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On the Complexity of Computing the Topology of Real Algebraic Space Curves
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Abstract
In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve. We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm. It is , where , are the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve. To our knowledge, this is the best bound among the existing work. It gains the existing results at least .
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In this paper, we present a new method for computing the topology of curves defined as the intersection of two implicit surfaces.
The main ingredients are projection tools, based on resultant constructions and 0-dimensional polynomial system solvers. We
describe a lifting method for points on the projection of the curve on a plane, even in the case of multiple preimages on
the 3D curve. Reducing the problem to the comparison of coordinates of so-called critical points, we propose an approach which
combines control and efficiency. An emphasis in this work is put on the experimental validation of this new method. Examples
treated with the tools of the library axel1 (Algebraic Software-Components for gEometric modeLing) are showing the potential of such techniques.
The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.
A local generic position method is proposed to isolate the real roots of a bivariate polynomial system Σ={f(x,y),g(x,y)}=0· In this method, the roots of the system are represented as linear combinations of the roots of two univariate polynomial equations t(x)=0 and T(X)=0: x=α,y=β-α s|α∈V(t(x)),β∈V(T(X)),|β-α|<S, where s,S are constants satisfying certain conditions. The multiplicities of the roots of Σ=0 are the same as those of the corresponding roots of T(X)=0. This representation leads to an efficient and stable algorithm to isolate the real roots of Σ=0
An algorithm is proposed to determine the topology of an implicit real algebraic surface inR3. The algorithm consists of four steps: surface pro- jection, projection curve topology determination, space curve segmenta- tion and surface patch composition, combination of surface patches and surface topology representation. The topology is represented by a set of surface patches, and each surface patch is presented by an ordered list of space curve segments. The relationship between the surface patches can be found by their space curve segments. Some examples show that our algorithm is efiective.
In this paper we describe an algorithm for computing the dual of a projective plane curve. The algorithm requires no extension of the field of coefficients of the curve and runs in polynomial time.
In this paper we give a new projection-based algorithm for computing the topology of a real algebraic space curve given implicitly by a set of equations. Under some genericity conditions, which may be reached through a linear change of coordinates, we show that a plane projection of the given curve, together with a special polynomial in the ideal of the curve contains all the information needed to compute its topological shape. Our method is also designed in such a way to exploit important particular cases such as complete intersection curves or curves contained in nonsingular surfaces.
We introduce a new algorithm denoted DSC2 to isolate the real roots of a
univariate square-free polynomial f with integer coefficients. The algorithm
iteratively subdivides an initial interval which is known to contain all real
roots of f. The main novelty of our approach is that we combine Descartes' Rule
of Signs and Newton iteration. More precisely, instead of using a fixed
subdivision strategy such as bisection in each iteration, a Newton step based
on the number of sign variations for an actual interval is considered, and,
only if the Newton step fails, we fall back to bisection. Following this
approach, our analysis shows that, for most iterations, we can achieve
quadratic convergence towards the real roots. In terms of complexity, our
method induces a recursion tree of almost optimal size O(nlog(n tau)), where n
denotes the degree of the polynomial and tau the bitsize of its coefficients.
The latter bound constitutes an improvement by a factor of tau upon all
existing subdivision methods for the task of isolating the real roots. In
addition, we provide a bit complexity analysis showing that DSC2 needs only
\tilde{O}(n^3tau) bit operations to isolate all real roots of f. This matches
the best bound known for this fundamental problem. However, in comparison to
the much more involved algorithms by Pan and Sch\"onhage (for the task of
isolating all complex roots) which achieve the same bit complexity, DSC2
focuses on real root isolation, is very easy to access and easy to implement.
An algorithm is presented for the geometric analysis of an algebraic curve f(x,y)=0 in the real affine plane. It computes a cylindrical algebraic decomposition (CAD) of the plane, augmented with adjacency information. The adjacency information describes the curve's topology by a topologically equivalent planar graph. The numerical data in the CAD gives an embedding of the graph. The algorithm is designed to provide the exact result for all inputs but to perform only few symbolic operations for the sake of efficiency. In particular, the roots of at a critical x-coordinate are found with adaptive-precision arithmetic in all cases, using a variant of the Bitstream Descartes method~(Eigenwillig et~al., 2005). The algorithm may choose a generic coordinate system for parts of the analysis but provides its result in the original system. The algorithm has been implemented as C++ library \texttt{AlciX} in the EXACUS project. Running time comparisons with \texttt{top} by Gonzalez-Vega and Necula~(2002), and with \texttt{cad2d} by Brown demonstrate its efficiency.
An algorithm is presented for the computation of the topology of a non-reduced 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 symbolic-numeric 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 non-reduced 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.
The algorithmic problems of real algebraic geometry such as real root counting, deciding the existence of solutions of systems of polynomial equations and inequalities, or deciding whether two points belong in the same connected component of a semi-algebraic set occur in many contexts. In this first-ever graduate textbook on the algorithmic aspects of real algebraic geometry, the main ideas and techniques presented form a coherent and rich body of knowledge, linked to many areas of mathematics and computing.
Mathematicians already aware of real algebraic geometry will find relevant information about the algorithmic aspects, and researchers in computer science and engineering will find the required mathematical background.
Being self-contained the book is accessible to graduate students and even, for invaluable parts of it, to undergraduate students.
Based on an efficient generic position checking method and on a method to solve bivariate polynomial systems, we give a new algorithm to compute the topology of an algebraic space curve. Compared to the method presented by the authors, in a joint work with Lazard, the new algorithm is efficient because of two reasons. One is the bitsize of the coefficients that may appear in projections is improved. The other is that one projection is enough for most general case in the new algorithm. We also give an ε-meshing of the space curve after we obtain its topology. Many nontrivial experiments show the efficiency of the algorithm.
We assume that a real square-free polynomial A has a degree d, a maximum coefficient bitsize τ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the Double Exponential Sieve algorithm (also called the Bisection of the Exponents), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of t=2-L. The algorithm has Boolean complexity ÕB(d2 τ + d L ). Our algorithms support the same complexity bound for the refinement of r roots, for any r ≤ d.
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently presented algorithm Bisolve [2]. Bisolve is an elimination method which, in a first step, projects the solutions of a system onto the x- and y-axes and, then, selects the actual solutions from the so induced candidate set. However, unlike similar algorithms, Bisolve requires no genericity assumption on the input, and there is no need for any kind of coordinate transformation. Furthermore, extensive benchmarks as presented in [2] confirm that the algorithm is highly practical, that is, a corresponding C++ implementation in Cgal outperforms state of the art approaches by a large factor. In this paper, we focus on the theoretical complexity of Bisolve. For two polynomials f, g ∈ Z[x, y] of total degree at most n with integer coefficients bounded by 2τ, we show that Bisolve computes isolating boxes for all real solutions of the system f = g = 0 using O(n8 + n7τ) bit operations, thereby improving the previous record bound for the same task by several magnitudes.
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, d resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.
We present a new and complete algorithm for computing the topology of an algebraic surface S 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 al-gorithm is provided leading to a bound in e OB(d 15 τ) 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 τ . Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces.
This paper is devoted to present a new algorithm computing in a very efficient way the topology of a real algebraic plane curve defined implicitly. This algorithm proceeds in a seminumerical way by performing a symbolic preprocessing which allows later to accomplish the numerical computations in a very accurate way.
It was proved over a century ago that an algebraic curve C in the real projective plane, of degree n, has at most connected components. If C is nonslngular, then each of its commponents is a topological circle. A circle in the projectlve plane either separates it into a disk (the interior of the circle) and a Möbius band (the circle's exterior), or does not separate it. In the former case, the circle is an oval. If C is nonsingular, then all its components are ovals if n is even, and all except one are ovals if n is odd. An oval is included in another if it lies in the other's interior. The topological type of (a nonsingular) C is completely determined by (1) the parity of n, (2) how many ovals it has, and (3) the partial ordering of its ovals by inclusion. We present an algorithm which, given a homogeneous polinomial f(x,y,z) of degree n with integer coefficients, checks whether tlte curve defined hy f = 0 is nonsingular and if so, computes its topological type. The algorithm's maximum computing time is O(n27L(d)3), where d is the sum of the absolute values of the integer coofficients of f, and L(d) is the length of d.
A practically efficient algorithm for analyzing the topology of plane real algebraic curves is given. Given a bivariate polynomial, the algorithm produces a planar graph which is topologically equivalent to the real variety of the polynomial on the Euclidean plane.The method does not require the expensive computations of g.c.d., divisions, and root bounds of polynomials with real algebraic number coefficients. Further, it utilizes floating point arithmetic and interval arithmetic whenever possible. Experiments show that most benchmark curves found in the literature can be analyzed within a few seconds on a workstation. Timings on randomly generated polynomials also indicate that the algorithm is efficient to be useful in practice.
An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic B-spline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation keeps many important geometric features of the original curve such as the topology, convexity and sharp points. Our method is implemented and experiments show that it may achieve better approximation bound with less segments than previously known methods. We also extend the method to approximate spatial algebraic curves.
We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties.The method isolates singular points from regular parts and deals with them independently. The topology near singular points is guaranteed through topological degree computation. In either case the topology inside regions is recovered from information on the boundary of a cell of the subdivision.Obtained regions are segmented to provide an efficient insertion operation while dynamically maintaining an arrangement structure.We use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. It is finally shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation.
In this paper, an algorithm to compute a certified rational parametric
approximation for algebraic space curves is given by extending the local
generic position method for solving zero dimensional polynomial equation
systems to the case of dimension one. By certified, we mean the approximation
curve and the original curve have the same topology and their Hausdauff
distance is smaller than a given precision. Thus, the method also gives a new
algorithm to compute the topology for space algebraic curves. The main
advantage of the algorithm, inhering from the local generic method, is that
topology computation and approximation for a space curve is directly reduced to
the same tasks for two plane curves. In particular, the error bound of the
approximation space curve is obtained from the error bounds of the
approximation plane curves explicitly. Nontrivial examples are used to show the
effectivity of the method.
This paper is concerned with exact real solving of well-constrained,
bivariate polynomial systems. The main problem is to isolate all common real
roots in rational rectangles, and to determine their intersection
multiplicities. We present three algorithms and analyze their asymptotic bit
complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based
method, and \sOB(N^{12}) for two subresultant-based methods: this notation
ignores polylogarithmic factors, where N bounds the degree and the bitsize of
the polynomials. The previous record bound was \sOB(N^{14}).
Our main tool is signed subresultant sequences. We exploit recent advances on
the complexity of univariate root isolation, and extend them to sign evaluation
of bivariate polynomials over two algebraic numbers, and real root counting for
polynomials over an extension field. Our algorithms apply to the problem of
simultaneous inequalities; they also compute the topology of real plane
algebraic curves in \sOB(N^{12}), whereas the previous bound was
\sOB(N^{14}).
All algorithms have been implemented in MAPLE, in conjunction with numeric
filtering. We compare them against FGB/RS, system solvers from SYNAPS, and
MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is
among the most robust, and its runtimes are comparable, or within a small
constant factor, with respect to the C/C++ libraries.
Key words: real solving, polynomial systems, complexity, MAPLE software
An algorithm for computing the topology of a real algebraic space curve C, implicitly defined as the intersection of two surfaces, is presented. Given C, the algorithm generates a space graph which is topologically equivalent to the real variety on the Euclidean space. The algorithm is based on the computation of the graphs of at most two projections of C. For this purpose, we introduce the notion of space general position for space curves, we show that any curve under the above conditions can always be linearly transformed to be in general position, and we present effective methods for checking whether space general position has been reached.
Computing the topology of an algebraic plane curve means to
compute a combinatorial graph that is isotopic to and thus
represents its topology in . We prove that, for a polynomial of
degree n with coefficients bounded by , the topology of the induced
curve can be computed with bit operations
deterministically, and with bit operations with a
randomized algorithm in expectation. Our analysis improves previous best known
complexity bounds by a factor of . The improvement is based on new
techniques to compute and refine isolating intervals for the real roots of
polynomials, and by the consequent amortized analysis of the critical fibers of
the algebraic curve.
Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive
epsilon, we want to compute an epsilon-isotopic polygonal approximation to the
restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on
subdivision algorithms because of their adaptive complexity and ease of
implementation. Plantinga and Vegter gave a numerical subdivision algorithm
that is exact when the curve S is bounded and non-singular. They used a
computational model that relied only on function evaluation and interval
arithmetic. We generalize their algorithm to any bounded (but possibly
non-simply connected) region that does not contain singularities of S. With
this generalization as a subroutine, we provide a method to detect isolated
algebraic singularities and their branching degree. This appears to be the
first complete purely numerical method to compute isotopic approximations of
algebraic curves with isolated singularities.
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Exact and efficient 2d-arrangements of arbitrary algebraic curves
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