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

  • Rémi ImbachEmail author
  • Guillaume Moroz
  • Marc Pouget
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


Let \({{\mathcal C}_{P\cap Q}}\) be a smooth real analytic curve embedded in \(\mathbb R{^3}\), defined as the solutions of real analytic equations of the form \(P(x,y,z)=Q(x,y,z)=0\) or \(P(x,y,z)=\frac{\partial P}{\partial z}=0\). Our main objective is to describe its projection \(\mathcal C\) onto the (xy)-plane. In general, the curve \(\mathcal C\) is not a regular submanifold of \(\mathbb R^2\) and describing it requires to isolate the points of its singularity locus \(\varSigma \). 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 \(\varSigma \). We experimented our method on pairs of independent random polynomials (PQ) and on pairs of random polynomials of the form \((P,\frac{\partial P}{\partial z})\) and got promising results.


Topology of analytic real curve Apparent contour Singularities isolation Numeric certified methods 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LORIA LaboratoryINRIA Nancy Grand EstNancyFrance

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