novel method to determine these primitive shapes by combining clustering
analysis with approximate implicitization. The proposed method is automatic and
can recover algebraic hypersurfaces of any degree in any dimension. In exact
arithmetic, the algorithm returns exact results. All the required parameters,
such as the implicit degree of the patches and the number of clusters of the
model, are inferred using numerical approaches in order to obtain an algorithm
that requires as little manual input as possible. The effectiveness, efficiency
and robustness of the method are shown both in a theoretical analysis and in
numerical examples implemented in Python
Compactifications of rational maps, and the implicit equations of their images
In this paper we give different compactifications for the domain and the
codomain of an affine rational map which parametrizes a hypersurface. We
show that the closure of the image of this map (with possibly some other extra
hypersurfaces) can be represented by a matrix of linear syzygies. We compactify
into an -dimensional projective arithmetically
Cohen-Macaulay subscheme of some . One particular interesting
compactification of is the toric variety associated to the
Newton polytope of the polynomials defining . We consider two different
compactifications for the codomain of : and .
In both cases we give sufficient conditions, in terms of the nature of the base
locus of the map, for getting a matrix representation of its closed image,
without involving extra hypersurfaces. This constitutes a direct generalization
of the corresponding results established in [BuseJouanolou03],
[BuseChardinJouanolou06], [BuseDohm07], [BotbolDickensteinDohm09] and
[Botbol09].Comment: 2 images, 28 pages. To appear in Journal of Pure and Applied Algebr
Mini-Workshop: Surface Modeling and Syzygies
The problem of determining the implicit equation of the image of a rational map φ : P2 99K P3 is of theoretical interest in algebraic geometry, and of practical importance in geometric modeling. There are essentially three methods which can be applied to the problem: Gröbner bases, resultants, and syzygies. Elimination via Gröbner basis methods tends to be computationally intensive and, being a general tool, is not adapted to the geometry of specific problems. Thus, it is primarily the latter two techniques which are used in practice. This is an extremely active area of research where many different perspectives come into play. The mini-workshop brought together a diverse group of researchers with different areas of expertise
Tropical secant graphs of monomial curves
The first secant variety of a projective monomial curve is a threefold with
an action by a one-dimensional torus. Its tropicalization is a
three-dimensional fan with a one-dimensional lineality space, so the tropical
threefold is represented by a balanced graph. Our main result is an explicit
construction of that graph. As a consequence, we obtain algorithms to
effectively compute the multidegree and Chow polytope of an arbitrary
projective monomial curve. This generalizes an earlier degree formula due to
Ranestad. The combinatorics underlying our construction is rather delicate, and
it is based on a refinement of the theory of geometric tropicalization due to
Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular,
old Sections 4 and 5 are merged into a single section. Also, added Figure 3
and discussed Chow polytopes of rational normal curves in Section
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