On the isotopic meshing of an algebraic implicit surface (original) (raw)

A New Algorithm for Implicitizing a Parametric Algebraic Surface

International Journal of Pure and Apllied Mathematics, 2015

Given a parametric representation of an algebraic projective surface S of the ordinary space we give a new algorithm for finding the implicit cartesian equation of S. The algorithm is based on finding a suitable finite number of points on S and computing, by linear algebra, the equation of the surface of least degree that passes through the points.

Efficient topology determination of implicitly defined algebraic plane curves

Computer Aided Geometric Design, 2002

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.

On the computation of the topology of a non-reduced implicit space curve

Proceedings of the twenty-first international symposium on Symbolic and algebraic computation - ISSAC '08, 2008

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.

On the computation of the topology of plane curves

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation - ISSAC '14, 2014

Let P ∈ Z[X, Y ] be a square-free polynomial and C(P) := {(α, β) ∈ R 2 , P (α, β) = 0} be the real algebraic curve defined by P. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular points and critical points of the projection wrt the X-axis inÕ(d 6 τ +d 7) bit operations whereÕ means that we ignore logarithmic factors in d and τ. Compared to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of C(P) i.e a straight-line planar graph isotopic to C(P) inÕ(d 6 τ + d 7) bit operations.

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

Journal of Complexity, 1996

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 ‫ޑ‬ 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.

An Implicitization Algorithm for Rational Surfaces with no Base Points

Journal of Symbolic Computation, 2001

We present an implicitization algorithm which is free of extraneous factors if the rational parametric surface has no base points. This algorithm is based on the method of Sylvester for computing the resultant of three homogeneous polynomials in three variables. Some examples and computations illustrate the efficiency and limits of this method.

A delineability-based method for computing critical sets of algebraic surfaces

Journal of Symbolic Computation, 2007

In this paper, we address the problem of determining a real finite set of z-values where the topology type of the level curves of a (maybe singular) algebraic surface may change. We use as a fundamental and crucial tool McCallum's theorem on analytic delineability of polynomials (see . An improved projection operation for cylindrical algebraic decomposition. In: Caviness, B.F., Johnson, J.R. (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, pp. 242-268]). Our results allow to algorithmically compute this finite set by analyzing the real roots of a univariate polynomial; namely, the double discriminant of the implicit equation of the surface. As a consequence, an application to offsets is shown.

A First Approach Towards Normal Parametrizations of Algebraic Surfaces

International Journal of Algebra and Computation, 2010

In this paper we analyze the problem of deciding the normality (i.e. the surjectivity) of a rational parametrization of a surface [Formula: see text]. The problem can be approached by means of elimination theory techniques, providing a proper close subset [Formula: see text] where surjectivity needs to be analyzed. In general, these direct approaches are unfeasible because [Formula: see text] is very complicated and its elements computationally hard to manipulate. Motivated by this fact, we study ad hoc computational alternative methods that simplifies [Formula: see text]. For this goal, we introduce the notion of pseudo-normality, a concept that provides necessary conditions for a parametrization for being normal. Also, we provide an algorithm for deciding the pseudo-normality. Finally, we state necessary and sufficient conditions on a pseudo-normal parametrization to be normal. As a consequence, certain types of parametrizations are shown to be always normal. For instance, pseudo-...

Automatic parameterization of rational curves and surfaces IV: algebraic space curves

ACM Transactions on Graphics, 1989

For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f (x, y, z) = 0 and g(r, y, z) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P, whose genus is the same as that of C. Thus C is rational iff the genus of P is zero. Given an irreducible space curve C = ( f n g), with f and g not tangent along C, we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C. Together with [4], this method yields an algorithm to compute the genus of C, and if the genus is zero, the rational parametric equations for C. As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C.

Representing rational curve segments and surface patches using semi-algebraic sets

Computer Aided Geometric Design, 2019

We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such a voxels for representing surface patches.