Abstract
Here we apply our original scheme for the reconstruction of plane curves whose curvatures are specified by functions of the radial coordinate to the curves introduced by J.-A. Serret. These curves are associated with the natural numbers and we extend their definition in order to include them into a family of curves dependingon two continuous real parameters. The explicit parametrization of this new class of curves is presented as well.
Mathematics Subject Classification (2010). Primary 53A04; Secondary 53A55, 53A17.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J.-A. Serret, Sur la représentation géometrique des fonctions elliptiques et ultraelliptiques, J. Mathematiques Pures et Appliquées 10 (1845), 257–290.
J.-A. Serret, Sur les courbes elliptiques de la première classe, J.MathematiquesPures et Appliquées 10 (1845), 421–429.
J. Liouville, A note on the Serret’s article [1], J. Mathematiques Pures et Appliquées 10 (1845), 293–296.
G. Krohs, Die Serret’schen Kurven sind die einzigen algebraischen vom Geschlecht Null, deren Koordinaten eindeutige doppelperiodische Functionen des Bogens der Kurve sind, Inaugural Dissertation Halle-Wittenberg Universit¨at, 1891, 74 pp.
Lipkovski A., Serret’s Curves, In: Book of Abstracts of the International Topological Conference Topology and Applications, Phasis, Moscow 1996, pp. 191–192.
M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Springer, New York, 1988.
J. Oprea, Differential Geometry and Its Applications, Mathematical Association of America, Washington D.C., 2007.
P. Djondjorov, V. Vassilev and I. Mladenov, Plane Curves Associated with Integrable Dynamical Systems of the Frenet-Serret Type, In: Proc. 9th International Workshop on Complex Structures, Integrability and Vector Fields, World Scientific, Singapore 2009, pp. 56–62.
M. Hadzhilazova and I. Mladenov, On Bernoulli’s Lemniscate and Co-lemniscate, C. R. Bulg. Acad. Sci. 63 (2010), 843–848.
I. Mladenov, M. Hadzhilazova, P. Djondjorov and V. Vassilev, On the Plane Curves Whose Curvature Depends on the Distance from the Origin, AIP Conference Proceedings 1307 (2010), 112–118.
I. Mladenov, M. Hadzhilazova, P. Djondjorov and V. Vassilev, On Some Deformations of the Cassinian Oval, In: AIP Conference Proceedings 1340 (2011), 81–89.
I. Mladenov, M. Hadzhilazova, P. Djondjorov and V. Vassilev, On the Generalized Sturmian Spirals, C. R. Bulg .Acad. Sci. 64 (2011), 633–640.
V. Vassilev, P. Djondjorov and I. Mladenov, Integrable Dynamical Systems of the Frenet-Serret Type, In: Proc. 9th International Workshop on Complex Structures, Integrability and Vector Fields, World Scientific, Singapore 2009, pp. 234–244.
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
F. Olver, D. Lozier, R. Boisvert and Ch. Clark (Eds), NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge, 2010.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Mladenov, I.M., Hadzhilazova, M.T., Djondjorov, P.A., Vassilev, V.M. (2013). Serret’s Curves, their Generalization and Explicit Parametrizations. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_35
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0448-6_35
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0447-9
Online ISBN: 978-3-0348-0448-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)