Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 1–29 | Cite as

Changing Views on Curves and Surfaces

  • Kathlén KohnEmail author
  • Bernd Sturmfels
  • Matthew Trager


Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface and observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface. We derive formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and show how to compute exact representations for all visual event surfaces using algebraic methods.


Computer vision Projections Contour curve Enumerative geometry 

Mathematics Subject Classification (2010)

Primary 14Q10 65D19; Secondary 68W30 53A05 



This project started in May 2016 at the GOAL workshop in Paris. We thank Mohab Safey El Din, Jean-Charles Faugère, Jon Hauenstein, and Jean Ponce for their help in the initial stages. We are also grateful to Joachim Rieger for an inspiring discussion on singularity theory, and to Emre Sertöz for helpful comments on intersection theory. Kathlén Kohn was funded by the Einstein Foundation Berlin. Bernd Sturmfels received partial support from the US National Science Foundation (DMS-1419018) and the Einstein Foundation Berlin. Matthew Trager was supported in part by the ERC advanced grant VideoWorld, the Institut Universitaire de France, the Inria-CMU associated team GAYA, and the ANR grant RECAP.


  1. 1.
    Abdeljaoued, J., Diaz-Toca, G. M., Gonzalez-Vega, L.: Minors of Bézout matrices, subresultants and the parametrization of the degree of the polynomial greatest common divisor. Inter. J. Comput. Math. 81, 1223–1238 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arnol’d, V. I.: Singularities of smooth mappings. Russian Math. Surveys 23, 1–43 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arrondo, E., Bertolini, M., Turrini, C.: A focus on focal surfaces. Asian J. Math. 5, 535–560 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beltrametti, M. C., Carletti, E., Monti Bragadin, G., Gallarati, D.: Lectures on Curves, Surfaces and Projective Varieties: a Classical View of Algebraic Geometry. EMS Textbooks in Mathematics, vol. 9 European Mathematical Society (2009)Google Scholar
  5. 5.
    Bertin, M. -A.: On the singularities of the trisecant surface to a space curve. Matematiche (Catania) 53, 15–22 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bowyer, K. W., Dyer, C. R.: Aspect graphs: an introduction and survey of recent results. Inter. J. Imaging Syst. Technol. 2, 315–328 (1990)CrossRefGoogle Scholar
  7. 7.
    Colley, S.: Lines having specified contact with projective varieties. Proc. 1984 Vancouv. Conf. Algeb. Geomet. 6, 47–70 (1986)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Colley, S.: Enumerating stationary multiple points. Adv. Math. 66, 149–170 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dolgachev, I. V.: Classical Algebraic Geometry: A Modern View. Cambridge Univ Press (2012)Google Scholar
  10. 10.
    Edge, W. L.: The Theory of Ruled Surfaces. Cambridge University Press (1931)Google Scholar
  11. 11.
    Eisenbud, D., Harris, J.: 3264 and All That: A Second Course in Algebraic Geometry. Cambridge University Press (2016)Google Scholar
  12. 12.
    Eisenbud, D., Schreyer, F. -O.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16, 537–579 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Forsyth, D. A., Ponce, J.: Computer Vision: A Modern Approach, 2nd edn. Pearson (2012)Google Scholar
  14. 14.
    Gel’fand, I. M., Kapranov, M. M., Zelevinsky, A. V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    Johansen, P.: The Geometry of the Tangent Developable. Computational Methods for Algebraic Spline Surfaces, pp 95–106. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kergosien, Y. L.: La famille des projections orthogonales d’une surface et ses singularités. C. R. Acad. Sci. Paris Sér. I Math. 292, 929–932 (1981)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Koenderink, J. J.: Solid Shape. MIT Press (1990)Google Scholar
  19. 19.
    Koenderink, J. J., van Doorn, A. J.: The singularities of the visual mapping. Biol. Cybern. 24, 51–59 (1976)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kohn, K.: Coisotropic hypersurfaces in the Grassmannian. arXiv:1607.05932
  21. 21.
    Kohn, K., Ndland, B.I.U., Tripoli, P.: Secants, bitangents, and their congruences. In: Smith, G., Sturmfels, B. (eds.) Combinatorial Algebraic Geometry, Fields Institute (to appear)Google Scholar
  22. 22.
    Lee, H., Sturmfels, B.: Duality of multiple root loci. J. Algebra 446, 499–526 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pae, S., Ponce, J.: On computing structural changes in evolving surfaces and their appearance. Inter. J. Comput. Vis. 43(2), 113–131 (2001)CrossRefzbMATHGoogle Scholar
  24. 24.
    Petitjean, S.: The complexity and enumerative geometry of aspect graphs of smooth surfaces. Algorithms in algebraic geometry and applications (Santander, 1994), 317–352, Progr Math., vol. 143. Basel, Birkhäuser (1996)Google Scholar
  25. 25.
    Petitjean, S., Ponce, J., Kriegman, D.: Computing exact aspect graphs of curved objects: Algebraic surfaces. Inter. J. Comput. Vis. 9, 231–255 (1992)CrossRefGoogle Scholar
  26. 26.
    Piene, R.: Numerical characters of a curve in projective n-space. In: Holm, P. (ed.) Real and Complex Singularities (1976)Google Scholar
  27. 27.
    Piene, R.: Some formulas for a surface in \(\mathbb {P}^{3}\). In: Olson, L.D. (ed.) Algebraic Geometry. Lecture Notes in Mathematics, vol. 687. Springer, Berlin (1978)Google Scholar
  28. 28.
    Piene, R.: Cuspidal projection of space curves. Math. Ann. 256, 95–119 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Platonova, O. A.: Projections of smooth surfaces. J. Math. Sci. 35(6), 2796–2808 (1986)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ponce, J., Hebert, M.: On Image Contours of Projective Shapes. European Conference on Computer Vision Springer International Publishing (2014)Google Scholar
  31. 31.
    Ponce, J., Kriegman, D. J.: Computing exact aspect graphs of curved objects: parametric surfaces. Department of Computer Science University of Illinois at Urbana-Champaign (1990)Google Scholar
  32. 32.
    Ranestad, K., Sturmfels, B.: On the convex hull of a space curve. Adv. Geom. 12, 157–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rieger, J. H.: Global bifurcation sets and stable projections of nonsingular algebraic surfaces. Inter. J. Comput. Vis. 7, 171–194 (1992)CrossRefGoogle Scholar
  34. 34.
    Rieger, J. H.: Computing view graphs of algebraic surfaces. J. Symb. Comput. 16, 259–272 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Salmon, G.: A Treatise on the Analytic Geometry of Three Dimensions, 4th edn. Dublin (1882)Google Scholar
  36. 36.
    Seigal, A., Sturmfels, B.: Real rank two geometry. J. Algebra 484, 310–333 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sturmfels, B.: The Hurwitz form of a projective variety. J. Symb. Comput. 79, 186–196 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Thom, R.: Structural Stability and Morphogenesis. W. A. Benjamin (1972)Google Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.TU BerlinBerlinGermany
  2. 2.MPI LeipzigLeipzigGermany
  3. 3.UC BerkeleyBerkeleyUSA
  4. 4.Inria, École Normale Supérieure Paris, CNRSPSL Research UniversityParisFrance

Personalised recommendations