Main Article Content
The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.
- C.G. Gibson, Elementary Geometry of Differentiable Curves, Cambridge University Press, Cambridge, 2001.
- V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, Birkha äuser, Boston, 1985.
- V.I. Arnold, Singularities of Caustics and Wave Fronts, Kluwer Academic Publishers, Dordrecht, 1990.
- J.W. Bruce, P.J. Giblin, Curves and Singularities, a Geometrical Introduction to Singularity Theory, Cambridge University Press, Cambridge, 1992.
- S. Izumiya, D.H. Pei, T. Sano, E. Torii, Evolutes of Hyperbolic Plane Curves, Acta Math. Sinica. 20 (2004), 543–550. https://doi.org/10.1007/s10114-004-0301-y.
- T. Fukunaga, M. Takahashi, Existence and Uniqueness for Legendre Curves, J. Geom. 104 (2013), 297–307. https://doi.org/10.1007/s00022-013-0162-6.
- T. Fukunaga, M. Takahashi, Evolutes of Fronts in the Euclidean Plane, J. Singul. 10 (2014), 92-107. https://doi.org/10.5427/jsing.2014.10f.
- H. Yu, D. Pei, X. Cui, Evolutes of Fronts on Euclidean 2-Sphere, J. Nonlinear Sci. Appl. 8 (2015), 678-686.
- L. Chen, M. Takahashi, Dualities and Evolutes of Fronts in Hyperbolic and De Sitter Space, J. Math. Anal. Appl. 437 (2016), 133-159. https://doi.org/10.1016/j.jmaa.2015.12.029.
- X. Cui, D. Pei, H. Yu, Evolutes of Null Torus Fronts, J. Nonlinear Sci. Appl. 8 (2015), 866-876.