In this paper, parallel q-equidistant ruled surfaces are defined such that the binormal vectors of given two differentiable curves are parallel along the striction curves of their corresponding binormal ruled surfaces, and the distance between the asymptotic planes is constant at proper points, which is related to symmetry. The characterizations and some other useful relations are drawn for these surfaces as well. If the surfaces are considered to be closed, then the integral invariants such as the pitch, the angle of the pitch, and the drall of them are given. Finally, some examples are presented to indicate that the distance between the proper points on the corresponding asymptotic planes is always constant.
The paper introduces a new kind of special ruled surface. The base of each ruled surface is taken to be one of the Smarandache curves of a given curve according to Frenet frame, and the generator (ruling) is chosen to be the corresponding unit Darboux vector. The characteristics of these newly defined ruled surfaces are investigated by means of first and second fundamental forms and their corresponding curvatures. An example is provided by considering both the helix curve and the Viviani’s curve.
In this paper, we define the necessary and sufficient conditions for a parametric surface on which both the involute and evolute of any given curve lie to be geodesic, asymptotic and curvature line. Then, the first and second fundamental forms of these surfaces are calculated. By using the Gaussian and mean curvatures, the developability and minimality assumptions are drawn, as well.
Moreover we extended the idea to the ruled surfaces. Finally, we provide a set of examples to illustrate the corresponding surfaces.
In this study, the spherical indicatrices of Flc frame vectors were defined on unit sphere. The arc length parameters and the Frenet vectors of these indicatrix curves were calculated, as well. Last, we have provided the geodesic curvatures according to both Euclidean space E 3 and unit sphere S 2 .
The paper investigates some special Smarandache curves according to Flc-frame in Euclidean 3-space. The Frenet and Flc frame vectors, curvature and torsion of the new constructed curves are expressed by means of the initial curve invariants. For the sake of comparison in view, an example for Smarandache curves according to both Frenet and Flc frame is also presented at the end of paper.
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