科研之友
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摘要
In this paper, we introduce a class of functionals, in the three-dimensional semi-Euclidean space R-q(3), having an energy density that depends only on curvature and whose moduli space of trajectories consists of LW-curves, i.e., curves with curvature kappa and torsion tau for which there are three real constants lambda, mu, rho such that lambda kappa + mu tau = rho, with lambda(2) + mu(2) > 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 R-q(3) by using an arclength parametrized curve in a totally umbilical surface S-2, S-1(2), or H-2 and prove that every Bertrand curve in R-q(3) 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 S-alpha whose directrix is a certain curve alpha with non-zero curvature and whose rulings are generated by its modified Darboux vector field.
- 出版日期2013-4
