In this paper we study the problem of determining whether two points lie in the same connected component of a semi-algebraic set S. Although we are mostly concerned with sets S # , our algorithm can also decide if points in an arbitrary set S # R can be joined by a semi-algebraic path, for any real closed field R. Our algorithm computes a one-dimensional semi-algebraic subset ##S# of S (actually of an embedding of S in a space R for a certain real extension field R of the given field R#. ##S# is called the roadmap of S. The basis of this work is the roadmap algorithm described in [3], [4] whichworked only for compact, regularly stratified sets. We measure...

Recently there has been a lot of activity in ... In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quantifier elimination in pseudo-polynomial time. By purely symbolic, we mean that it is possible to eliminate a quantified variable from a system of polynomials no matter what the coefficient values are. The previous methods required the coefficients to be themselves polynomials in other variables

We present MAPC, a library for exact representation of geometric objects -- specifically points and algebraic curves in the plane. Our library makes use of several new algorithms, which we present here, including methods for nding the sign of a determinant, finding intersections between two curves, and breaking a curve into monotonic segments. These algorithms are used to speed up the underlying computations. The library provides C++ classes that can be used to easily instantiate, manipulate, and perform queries on points and curves in the plane. The point classes can be used to represent points known in a variety of ways (e.g. as exact rational coordinates or algebraic numbers) in a unified manner. The curve class can be used to represent a portion of an algebraic curve. We have used MAPC for applications dealing with algebraic points and curves, including sorting points along a curve, computing arrangement of curves, medial axis computations, and boundary evaluation on curved primitives. As compared to earlier algorithms and implementations utilizing exact arithmetic, our library is able to achieve more than an order of magnitude improvement in performance.

The Bentley-Ottmann sweep-line method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves. The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.

We present efficient and accurate algorithms to compute solutions of zero-dimensional multivariate polynomial equations in a given domain. The total number of solutions correspond to the Bezout bound for dense polynomial systems or the Bernstein bound for sparse systems. In most applications the actual number of solutions in the domain of interest is much lower than the Bezout or Bernstein bound. Our approach is based on global symbolic formulation of the problem using resultants and matrix computations and localizing it to find selected solutions based on numerical computations. The problem of finding roots is reduced to computing eigenvalues of a generalized companion matrix and we use the structure of the matrix to compute the solutions in the domain of interest only. The resulting algorithm combines symbolic preprocessing with numerical iterations and works well in practice. We discuss its performance on a number of applications. 1 Introduction Finding roots of polynomial system...

An important part of solid modeling systems based on curved primitives is the representation and manipulation of algebraic plane curves with rational coefficients and points with algebraic coordinates. These objects are often approximated by floating-point numbers and spline curves, which are easy to manipulate, but are subject to accuracy and robustness problems. Exact computation can eliminate these numerical robustness problems, but efficient exact methods have not been available. Moreover, it is widely believe that exact arithmetic is impractical for manipulating curved primitives.

Many geometric problems like generalized Voronoi diagrams, medial axis computations and boundary evaluation involve computation and manipulation of non-linear algebraic primitives like curves and surfaces. The algorithms designed for these problems make decisions based on signs of geometric predicates or on the roots of polynomials characterizing the problem. The reliability of the algorithm depends on the accurate evaluation of these signs and roots. In this paper, we present a naive precision-driven computational model to perform these computations reliably and demonstrate its effectiveness on a certain class of problems like sign of determinants with rational entries, boundary evaluation and curve arrangements. We also present a novel algorithm to compute all the roots of a univariate polynomial to any desired accuracy. The computational model along with the underlying number representation, precision-driven arithmetic and all the algorithms are implemented as part of a stand-alone software library, PRECISE. 1.

We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event points in these arrangements, including tangential intersections and singular points. By introducing an additional curve, which we call the Jacobi curve, we are able to find non-singular tangential intersections. We show that the coordinates of the singular points in our special projected planar arrangements are roots of quadratic polynomials. The coefficients of these polynomials are usually rational and contain at most a single square root. A prototypical implementation indicates that our approach leads to good performance in practice.

Many geometric algorithms involve dealing with numeric data corresponding to high degree algebraic numbers. They come up in computing generalized Voronoi diagrams of lines and planes, medial axis of a polyhedron and geometric computation on non-linear primitives described using algebraic functions. Earlier algorithms dealing with algebraic numbers either use fixed precision arithmetic or techniques from symbolic computation. While the former can be inaccurate, the latter is too slow in practice. We present efficient representations and algorithms for reliable computations with algebraic numbers. We use these representations to efficiently perform geometric queries like inside/outside tests, which-side or orientation tests. The overall approach combines different techniques from symbolic computation based on exact arithmetic with floating point arithmetic. We demonstrate its applications to efficient and reliable computation of curve and surface intersections. In practice, it is about one order of magnitude faster as compared to earlier implementations that produce reliable results.

The Bentley-Ottmann sweep-line method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves. The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.