Numeric and Certified Isolation of the Singularities of the Projection of a Smooth Space Curve

Rémi Imbach (1) , Guillaume Moroz (1) , Marc Pouget (1)

1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility

Let CP ∩Q be a smooth real analytic curve embedded in R 3 , defined as the solutions of real analytic equations of the form P (x, y, z) = Q(x, y, z) = 0 or P (x, y, z) = ∂P ∂z = 0. Our main objective is to describe its projection C onto the (x, y)-plane. In general, the curve C is not a regular submanifold of R 2 and describing it requires to isolate the points of its singularity locus Σ. After describing the types of singularities that can arise under some assumptions on P and Q, we present a new method to isolate the points of Σ. We experimented our method on pairs of independent random polynomials (P, Q) and on pairs of random polynomials of the form (P, ∂P ∂z) and got promising results.

Mots clés: Topology of Projected Real Curve; Topology of Apparent Contour; Singularities Isolation; Topology Computation; Numeric Certi-fied Methods

Domaines: Géométrie algorithmique [cs.CG]; Calcul formel [cs.SC]

Rémi Imbach, Guillaume Moroz, Marc Pouget. Numeric and Certified Isolation of the Singularities of the Projection of a Smooth Space Curve. Proceedings of the 6th International Conferences on Mathematical Aspects of Computer and Information Sciences, Oct 2015, Berlin, Germany. ⟨hal-01239447⟩