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29
On the computation of the topology of a nonreduced implicit space curve
, 2008
"... An algorithm is presented for the computation of the topology of a nonreduced space curve defined as the intersection of two implicit algebraic surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the original space curve. The algorithm is designed to provide the exact result for al ..."
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Cited by 7 (1 self)
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An algorithm is presented for the computation of the topology of a nonreduced space curve defined as the intersection of two implicit algebraic surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the original space curve. The algorithm is designed to provide the exact result for all inputs. It’s a symbolicnumeric algorithm based on subresultant computation. Simple algebraic criteria are given to certify the output of the algorithm. The algorithm uses only one projection of the nonreduced space curve augmented with adjacency information around some “particular points ” of the space curve. The algorithm is implemented with the Mathemagix Computer Algebra System (CAS) using the SYNAPS library as a backend.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 7 (1 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm.
An Efficient Algorithm for the Stratification and Triangulation of Algebraic Surfaces
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS 43 (2010) 257–278. SPECIAL ISSUE ON SOCG’08
, 2010
"... We present a method to compute the exact topology of a real algebraic surface S, implicitly given by a polynomial f ∈ Q[x,y,z] of arbitrary total degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical po ..."
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Cited by 7 (6 self)
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We present a method to compute the exact topology of a real algebraic surface S, implicitly given by a polynomial f ∈ Q[x,y,z] of arbitrary total degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical points. We compute a stratification ΩS of S into O(N 5) nonsingular cells, including the complete adjacency information between these cells. This is done by a projection approach. We construct a special planar arrangement AS with fewer cells than a 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. Based on ΩS we also compute a simplicial complex which is isotopic to S. A complete C++implementation of the stratification algorithm is presented. It shows good performance for many wellknown examples from algebraic geometry.
Efficient Real Root Approximation
"... Weconsidertheproblemofapproximatingallrealrootsofasquarefree polynomial f. Given isolating intervals, our algorithm refines each of them to a width at most 2 −L, that is, each of the roots is approximated toLbitsafterthe binarypoint. Ourmethod provides a certified answer for arbitrary real polynomia ..."
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Cited by 4 (2 self)
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Weconsidertheproblemofapproximatingallrealrootsofasquarefree polynomial f. Given isolating intervals, our algorithm refines each of them to a width at most 2 −L, that is, each of the roots is approximated toLbitsafterthe binarypoint. Ourmethod provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precisionadaptively. Inthis way, weget acorrect algorithmthatissimpletoimplementandpracticallyefficient. Our algorithm uses the quadratic interval refinement method; we adapt thatmethodtobeabletocopewithinaccuracieswhenevaluating f, without sacrificingits quadratic convergence behavior. We prove a boundonthebitcomplexityofouralgorithmintermsofdegree,coefficientsizeanddiscriminant. Ourboundimproves previous work onintegerpolynomials byafactorofdeg f andessentiallymatches best known theoretical bounds on root approximation which are obtained by verysophisticated algorithms. Categories andSubject Descriptors
On the topology of real algebraic plane curves
, 2010
"... We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coo ..."
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Cited by 4 (2 self)
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We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use subresultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in nongeneric positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on highdegree nongeneric curves.
Arrangements on parametric surfaces II: Concretizations and applications
 IN COMPUTER SCIENCE
, 2010
"... We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamental ..."
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We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.
On the Complexity of Reliable Root Approximation
"... This is an authorprepared version of the article. The original publication is available at www.springerlink.com This work addresses the problem of computing a certified ǫapproximation of all real roots of a squarefree integer polynomial. We proof an upper bound for its bit complexity, by analyzin ..."
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Cited by 3 (2 self)
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This is an authorprepared version of the article. The original publication is available at www.springerlink.com This work addresses the problem of computing a certified ǫapproximation of all real roots of a squarefree integer polynomial. We proof an upper bound for its bit complexity, by analyzing an algorithm that first computes isolating intervals for the roots, and subsequently refines them using Abbott’s Quadratic Interval Refinement method. We exploit the eventual quadratic convergence of the method. The threshold for an interval width with guaranteed quadratic convergence speed is bounded by relating it to wellknown algebraic quantities. 1
A generic and flexible framework for the geometrical and topological analysis of (algebraic) surfaces
 Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling, ACM, Stony Brook
, 2008
"... We present a generic framework on a set of surfaces S in R3 that provides their geometric and topological analysis in order to support various algorithms and applications in computational geometry. Our implementation follows the generic programming paradigm, i.e., to support a certain family of sur ..."
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We present a generic framework on a set of surfaces S in R3 that provides their geometric and topological analysis in order to support various algorithms and applications in computational geometry. Our implementation follows the generic programming paradigm, i.e., to support a certain family of surfaces, we require a small set of types and some basic operations on them, all collected in a model of the newly presented SURFACETRAITS 3 concept. The framework obtains geometric and topological information on a nonempty set of surfaces in two steps. First, important 0and 1dimensional features are projected onto the xyplane, obtaining an arrangement AS with certain properties. Second, for each of its components, a sample point is lifted back to R3 while detecting intersections with the given surfaces. This idea is similar to Collins ’ cylindrical algebraic decomposition (cad). In contrast, we reduce the number of liftings using CGAL’s Arrangement 2 package as a basic tool. Properly instantiated, the framework provides main functionality required to support the computation of a Piano Mover’s instance. On the other hand, the complexity of the output is high, and thus, we particularly regard the framework as key ingredient for querying information on and constructing geometric objects from a small set of surfaces. Examples are meshing of single surfaces, the computation of spacecurves defined by two surfaces, to compute lower envelopes of surfaces, or as a basic step to compute an efficient representation of a threedimensional arrangement. We also inspirit the framework in two steps. First, we show that the wellknown family of algebraic surfaces fulfils the framework’s requirements. As robust implementations on these surfaces are lacking these days, we consider the framework to be an important step to fill this gap. Second, we instantiate the framework by a fullyfledged model for special algebraic surfaces, namely quadrics. This instantiation already supports main tasks demanded from rotational robot motion planning [Latombe 1993]. How to provide a model for algebraic surfaces of arbitrary degree, is partly discussed in [Berberich et al. 2008].
Complete, Exact and Efficient Implementation for Computing the Adjacency Graph of an Arrangment of Quadrics
"... We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, surfaces of algebraic degree 2. This is a major step towards the computation of the full arrangement. We enhanced an implementation for an exact parameterization of the intersectio ..."
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Cited by 2 (0 self)
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We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, surfaces of algebraic degree 2. This is a major step towards the computation of the full arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kind of inputs including all degenerate ones where intersection curves have singularities or pairs of curves intersect with high multiplicity. It is exact in that it always computes the mathematical correct result. It is efficient measured in running times, i.e. we compare it with a previous implementation based on planar arrangements of the projected intersection curves.
Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast
"... Abstract. Given a Cylindrical Algebraic Decomposition of an implicit algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize disti ..."
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Abstract. Given a Cylindrical Algebraic Decomposition of an implicit algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct connected arcs of an algebraic curve efficiently and precise (at a given resolution), irrespective of how close to each other they actually pass. Our hybrid method inherits the ideas of subdivision and curvetracking methods. With an adaptive mixedprecision model we can render the majority of algebraic curves using floatingpoint arithmetic without sacrificing the exactness of the final result. The correctness and applicability of our algorithm is borne out by the success of our webdemo 1 presented in [10].