Skip to main content
SpringerLink
Log in

On Curves with Affine-Congruent Arcs in an \( n \)-Dimensional Affine Space

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Considering an \( n \)-dimensional affine space, we demonstrate that enics (moment curves) are the only nondegenerate curves in the class of \( C^{n} \)-smooth curves every two oriented arcs of which are affine congruent. The proof is reduced to a system of functional-differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akivis M. A. and Goldberg V. V., Projective Differential Geometry of Submanifolds, North-Holland, Amsterdam (1993).

    MATH  Google Scholar 

  2. Nomizu K., “A characterization of Veronese varieties,” Nagoya Math. J., vol. 60, 181–188 (1976).

    Article  MathSciNet  Google Scholar 

  3. Nomizu K. and Yano K., “On circles and spheres in Riemannian geometry,” Math. Ann., vol. 210, 163–170 (1974).

    Article  MathSciNet  Google Scholar 

  4. Sasaki T., “On the Veronese embedding and related system of differential equations,” in: Global Differential Geometry and Global Analysis. Proc. Conf. (Berlin, 1990), vol. 1481, Springer, Berlin (1991), 210–247.

  5. Alcázar J. G., “Efficient detection of symmetries of polynomially parametrized curves,” J. Comput. Appl. Math., vol. 255, no. C, 715–724 (2014).

    Article  MathSciNet  Google Scholar 

  6. Alcázar J. G., Hermoso C., and Muntingh G., “Symmetry detection of rational space curves from there curvature and torsion,” Comput.-Aided Geom. Des., vol. 33, 51–65 (2015).

    Article  Google Scholar 

  7. Alcázar J. G., Hermoso C., and Muntingh G., “Similarity detection of rational space curves,” J. Symb. Comput., vol. 83, 4–24 (2018).

    Article  MathSciNet  Google Scholar 

  8. Alcázar J. G., Lávička M., and Vršek J., “Symmetries and similarities of planar algebraic curves using harmonic polynomials,” J. Comput. Appl. Math., vol. 357, 302–318 (2019).

    Article  MathSciNet  Google Scholar 

  9. Hauer M. and Juttler B., “Projective and affine symmetries and equivalences of rational curves in arbitrary dimension,” J. Symb. Comput., vol. 87, 68–86 (2018).

    Article  MathSciNet  Google Scholar 

  10. Bizzarri M., Lávička M., and Vršek J., Computing Projective Equivalences of Special Algebraic Varieties. arXiv: 1806.05827v1 [cs.GG] (2018).

    MATH  Google Scholar 

  11. Polikanova I. V., “On the curves of the \( n \)-dimensional affine space with affine congruent arcs,” in: MAK-2015: “Mathematicians to Altai Krai,” Proceedings of the All-Russia Conference of Mathematics, Altai Univ., Barnaul (2015), 34–38.

  12. Anisov S. S., “Convex curves in \( RP^{n} \),” Proc. Steklov Inst. Math., vol. 221, 3–39 (1998).

    Google Scholar 

  13. Polikanova I. V., “On the flat curves with affine congruent arcs,” Sib. Electr. Math. Reports, vol. 15, 882–889 (2018).

    MathSciNet  MATH  Google Scholar 

  14. Polikanova I. V., “On the curves with affine congruent arcs in an affine space,” Sib. Electr. Math. Reports, vol. 16, 1612–1622 (2019).

    MathSciNet  MATH  Google Scholar 

  15. Kazaryan M. E., Lando S. K., and Prasolov V. V., Algebraic Curves: Towards moduli spaces, Springer, Moscow (2019).

    MATH  Google Scholar 

  16. Konnov V. V., “On differential-geometric characteristics of Veronese curves,” Sb. Math., vol. 191, no. 7, 1015–1031 (2000).

    Article  MathSciNet  Google Scholar 

  17. Davis D., “Generic affine differential geometry of curves in \( R^{n} \),” Proc. R. Soc. Edinb., Sect. A, Math., vol. 136, no. 6, 1195–1205 (2006).

    Article  MathSciNet  Google Scholar 

  18. Polikanova I. V., “Definition of enic by a finite set of points,” in: MAK-2017: “Mathematicians to Altai Krai,” Altai Univ., Barnaul (2017), 37–40.

  19. Aczél J., “Some general methods in the theory of functional equations in one variable. New applications of functional equations,” Uspekhi Mat. Nauk, vol. 11, no. 3, 3–68 (1956).

    MathSciNet  Google Scholar 

  20. Likhtarnikov L. M., Elementary Introduction to Functional Equations, Lan’, St. Petersburg (1998) [Russian].

    Google Scholar 

  21. Aczél J. and Dhombres J., Functional Equations in Several Variables with Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Cambridge University, Cambridge (2008) (Encyclopaedia of Mathematics and Its Applications).

    MATH  Google Scholar 

  22. Johnson W. P., “The curious history of Faa di Bruno’s formula,” Amer. Math. Monthly, vol. 109, no. 3, 217–234 (2002).

    MathSciNet  MATH  Google Scholar 

  23. Shabat A. B. and Efendiev M. Kh., “On applications of Faà-di-Bruno formula,” Ufa Math. J., vol. 9, no. 3, 131–136 (2017).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Altai Pedagogical University, Barnaul, Russia

    I. V. Polikanova

Authors
  1. I. V. Polikanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Polikanova.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 1, pp. 180–196. https://doi.org/10.33048/smzh.2022.63.112

Dedicated to the blessed memory of my mother Roza Mikhailovna Polikanova.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Polikanova, I.V. On Curves with Affine-Congruent Arcs in an \( n \)-Dimensional Affine Space. Sib Math J 63, 149–162 (2022). https://doi.org/10.1134/S0037446622010128

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446622010128

Keywords

UDC

Search

Navigation