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Evolutes and involutes of frontals in the euclidean plane

Fukunaga T., Takahashi M.

Demonstratio Mathematica

2015

Vol. 48, nr 2

147--166

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.

Bertrand mate of timelike biharmonic legendre curves in Lorentzian Heisenberg group Heis³

Turhan E., Körpinar T.

Demonstratio Mathematica

2011

Vol. 44, nr 2

273-283

In this paper, we study Bertrand mate of timelike biharmonic Legendre curves in the Lorentzian Heisenberg group Heis³. We characterize timelike biharmonic Legendre curves in terms of their curvature and torsion. Moreover, we obtain the position vectors of timelike biharmonic Legendre curves and we construct parametric equations of Bertrand mate of timelike biharmonic Legendre curves in the Lorentzian Heisenberg group Heis³.