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Evolutes and involutes of frontals in the euclidean plane

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EN
Abstrakty
EN
We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. The definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals.
Wydawca
Rocznik
Strony
147--166
Opis fizyczny
Bibliogr. 18 poz., rys.
Twórcy
autor
autor
Bibliografia
  • [1] V. I. Arnol’d, Critical points of functions on a manifolds with boundary, the simple Lie groups Bk, Ck, and F4 and singularities of evolvents, Russian Math. Surveys 33 (1978), 99–116.
  • [2] V. I. Arnol’d, Singularities of Caustics and Wave Fronts, Mathematics and its Applications, 62, Kluwer Academic Publishers, 1990.
  • [3] V. I. Arnol’d, Topological properties of Legendre projections in contact geometry of wave fronts, St. Petersburg Math. J. 6 (1995), 439–452.
  • [4] V. I. Arnol’d, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps, vol. I, Birkhäuser, 1986.
  • [5] J. W. Bruce, T. J. Gaffney, Simple singularities of mappings C, 0 → C2, 0, J. London Math. Soc. (2) 26 (1982), 465–474.
  • [6] J. W. Bruce, P. J. Giblin, Curves and Singularities, A Geometrical Introduction to Singularity Theory, Second edition, Cambridge University Press, Cambridge, 1992.
  • [7] T. Fukunaga, M. Takahashi, Existence and uniqueness for Legendre curves, J. Geom. 104 (2013), 297–307.
  • [8] T. Fukunaga, M. Takahashi, Evolutes of fronts in the Euclidean plane, Journal of Singularities 10 (2014), 92–107.
  • [9] T. Fukunaga, M. Takahashi, Involutes of fronts in the Euclidean plane, Preprint, Hokkaido University Preprint Series, No. 1045, 2013.
  • [10] C. G. Gibson, Elementary Geometry of Differentiable Curves, An Undergraduate Introduction, Cambridge University Press, Cambridge, 2001.
  • [11] A. Gray, E. Abbena, S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third edition, Studies in Advanced Mathematics, Chapman and Hall/CRC, Boca Raton, FL, 2006.
  • [12] E. Hairer, G. Wanner, Analysis by its History, Springer-Verlag, New York, 1996.
  • [13] C. Huygens, Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae, 1673.
  • [14] G. Ishikawa, Zariski’s moduli problem for plane branches and the classification of Legendre curve singularities, Real and Complex Singularities, World Sci. Publ. Hackensack, NJ, 2007, 56–84.
  • [15] G. Ishikawa, S. Janeczko, Symplectic bifurcations of plane curves and isotropic liftings, Q. J. Math. 54 (2003), 73–102.
  • [16] J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York, 1975.
  • [17] G. de l’Hospital, Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes, 1696.
  • [18] O. P. Shcherbak, Singularities of families of evolvents in the neighborhood of an inflection point of the curve, and the group H3 generated by reflections,Functional Anal. Appl. 17 (1983), 301–303.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-95443f12-8ef7-467c-a597-913297eb53a9
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