Cylindrical Decomposition Research Articles

Exact symbolic–numeric computation of planar algebraic curves

We present a certified and complete algorithm to compute arrangements of real planar algebraic curves. It computes the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to compute the topology of a single algebraic curve.Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of symbolic operations involved, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry.We have implemented our algorithms as prototypical contributions to the C++-project Cgal. We exploit graphics hardware to expedite the remaining symbolic computations. We have also compared our implementation with the current reference implementations, that is, Lgp and Maple’s Isolate for polynomial system solving, and Cgal’s bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.

Singularity-free design of the translational parallel manipulator IRSBot-2

The IRSBot-2 is a two degree-of-freedom translational parallel manipulator dedicated to fast and accurate pick-and-place operations. This paper deals with the determination of the design parameters of the manipulator for the IRSBot-2 to be free of parallel singularity. First, the robot architecture is introduced. The IRSBot-2 is composed of two identical spatial limbs, each one containing a proximal module and a distal module. Then, its actuation singularities and constraint singularities are analyzed. The latter are analyzed in its distal parameter space with a method based on the notion of discriminant varieties and cylindrical algebraic decomposition. Moreover, a deep analysis is carried out in order to determine the set of design parameters of the distal modules that prevents the IRSBot-2 from reaching any constraint singularity. To the best of our knowledge, such an analysis is performed for the first time. Finally, a design methodology is proposed to determine the set of design parameters associated with the proximal modules for the IRSBot-2 to be assembled and free of parallel singularity.

A repository for CAD examples

Cylindrical Algebraic Decomposition (CAD, first introduced in [Col75]) of Euclidean space has become an important tool in mathematics and allows for practical quantifier elimination (QE) over the reals. Much research has gone into improving the projection operator (e.g. [McC85]), the use of partial CADs (e.g. [CH91]), or into alternative algorithms (e.g. [CDM+09]). A problem of fundamentally high complexity (doubly-exponential in the number of variables [DH88]) it can be difficult to judge when a problem will be solved quickly, and easy to write down problems that are computationally infeasible. It can therefore be difficult to test new advances in this field, and even harder to 'experiment' with new ideas. We have created a unified machine-readable repository of examples to be used when considering CADs. This allows for quick access to a host of examples, pulled from various theoretical questions and applications. As well as enabling more efficient research, the creation of the repository has prompted us to look deeper at certain questions related to CADs. The example bank currently facilitates quick input for Maple and Qepcad implementations of CAD and QE. Whilst CAD and QE procedures exist in other software, they are not directly involved in our research.

Solving non-linear arithmetic

We propose a new decision procedure for the existential theory of the reals. It performs a backtracking search for a model in R, where the backtracking is powered by a novel conflict resolution procedure based on cylindrical algebraic decomposition. The initial experimental results are very encouraging. The full article has been accepted at the 6th International Joint Conference on Automated Reasoning (IJCAR 2012).

Computing Intergrals Over Polynomially Defined Regions and Their Boundaries in 2 and 3 Dimensions

This study uses the cylindrical algebraic decomposition algorithms implemented in Mathematica to produce procedures to analytically compute integrals over polynomially defined regions and their boundaried in two and three dimensions. Using these results, we can implement the divergence theorem in three dimensions or the Green's theorems in two dimensions. These theorems are of central importance in the applications of multidimensional integration. They also provide a strong correctness test for the implementation of our results in a computer algebra system. The resulting software can solve many of the two and some of the three dimensions lintegration problems in vector calculu stextbooks. The three dimensional results are being extended. The results in this paper are being included in an automated student assistant for vector calculus.

Study of D-decompositions by the methods of computational real-valued algebraic geometry

New methods to study the D-decomposition with the use of the computational realvalued algebraic geometry were proposed. The number of domains of D-decomposition for the polynomial parametric families of polynomials and matrices was estimated. This technique which requires construction of the Gröbner bases and cylindrical decomposition sometimes proves to be more precise than the traditional technique. The symbolic calculation system Maple v.14 and, in particular, its package RegularChains are used.

An effective implementation of symbolic–numeric cylindrical algebraic decomposition for quantifier elimination

With many applications in engineering and scientific fields, quantifier elimination (QE) has received increasing attention. Cylindrical algebraic decomposition (CAD) is used as a basis for a general QE algorithm. In this paper we present an effective symbolic–numeric cylindrical algebraic decomposition (SNCAD) algorithm for QE incorporating several new devices, which we call “quick tests”. The simple quick tests are run beforehand to detect an unnecessary procedure that might be skipped without violating the correctness of results and they thus considerably reduce the computing time. The effectiveness of the SNCAD algorithm is examined in a number of experiments including practical engineering problems, which also reveal the quality of the implementation. Experimental results show that our implementation has significantly improved efficiency compared with our previous work.

Moment-Method Analysis of Normal-to-Body Antennas Using a Green's Function Approach

The cylindrical wave decomposition of fields from a normal-to-cylinder point source is derived using a modified addition theorem. The fields scattered from a lossy cylinder, as an approximation of the human body, are also calculated and tabulated for use in an integral equation study of dipoles. The transmittance between two optimized radially polarized dipoles is evaluated on the body trunk surface, using the method of moments (MoM) in the 2.45 GHz ISM band. An anechoic chamber measurement is performed and the measured transmittance between two dipoles on the human body has been compared to the simulated one on the lossy cylinder. Although the abdominal cross section of the body does not have exactly a circular shape, strong similarities have been observed in their path-loss trends around the body. A very good agreement has been obtained between the dipoles transmittances provided by the analytical/MoM model, CST Microwave Studio simulations, and experiments, including interference effects attributed to the body-attached radiating waves.

Optimal designs for two-level factorial experiments with binary response

We consider the problem of obtaining locally D-optimal designs for factorial experiments with qualitative factors at two levels each and with binary response. For the 2 factorial experiment with main-effects model, we obtain optimal designs analytically in special cases and demonstrate how to obtain a solution in the general case using cylindrical algebraic decomposition. The optimal designs are shown to be robust to the choice of the assumed values of the prior, and when there is no basis to make an informed choice of the assumed values we recommend the use of the uniform design that assigns equal number of observations to each of the four points. For the general 2 case we show that the uniform design has a maximin property.

Generating Invariants for Hybrid Systems by Computing Vanishing Ideals of Sample Points

In this paper, we investigate the problem of generating invariants of hybrid systems. We present a new approach, for generating polynomial equation invariants of hybrid systems through computing vanishing ideal of sample points. The challenge is to deal with the continuous consecution of hybrid systems because of the existence of differential equations. Our approach avoids first-order quantifier elimination and cylindrical algebraic decomposition (CAD). From the preliminary experiment results, we demonstrate the feasibility of our approach.

An approach to geometric interpolation by Pythagorean-hodograph curves

The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d???2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.

Cylindrical decomposition for systems transcendental in the first variable

We present a generalization of the Cylindrical Algebraic Decomposition (CAD) algorithm to systems of equations and inequalities in functions of the form p ( x , f 1 ( x ) , … , f m ( x ) , y 1 , … , y n ) , where p ∈ Q [ x , t 1 , … , t m , y 1 , … , y n ] and f 1 ( x ) , … , f m ( x ) are real univariate functions such that there exists a real root isolation algorithm for functions from the algebra Q [ x , f 1 ( x ) , … , f m ( x ) ] . In particular, the algorithm applies when f 1 ( x ) , … , f m ( x ) are real exp–log functions or tame elementary functions.

Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions

We use the cylindrical algebraic decomposition algorithms implemented in Mathematica to produce procedures to analytically compute integrals over polynomially defined regions and their boundaries in two and three dimensions. Using these results, we can implement the divergence theorem in three dimensions or the Green's theorems in two dimensions. These theorems are of central importance in the applications of multidimensional integration. They also provide a strong correctness test for the implementation of our results in a computer algebra system. The resulting software can solve many of the two and some of the three dimensional integration problems in vector calculus textbooks. The three dimensional results are being extended. The results in this paper are being included in an automated student assistant for vector calculus.

Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

We study the problem of optimizing over parameters a particular real root of a polynomial with parametric coefficients. We propose an efficient symbolic method for solving the optimization problem based on a special cylindrical algebraic decomposition algorithm, which asks for a semi-algebraic decomposition into cells in terms of number-of-roots-invariance.

Cylindrical algebraic decomposition for reasoning about qualitative spatial knowledge

Cylindrical algebraic decomposition for reasoning about qualitative spatial knowledge

Foreword

In the last years, the field of non-linear Computational Geometry has produced very relevant algorithmic advances, for instance, by adapting to low-degree curved objects techniques dealing with linear primitives. Moreover the introduction of new algorithms and approaches for dealing efficiently with curves and surfaces has shown a considerable impact in fields such as Computer-Aided Design where final users are continuously demanding more efficiency, accuracy, robustness, etc. New approaches for solving geometric problems, coming from Computer-Aided Geometric Design and involving curves and surfaces with a Computational Geometry flavour is the main focus of this special issue. Contributions were asked in areas such as: arrangements of low degree curves and surfaces; Voronoi diagrams and medial axes involving curved ob jects; offset computations and their topology; distance predicates for curves and surfaces; closed formulae for geometric queries involving curves and surfaces; intersection detection for moving curved ob jects; adapting algebraic or geometrical techniques for solving algorithmic problems involving curves and surfaces. This special issue is a direct consequence of the Second International Conference on Mathematical Aspects of Computer and Information Sciences - MACIS 2007 (see http:// www- spiral.lip6.fr/ MACIS2007) held in Paris in December 2007 and where Computational Geometry and Computer Aided Geometric Design were one of the main topics (among other two, Polynomial System Solving and Algorithms and Complexity). Finally six papers (out of eight papers submitted by the March 2009 deadline) were finally accepted. The two papers by C. D'Andrea et al. and I. Z. Emiris et al. deal with the problem of analyzing the Newton polygon of the implicit equation of a rational curve starting from its parameterization: both papers are devoted to study from different viewpoints the properties of the set of monomials appearing in the implicit equation of the considered curve without computing it. The two papers by E. Berberich et al. introduce a new approach to deal with the problem of determining the shape of an arrangement of curves on several classes of parametric surfaces which is then applied to compute, for example, the Minkowski sum of polytopes or the envelope of surfaces. The two papers by D. Lazard and J. Cheng et al. deal with the problem of computing the topology of a semialgebraic set (i.e. a set in $\mathbb{R}^n$ defined by polynomial equations and inequations): the first one shows how to deal with the general case by using the so called Cylindrical Algebraic Decomposition and the second one introduces an extremely fast algorithm computing the topology of a real algebraic plane curve defined implicitely. We would like finally to thank all the reviewers who definitively helped with their work to improve the papers that appear into this special issue.

On the Topology of Real Algebraic Plane Curves

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 sub-resultant 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 non-generic 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 high-degree non-generic curves.

CAD and Topology of Semi-Algebraic Sets

Although Cylindrical Algebraic Decomposition (CAD) is widely used to study the topology of semi-algebraic sets (especially algebraic curves), there are very few studies of the topological properties of the output of the CAD algorithms. In this paper three possible bad topological properties of the output of CAD algorithms are described. It is shown that these properties may not occur after a generic change of coordinates and that they may be avoided, in dimension not greater than three, with a modification of the CAD algorithm. As this modification of the CAD algorithm requires to solve some polynomial systems, it is also shown that the computation with real algebraic numbers may be advantageously replaced, in all the variants of the CAD algorithm, by solving systems of polynomial equations and inequalities.

A package for solving parametric polynomial systems

A new Maple package for solving parametric systems of polynomial equations and inequalities is described. The main idea for solving such a system is as follows. The parameter space Rd is divided into two parts: the discriminant variety W and its complement R d nW . The discriminant variety is a generalization of the well-known discriminant of a univariate polynomial and contains all those parameter values leading to non-generic solutions of the system. The complement R d nW can be expressed as a finite union of open cells such that the number of real solutions of the input system is constant on each cell. In this way, all parameter values leading to generic solutions of the system can be described systematically. The underlying techniques used are Gröbner bases, polynomial real root finding, and cylindrical algebraic decomposition. This package offers a friendly interface for scientists and engineers to solve parametric problems, as illustrated by an example from control theory.

Simple and Efficient Algorithms for Computing Smooth, Collision-free Feedback Laws Over Given Cell Decompositions

This paper presents a novel approach to computing feedback laws in the presence of obstacles. Instead of computing a trajectory between a pair of initial and goal states, our algorithms compute a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state. As a result, the vector field globally solves the navigation problem and provides robustness to disturbances in sensing and control. The vector field's integral curves (system trajectories) are guaranteed to avoid obstacles and are C ∞ smooth. We construct a vector field with these properties by partitioning the space into simple cells, defining local vector fields for each cell, and smoothly interpolating between them to obtain a global vector field. We present an algorithm that computes these feedback controls for a kinematic point robot in an arbitrary dimensional space with piecewise linear boundary; the algorithm requires minimal preprocessing of the environment and is extremely fast during execution. For many practical applications in two-dimensional environments, full computation can be done in milliseconds. We also present an algorithm for computing feedback laws over cylindrical algebraic decompositions, thereby solving a smooth feedback version of the generalized piano movers' problem.