Advertisement

Mathematische Annalen

, Volume 373, Issue 3–4, pp 1299–1327 | Cite as

Surfaces containing two circles through each point

  • M. SkopenkovEmail author
  • R. Krasauskas
Article

Abstract

We find all analytic surfaces in 3-dimensional Euclidean space such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface (and analytically depending on the point). The search for such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of inversions:
  • the set \(\{p+q:p\in \alpha ,q\in \beta \}\), where \(\alpha ,\beta \) are two circles in \(\mathbb {R}^3\);

  • the set \(\{2\frac{p \times q^{}}{|p+q|^2}:p\in \alpha ,q\in \beta ,p+q\ne 0\}\), where \(\alpha ,\beta \) are circles in the unit sphere \({S}^2\);

  • the set \(\{(x,y,z): Q(x,y,z,x^2+y^2+z^2)=0\}\), where \(Q\in \mathbb {R}[x,y,z,t]\) has degree 2 or 1.

    The proof uses a new factorization technique for quaternionic polynomials.

Mathematics Subject Classification

51B10 13F15 16H05 

Notes

Acknowledgements

This results have been presented at Moscow Mathematical Society seminar, SFB “Discretization in geometry and dynamics” colloquim in Berlin, Ya. Sinai—G. Margulis conference, and the conference “Perspectives in real geometry” in Luminy. The authors are especially grateful to A. Pakharev for joint numerical experiments, related studies [26], and for pointing out that numerous surfaces we tried to invent all have form (4). The authors are grateful to N. Lubbes and L. Shi for parts of Fig. 1, to A. Gaifullin and S. Ivanov for finding gaps in earlier versions of the proofs, to A. Bobenko, J. Capco, S. Galkin, A. Kanunnikov, O. Karpenkov, A. Klyachko, J. Kollár, W. Kühnel, A. Kuznetsov, N. Moshchevitin, S. Orevkov, F. Petrov, R. Pignatelli, F. Polizzi, H. Pottmann, G. Robinson, I. Sabitov, J. Schicho, K. Shramov, S. Tikhomirov, V. Timorin, M. Verbitsky, E. Vinberg, J. Zahl, S. Zubė for useful discussions. The first author is grateful to King Abdullah University of Science and Technology for hosting him during the start of the work over the paper.

References

  1. 1.
    Beauregard, R.: When is F[x, y] a unique factorization domain? Proc. Am. Math. Soc. 117(1), 67–70 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Beauville, A.: Complex algebraic surfaces, London Math. Soc. Student Texts, vol. 34, 2nd edn. Cambridge University Press, Cambridge, p. 132 (1996)Google Scholar
  3. 3.
    Brauner, H.: Die windschiefen Kegelschnittfläschen. Math. Ann. 183, 33–44 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cass, D., Arpaia, P.J.: Matrix generation of Pythagorean n-tuples. Proc. Am. Math. Soc. 109(1), 1–7 (1990)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Coolidge, J.L.: A Treatise on the Circle and Sphere, p. 603. The Clarendon Press, Oxford (1916)zbMATHGoogle Scholar
  6. 6.
    Degen, W.: Die zweifachen Blutelschen Kegelschnittflächen. Manuscr. Math. 55(1), 9–38 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dietz, R., Hoschek, J., Juettler, B.: An algebraic approach to curves and surfaces on the sphere and on other quadrics. Comput. Aided Geom. Design 10, 211–229 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eilenberg, S., Niven, I.: The “fundamental theorem of algebra” for quaternions. Bull. Am. Math. Soc. 50, 246–248 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193, 56–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gentili, G., Stoppato, C., Struppa, D.: Regular Functions of a Quaternionic Variable. Springer Monographs in Math. Springer, Berlin, Heidelberg, p. 185 (2013)Google Scholar
  11. 11.
    Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic variable. Mich. Math. J. 56, 655–667 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gohberg, I., Lancaster, P.J., Bodman, L.: Matrix Polynomials. Academic Press Inc, Cambridge (1982)Google Scholar
  13. 13.
    Guth, L., Zahl, J.: Algebraic curves, Rich Points, and Doubly-Ruled Surfaces. arXiv:1503.02173
  14. 14.
    Gordon, B., Motzkin, T.S.: On the zeros of polynomials over division rings. Trans. Amer. Math. Soc. 116, 218–226 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ivey, T.: Surfaces with orthogonal families of circles. Proc. Am. Math. Soc. 123(3), 865–872 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kocik, J.: Clifford algebras and Euclid’s parametrization of Pythagorean triples. Adv. Appl. Clifford Algebras 17(1), 71–93 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kollár, J.: Rational Curves on Algebraic Varieties. Springer, Berlin, Heidelberg (1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kollár, J.: Quadratic Solutions of Quadratic Forms. arXiv:1607.01276
  19. 19.
    Krasauskas, R., Zubė, S.: Rational Bézier formulas with quaternion and Clifford algebra weights. In: Dokken, T., Muntingh, G. (eds.) SAGA–Advances in Shapes, Geometry, and Algebra, Geometry and Computing, vol. 10, pp. 147–166. Springer, New York (2014)Google Scholar
  20. 20.
    Lavicka, R., O’Farrell, A.G., Short, I.: Reversible maps in the group of quaternionic Moebius transformations. Math. Proc. Camb. Philos. Soc. 143(1), 57–69 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lubbes, N.: Minimal families of curves on surfaces. J. Symb. Comput. 65, 29–48 (2014). arXiv:1302.6687 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lubbes, N.: Families of circles on surfaces. arXiv:1302.6710
  23. 23.
    Lubbes, N.: Clifford and Euclidean Translations of Circles. arXiv:1306.1917v11
  24. 24.
    Nilov, F., Skopenkov, M.: A surface containing a line and a circle through each point is a quadric. Geom. Dedicata 163(1), 301–310 (2013). arXiv:1110.2338 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ore, O.: Theory of non-commutative polynomials. Ann. Math. (II) 34, 480–508 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pakharev, A., Skopenkov, M.: Surfaces containing two circles through each point and decomposition of quaternionic matrices. Russ. Math. Surv. 72(2(434)), 195–196 (2017). arXiv:1510.06510 zbMATHGoogle Scholar
  27. 27.
    Postnikov, M.: The Fermat Theorem. An Introduction to Algebraic Numbers Theory. Nauka, Moscow, p. 128 (1978)Google Scholar
  28. 28.
    Pottmann, H., Shi, L., Skopenkov, M.: Darboux cyclides and webs from circles. Comput. Aided Geom. D. 29(1), 77–97 (2012). arXiv:1106.1354 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schicho, J.: The multiple conical surfaces. Contrib. Algebra Geom. 42(1), 71–87 (2001)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Shafarevich, I.R.: Algebraic Geometry, vol. 1. Springer, New York (1974)zbMATHGoogle Scholar
  31. 31.
    Smertnig, D.: Factorizations of elements in noncommutative rings: a survey. In: Chapman, S., Fontana, M., Geroldinger, A., Olberding, B. (eds) Multiplicative Ideal Theory and Factorization Theory. Springer Proceedings in Mathematics and Statistics, vol. 170. Springer, Cham (2016)Google Scholar
  32. 32.
    Takeuchi, N.: Cyclides. Hokkaido Math. J. 29, 119–148 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Timorin, V.A.: Rectifiable pencils of conics. Mosc. Math. J. 7(3), 561–570 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Research University Higher School of Economics (Faculty of Mathematics) and Institute for Information Transmission Problems of the Russian Academy of SciencesMoscowRussia
  2. 2.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

Personalised recommendations