A variational characterization and geometric integration for Bertrand curves

Abstract

In this paper, we introduce a class of functionals, in the three-dimensional semi-Euclidean space \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3_q$\end{document}Rq3, having an energy density that depends only on curvature and whose moduli space of trajectories consists of LW-curves, i.e., curves with curvature κ and torsion τ for which there are three real constants λ, μ, ρ such that λκ + μτ = ρ, with λ2 + μ2 > 0. This family of curves includes plane curves, helices, curves of constant curvature, curves of constant torsion, Lancret curves (also called generalized helices), and Bertrand curves. We present an algorithm to construct Bertrand curves in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3_q$\end{document}Rq3 by using an arclength parametrized curve in a totally umbilical surface \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^2$\end{document}S2, \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^2_1$\end{document}S12, or \documentclass[12pt]{minimal}\begin{document}$\mathbb {H}^2$\end{document}H2 and prove that every Bertrand curve in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3_q$\end{document}Rq3 can be obtained in this way. A second algorithm is presented for the construction of LW-curves by using a curve of constant slope in the ruled surface Sα whose directrix is a certain curve α with non-zero curvature and whose rulings are generated by its modified Darboux vector field.