47 research outputs found

    Real algebraic surfaces with isolated singularities

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    Given a real algebraic surface S in RP3, we propose a constructive procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP3, S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the weighted 2-adjacency graph of the surface

    Index to Volumes 37 and 38

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    Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities

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    A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a symbolic-numeric method to compute a certified meshing for the surface inside a box containing singularities and use a modified Plantinga-Vegter marching cube method to compute a certified meshing for the surface inside a box without singularities. Nontrivial examples are given to show the effectiveness of the algorithm. To our knowledge, this is the first method to compute a certified meshing for surfaces with singularities.Comment: 34 pages, 17 Postscript figure

    Exact geometric-topological analysis of algebraic surfaces

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    We present a method to compute the exact topology of a real algebraic surface SS, implicitly given by a polynomial f∈Q[x,y,z]f \in \mathbb{Q}[x,y,z] of arbitrary degree NN. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of SS including critical points. We use a projection approach, similar to Collins' cylindrical algebraic decomposition (cad). In comparison we reduce the number of output cells to O(N5)O(N^5) by constructing a special planar arrangement instead of a full cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. We provide a complete implementation of the algorithm, written in C++. It shows good performance for many well known examples from algebraic geometry

    Isotopic triangulation of a real algebraic surface

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    International audienceWe present a new algorithm for computing the topology of a real algebraic surface SS in a ball BB, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to SS, and even a simplicial complex isotopic to SS by exploiting properties of the contour curve of SS. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of SS, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of SS from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper

    On the isotopic meshing of an algebraic implicit surface

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    International audienceWe present a new and complete algorithm for computing the topology of an algebraic surface given by a squarefree polynomial in Q[X, Y, Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in [15], on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S. We exploit simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in OB (d15 Ï„ ) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficients polynomial of degree d and coefficients size Ï„ . Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces

    Topologically certified approximation of umbilics and ridges on polynomial parametric surface

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    Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and encode important informations used in surface analysis or segmentation. But reporting the ridges of a surface requires manipulating third and fourth order derivatives whence numerical difficulties. Additionally, ridges have self-intersections and complex interactions with the umbilics of the surface whence topological difficulties. In this context, we make two contributions for the computation of ridges of polynomial parametric surfaces. First, by instantiating to the polynomial setting a global structure theorem of ridge curves proved in a companion paper, we develop the first certified algorithm to produce a topological approximation of the curve P encoding all the ridges of the surface. The algorithm exploits the singular structure of P umbilics and purple points, and reduces the problem to solving zero dimensional systems using Gröbner basis. Second, for cases where the zero-dimensional systems cannot be practically solved, we develop a certified plot algorithm at any fixed resolution. These contributions are respectively illustrated for Bezier surfaces of degree four and five

    The role of data in model building and prediction: a survey through examples

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    The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play-and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components
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