On constructions of surfaces using a geodesic in Lie group
TL;DR: In this paper, the authors investigate the problem of characterizing a surface family from a given common geodesic curve in a 3-dimensional Lie group G. They obtain a parametric representation of the surface as a linear combination of the Frenet frame in G.
Abstract: In this study, we investigate the problem how to characterize a surface family from a given common geodesic curve in a 3-dimensional Lie group G. We obtain a parametric representation of the surface as a linear combination of the Frenet frame in G. Also, we give the relation about developability along the common geodesic of the members of surface family. Finally, we illustrate some examples to verify this method.
Citations
More filters
DOI•
01 Sep 2020
TL;DR: In this paper, the authors construct timelike ruled surfaces whose Disteli-axis is constant in Minkowski 3-space and attain a general system characterizing these surfaces.
Abstract: In this study, we construct timelike ruled surfaces whose Disteli-axis is constant in Minkowski 3-space □. Then we attain a general system characterizing these surfaces, and also give neces sary and sufficient conditions for a timelike ruled surface to get a constant Disteli-axis.
12 citations
Additional excerpts
...Despite its long history, the ruled surface has been highlighted by researchers as well as mathematicians because it has numerous applications such as problems of designs in spatial mechanisms and physics, kinematics and computer aided design (CAD) [4, 12, 23]....
[...]
TL;DR: In this paper, the rotation minimizing frames that are related to the space curves and the sweeping surfaces that are traced by these frames in the three-dimensional Lie group were investigated, and sufficient and necessary conditions for the sweeping surface to be a developable ruled surface were obtained.
Abstract: This paper investigated the rotation minimizing frames that are related to the space curves and the sweeping surfaces that are traced by these frames in the three-dimensional Lie group. Then, the sufficient and necessary conditions for the sweeping surface to be a developable ruled surface were obtained. In particular, we mostly focused on the study of the resulting developable surface is a cylinder, cone, or tangent surface. Meanwhile, to support the results in the paper, some illustrative examples are presented.
1 citations
TL;DR: In this paper , an alternative moving frame in the 3D Lie group is used to construct the problem of how to characterize a surface family and derive the conditions from a given common geodesic curve as an isoparametric curve.
Abstract: Our purpose in this research is to use an alternative moving frame in the 3-dimensional Lie group to construct the problem of how to characterize a surface family and derive the conditions from a given common geodesic curve as an isoparametric curve. We also derive the relation about developability along the common geodesic of a ruled surface as a member of the surface family. Finally, we will give some examples to show some applications of the method.
TL;DR: In this article , the problem of constructing hypersurfaces from a given geodesic curve in 4D Euclidean space E4 was addressed using the Serret-Frenet frame.
Abstract: In this paper, we attain the problem of constructing hypersurfaces from a given geodesic curve in 4D Euclidean space E4. Using the Serret–Frenet frame of the given geodesic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be geodesic. We illustrate this method by presenting some examples.
TL;DR: In this article, the authors constructed timelike ruled surfaces with constant Disteli-axis based upon the curvature theory of dual hyperbolic (respectively, Lorentzian) spherical curvatures.
Abstract: In this work, by using the E. Study map, we construct timelike ruled surfaces with constant Disteli-axis based upon the curvature theory of dual hyperbolic (respectively, Lorentzian) spherical curv...
References
More filters
TL;DR: A parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve is derived by utilizing the Frenet trihedron frame along the given geodesics.
Abstract: In this paper, we study the problem of constructing a family of surfaces from a given spatial geodesic curve. We derive a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. By utilizing the Frenet trihedron frame along the given geodesic, we express the surface pencil as a linear combination of the components of this local coordinate frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements. We illustrate and verify the method by finding exact surface pencil formulations for some simple surfaces, such as surfaces of revolution and ruled surfaces. Finally, we demonstrate the use of this method in a garment design application. q 2003 Elsevier Ltd. All rights reserved.
117 citations
TL;DR: It is obtained that slant helices in three dimensional Lie groups with a bi-invariant metric and their involutes, spherical images, are defined and some relations between them are given.
49 citations
TL;DR: In this article, a general helices in a three dimensional Lie group with a bi-invariant metric is defined and a generalization of Lancret's theorem is obtained.
48 citations
TL;DR: The Frenet trihedron frame of the curve in Minkowski space is used to express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements.
46 citations
TL;DR: In this article, it was shown that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map of hypersurfaces in Euclidean spaces.
Abstract: Given an orientable hypersurface M of a Lie group 𝔾 with a bi-invariant metric we consider the map N : M → 𝕊 n that translates the normal vector field of M to the identity, which is a natural extension of the usual Gauss map of hypersurfaces in Euclidean spaces; it is proved that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map. One may then conclude that M has constant mean curvature (cmc) if and only if N is harmonic; some other aplications to cmc hypersurfaces of 𝔾 are also obtained.
38 citations