On the isotopic meshing of an algebraic implicit surface

Abstract

International audienceWe present a new and complete algorithm for computing the topology of an algebraic surface 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 algorithm is provided leading to a bound in OB (d15 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

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

“…Subdivision and parametrization are the most common tools in these domains. In many cases special requirements are imposed, in particular, topological consistence (compare [1,26,47]), or an optimal fitting to certain computational requirements (see [72,73,74]) and references therein.…”

On the Complexity of Computing the Topology of Real Algebraic Space Curves

“…Algebraic space curves are used in computer aided (geometric) design, and geometric modeling. Computing the topology of an algebraic curve is also a basic step to compute the topology of algebraic surfaces [10,16]. There have been many papers studied the guaranteed topology and meshing for plane algebraic curves [1,3,5,8,14,18,19,23,28,33].…”

“…Most of the existing work ([2, 11, 12, 15, 25]) of computing the topology of algebraic space curves require the space curve to be in a generic position. But checking whether an algebraic space curve is in a generic position or not is not a trivial task, see [2,12,16]. In this paper, we will give a deterministic algorithm to find a generic position for an algebraic space curve, which is another contribution of the paper.…”

Computation of topological invariants for real projective surfaces with isolated singularities

“…Other algorithms for studying the topology of affine surfaces of degree d in R 3 with arbitrary singularities have been presented in [3], [1], [9] and [6]. The first three compute triangulations of the surface using O(d 7 ) cells, while the latter one computes a curvilinear wireframe model.…”

“…All of these methods perform much better than CAD, e.g. [1] and [9] show that their method requires O(d 7 ) points to compute the triangulation while CAD would require O(d 13 ). It is not easy to decide whether two pairs (RP 3 , S) and (RP 3 , S ′ ) are homeomorphic or not, even after computing a combinatorial description of each pair.…”