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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.
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. 1
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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Cited by 4 (4 self)
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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.
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 3 (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. 1
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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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
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].
Arrangements on Surfaces of Genus One: Tori and Dupin Cyclides
"... An algorithm is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide, including the special case of the torus. The intersection of an algebraic surface of degree n with a reference cyclide is represented as a real algebraic curve of ..."
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An algorithm is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide, including the special case of the torus. The intersection of an algebraic surface of degree n with a reference cyclide is represented as a real algebraic curve of bidegree (2n, 2n) in the cyclide’s twodimensional parameter space. We use Eigenwillig and Kerber [11] to compute a planar arrangement of such curves and extend their approach to obtain more asymptotic information about curves approaching the boundary of the cyclide’s parameter space. With that, we can base our implementation on a general software framework by Berberich et. al. [3] to construct the arrangement on the cyclide. Our contribution provides the demanded techniques to model the special topology of the reference surface of genus one. Our experiments show no combinatorial overhead of the framework, i.e., the overall performance is strongly coupled to the efficiency of the implementation for arrangements of algebraic plane curves. 1
Geometric Analysis of Algebraic Surfaces Based on Planar Arrangements
"... 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 degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical points ..."
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Cited by 1 (0 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 degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S 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(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 C++implementation of the algorithm that shows good performance for many wellknown examples from algebraic geometry. 1
ACS Algorithms for Complex Shapes with Certified Numerics and Topology An implementation for the 2D Algebraic Kernel
"... Project cofunded by the European Commission within FP6 (2002–2006) We report on a model for CGAL’s AlgebraicKernelWithAnalysis d 2 concept which refines AlgebraicKernel d 2. Our implementation handles bivariate polynomials in full generality, i.e., with no restriction on their degree. Moreover, it ..."
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Project cofunded by the European Commission within FP6 (2002–2006) We report on a model for CGAL’s AlgebraicKernelWithAnalysis d 2 concept which refines AlgebraicKernel d 2. Our implementation handles bivariate polynomials in full generality, i.e., with no restriction on their degree. Moreover, it allows both integers and nested squareroot numbers as coefficient type. The Curve analysis and Curve pair analysis required by the concept are realized using recent work of Eigenwillig, Kerber and Wolpert (’Fast and exact geometric analysis...’, ISSAC 2007 and ’Exact and efficient 2DArrangements... ’. SODA 2008). The consequent use of certified numerical methods leads to significant speedups without spoiling exactness. We present benchmark results about the performance of several key methods. 1