Advertisement

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)

Abstract

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.

Keywords

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

References

  1. 1.
    Alberti, L., Mourrain, B., Técourt, J.P.: Isotopic triangulation of a real algebraic surface. J. Symb. Comput. 44(9), 1291–1310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, V.I., Varchenko, A., Gusein-Zade, S.: Singularities of Differentiable Maps: Volume I: The Classification of Critical Points Caustics and Wave Fronts. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  3. 3.
    Bank, B., Giusti, M., Heintz, J., Lecerf, G., Matera, G., Solernó, P.: Degeneracy loci and polynomial equation solving. Found. Comput. Math. 15(1), 159–184 (2015). http://dx.doi.org/0.1007/s10208-014-9214-z MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Adaptive multiprecision path tracking. SIAM J. Numer. Anal. 46(2), 722–746 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berberich, E., Kerber, M., Sagraloff, M.: An efficient algorithm for the stratification and triangulation of an algebraic surface. Comput. Geom. Theory Appl. 43(3), 257–278 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheng, J., Lazard, S., Peñaranda, L., Pouget, M., Rouillier, F., Tsigaridas, E.: On the topology of real algebraic plane curves. Math. Comput. Sci. 4, 113–137 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daouda, D.N., Mourrain, B., Ruatta, O.: On the computation of the topology of a non-reduced implicit space curve. In: Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation, ISSAC 2008, pp. 47–54. ACM, New York (2008)Google Scholar
  8. 8.
    Dedieu, J.: Points fixes, zéros et la méthode de Newton. Mathématiques et Applications. Springer, Heidelberg (2006)Google Scholar
  9. 9.
    Delanoue, N., Lagrange, S.: A numerical approach to compute the topology of the apparent contour of a smooth mapping from \(R^2\) to \(R^2\). J. Comput. Appl. Math. 271, 267–284 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demazure, M.: Bifurcations and Catastrophes: Geometry of Solutions to Nonlinear Problems. Universitext. Springer, Heidelberg (2000). École PolytechniqueCrossRefzbMATHGoogle Scholar
  11. 11.
    El Kahoui, M.: Topology of real algebraic space curves. J. Symb. Comput. 43(4), 235–258 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.C.: On location and approximation of clusters of zeros: Case of embedding dimension one. Found. Comput. Math. 7(1), 1–58 (2007). http://dx.doi.org/10.1007/s10208-004-0159-5 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hauenstein, J.D., Mourrain, B., Szanto, A.: Certifying isolated singular points and their multiplicity structure. In: Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2015, pp. 213–220. ACM, New York (2015).http://doi.acm.org/10.1145/2755996.2756645
  14. 14.
    Imbach, R., Moroz, G., Pouget, M.: Numeric certified algorithm for the topology of resultant and discriminant curves. Research Report RR-8653. Inria, April 2015Google Scholar
  15. 15.
    Kearfott, R., Xing, Z.: An interval step control for continuation methods. SIAM J. Numer. Anal. 31(3), 892–914 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Basler Lehrbücher, vol. 4. Birkhäuser Basel, Boston (1992)CrossRefzbMATHGoogle Scholar
  17. 17.
    Leykin, A., Verschelde, J., Zhao, A.: Newton’s method with deflation for isolated singularities of polynomial systems. Theoret. Comput. Sci. 359(13), 111–122 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mantzaflaris, A., Mourrain, B.: Deflation and certified isolation of singular zeros of polynomial systems. In: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011, pp. 249–256. ACM, New York (2011). http://doi.acm.org/10.1145/1993886.1993925
  19. 19.
    Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: Certified parallelotope continuation for one-manifolds. SIAM J. Numer. Anal. 51(6), 3373–3401 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    Morgan, A.: Solving Polynominal Systems Using Continuation for Engineering and Scientific Problems. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Mourrain, B., Pavone, J.: Subdivision methods for solving polynomial equations. J. Symbolic Comput. 44(3), 292–306 (2009). http://www.sciencedirect.com/science/article/pii/S0747717108001168 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mourrain, B., Pion, S., Schmitt, S., Técourt, J.P., Tsigaridas, E.P., Wolpert, N.: Algebraic issues in computational geometry. In: Boissonnat, J.D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces, chap. 3. Mathematics and Visualization, pp. 117–155. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  25. 25.
    Ojika, T., Watanabe, S., Mitsui, T.: Deflation algorithm for the multiple roots of a system of nonlinear equations. J. Math. Anal. Appl. 96(2), 463–479 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Plantinga, S., Vegter, G.: Isotopic approximation of implicit curves and surfaces. In: Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, SGP 2004, pp. 245–254 (2004)Google Scholar
  27. 27.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. J. Appl. Algebra Eng. Commun. Comput. 9(5), 433–461 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stahl, V.: Interval Methods for Bounding the Range of Polynomials and Solving Systems of Nonlinear Equations. Ph.D. thesis, Johannes Kepler University, Linz (1995)Google Scholar
  29. 29.
    Whitney, H.: On singularities of mappings of euclidean spaces. I. mappings of the plane into the plane. Ann. Math. 62(3), 374–410 (1955)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LORIA LaboratoryINRIA Nancy Grand EstNancyFrance

Personalised recommendations