Abstract
For singular plane curves, the classical definitions of envelopes are vague. In order to define envelopes for singular plane curves, we introduce a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane and the curvature. Then we give a definition of an envelope for the one-parameter family of Legendre curves. We investigate properties of the envelopes. For instance, the envelope is also a Legendre curve. Moreover, we consider bi-Legendre curves and give a relationship between envelopes.
References
Arnol’d, V.I.: Singularities of Caustics and Wave Fronts. Mathematics and Its Applications, vol. 62. Kluwer Academic Publishers, Dordrecht (1990)
Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Basel (1986)
Bruce, J.W., Giblin, P.J.: What is an envelope? Math. Gaz. 65, 186–192 (1981)
Bruce, J.W., Giblin, P.J.: Curves and Singularities. A Geometrical Introduction to Singularity Theory, 2nd edn. Cambridge University Press, Cambridge (1992)
Bruce, J.W., Giblin, P.J., Gibson, C.G.: Caustics through the looking glass. Math. Intell. 6, 47–58 (1984)
Capitanio, G.: On the envelope of 1-parameter families of curves tangent to a semicubic cusp. C. R. Math. Acad. Sci. Paris. 335, 249–254 (2002)
Ei, S., Fujii, K., Kunihiro, T.: Renormalization-group method for reduction of evolution equations; invariant manifolds and envelopes. Ann. Phys. 280, 236–298 (2000)
Fukunaga, T., Takahashi, M.: Existence and uniqueness for Legendre curves. J. Geom. 104, 297–307 (2013)
Fukunaga, T., Takahashi, M.: Evolutes of fronts in the Euclidean plane. J. Singul. 10, 92–107 (2014)
Fukunaga, T., Takahashi, M.: Evolutes and involutes of frontals in the Euclidean plane. Demonstr. Math. 48, 147–166 (2015)
Gibson, C.G.: Elementary Geometry of Differentiable Curves. An Undergraduate Introduction. Cambridge University Press, Cambridge (2001)
Gray, A., Abbena, E., Salamon, S.: Modern Differential Geometry of Curves and Surfaces with Mathematica, Studies in Advanced Mathematics, 3rd edn. Chapman and Hall/CRC, Boca Raton (2006)
Ishikawa, G.: Zariski’s Moduli Problem for Plane Branches and the Classification of Legendre Curve Singularities. Real and Complex Singularities, pp. 56–84. World Sci. Publ., Hackensack (2007)
Ishikawa, G.: Singularities of Curves and Surfaces in Various Geometric Problems. CAS Lecture Notes, vol. 10, Exact Sciences (2015)
Izumiya, S.: Singular solutions of first-order differential equations. Bull. Lond. Math. Soc. 26, 69–74 (1994)
Izumiya, S.: On Clairaut-type equations. Publ. Math. Debrecen 45, 159–166 (1995)
Izumiya, S., Romero-Fuster, M.C., Ruas, M.A.S., Tari, F.: Differential Geometry from a Singularity Theory Viewpoint. World Scientific Pub. Co Inc, Singapore (2015)
Kunihiro, T.: A geometrical formulation of the renormalization group method for global analysis. Prog. Theor. Phys. 94, 503–514 (1995)
Rutter, J.W.: Geometry of Curves. Chapman & Hall/CRC, Boca Raton (2000)
Takahashi, M.: On completely integrable first order ordinary differential equations. In: Proceedings of the Australian-Japanese Workshop on Real and Complex Singularities, pp. 388–418 (2007)
Thom, R.: Sur la thorie des enveloppes. J. Math. Pures Appl. (9) 41, 177–192 (1962)
Author information
Authors and Affiliations
Corresponding author
Additional information
This is a partially supported by JSPS KAKENHI Grant Number JP 26400078.
Rights and permissions
About this article
Cite this article
Takahashi, M. Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane. Results Math 71, 1473–1489 (2017). https://doi.org/10.1007/s00025-016-0619-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0619-7