Framed Curves Research Articles

Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Characterization of framed curves arising from local isometric immersions

We characterize the class of framed curves that are induced by local isometric immersions (bending deformations) defined in a neighbourhood of a curve in the reference configuration. This characterization is sharp; in particular, for every framed curve belonging to the class, we construct a local isometric immersion from which it arises.

On generalised framed surfaces in the Euclidean space

<abstract><p>We have introduced framed surfaces as smooth surfaces with singular points. The framed surface is a surface with a moving frame based on the unit normal vector of the surface. Thus, the notion of framed surfaces (respectively, framed base surfaces) is locally equivalent to the notion of Legendre surfaces (respectively, frontals). A more general notion of singular surfaces, called generalised framed surfaces, is introduced in this paper. The notion of generalised framed surfaces includes not only the notion of framed surfaces, but also the notion of one-parameter families of framed curves. It also includes surfaces with corank one singularities. We investigate the properties of generalised framed surfaces.</p></abstract>

Circular evolutes and involutes of spacelike framed curves and their duality relations in Minkowski 3-space

<abstract><p>In the present paper, we defined the circular evolutes and involutes for a given spacelike framed curve with respect to Bishop directions in Minkowski 3-space. Then, we studied the essential duality relations among parallel curves, normal surfaces, and circular evolutes and involutes. Furthermore, we also studied the duality relations of their singularities. Based on these studies, we found that it is crucially important to consider the duality relations among different geometric objects for the research of submanifolds with singularities.</p></abstract>

Spacelike Framed Curves with Lightlike Components and Singularities of Their Evolutes and Focal Surfaces in Minkowski 3-space

Spacelike Framed Curves with Lightlike Components and Singularities of Their Evolutes and Focal Surfaces in Minkowski 3-space

Framed Natural Mates of Framed Curves in Euclidean 3-Space

In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we present the relationships between a framed curve and its framed natural mates. In particular, we establish some necessary and sufficient conditions for the framed natural mates of specific framed curves, such as framed spherical curves, framed helices, framed slant helices, and framed rectifying curves. Finally, we support the concept with some examples.

Framed Curve Families Induced by Real and Complex Coupled Dispersionless-Type Equations

In this study, we investigate coupled real and complex dispersionless equations for curve families, even if they have singular points. Even though the connections with the differential equations and regular curves were considered in various ways in the past, since each curve does not need to be regular, we establish the connections for framed base curves, which generalize regular curves with linear independent conditions. Also, we give the Lax pairs of the real and complex coupled dispersionless equations from the motions of any framed curve. These give us significant conditions based on the framed curvatures and associated curvatures of the framed curves for integrability since it is well known that the Lax pair provides the integrability of differential equations.

Framed Curves, Ribbons, and Parallel Transport on the Sphere

We consider curves γ:[0,1]→R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : [0, 1]\\rightarrow {\\mathbb {R}}^3$$\\end{document} endowed with an adapted orthonormal frame r:[0,1]→SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r: [0, 1]\\rightarrow SO(3)$$\\end{document}. We wish to deform such framed curves (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of r′rT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r'r^T$$\\end{document}), and a global constraint prescribing the initial and terminal values of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.

Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

On special singular curve couples of framed curves in 3D Lie groups

In this paper, we introduce Bertrand and Mannheim curves of framed curves, which are a special singular curve in 3D Lie groups. We explain the conditions for framed curves to be Bertrand curves and Mannheim curves in 3D Lie groups. We give relationships between framed curvatures and Lie curvatures of Bertrand and Mannheim curves of framed curves. In addition, we obtain the characterization of Bertrand and Mannheim curves according to the various frames of framed curves in 3D Lie groups.

Framed clad helices in Euclidean 3-space

In this study, we introduce framed clad helices, which are a generalization of clad (i.e. C-slant or 2-slant) helices in Euclidean 3-space. They also include framed helices and framed slant helices. After, we give a characterization of the framed clad helices using the alternative adapted frame, which is more useful than the adapted frame of a framed curve. Moreover, we prove the existence of framed spherical images of any framed curve using alternative adapted frames. Additionally, we obtain interesting results regarding the relationship between a framed clad helix and its framed spherical images. Finally, we support the concept with some nice graphics.

Framed curves in three-dimensional Lie groups and a Berry phase model

In this study, we introduce framed curves which can have singular points in three-dimensional Lie groups. We give Frenet-Serret type formulas of framed curves with the help of a Lie curvature in three-dimensional Lie groups. Then, we define adapted frames of framed curves in three-dimensional Lie groups such that Frenet-Serret type frame and Bishop frame. Finally, as a physical application, we give Berry phase model of polarized light wave along an optical fiber in Lie groups.

Evolutes of Fronts in de Sitter and Hyperbolic Spheres

The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.

ON FRAMED TZITZEICA CURVES IN EUCLIDEAN SPACE

Investigations are very important for non-regular curves in differential geometry. Framed curves have been used recently to study singular curves, and they have many contributions to singularity theory. In this study, framed Tzitzeica curves are introduced with the help of framed curves. In addition, some framed special curves that satisfy the Tzitzeica condition are given. New results have been obtained among the framed curves of these curves.

Singularities of the Ruled Surfaces According to RM Frame and Natural Lift Curves

In this study, the ruled surface generated by the natural lift curve in IR^3 is obtained by using the isomorphism between unit dual sphere, DS^2 and the subset of the tangent bundle of unit 2-sphere, T\bar{M}. Then, exploitting E. Study mapping and the isomorphism mentioned below, each natural lift curve on T\bar{M} is corresponded to the ruled surface in IR^3. Moreover, the singularities of this ruled surface are examined according to RM vectors and these ruled surfaces have been classified. Some examples are given to support the main results.

Framed Curves and Their Applications Based on a New Differential Equation

Characterization is very important for non-regular curves in differential geometry. Recently, the concept of framed curve has been proposed to examine a non-regular curve. Framed curves are defined as smooth curves with a moving frame that can have singular points. In this paper, the differential equation is obtained by using distance squared functions for framed curves with for each $p,q\neq 0$ framed curvatures in Euclidean space. Next, we give the relationships between this differential equation and some special curves. Moreover we intoduce a new equation of framed helices with the help of this differential equation. Moreover, these are support by some examples and figures.

Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.

Spinor representation of framed Mannheim curves

In this paper, we obtain spinor with two complex components representations of Mannheim curves of framed curves. Firstly, we give the spinor formulas of the frame corresponding to framed Mannheim curve. Later, we obtain the spinor formulas of the frame corresponding to framed Mannheim partner curve. Moreover, we explain the relationships between spinors corresponding to framed Mannheim pairs and their geometric interpretations. Finally, we present some geometrical results of spinor representations of framed Mannheim curves.

Characterizations of Framed Curves in Four-Dimensional Euclidean Space

Framed curves in Euclidean space are used to investigate singular curves and are important for singularity theory. In this study, framed curves in four-dimensional Euclidean space are introduced and new results are obtained. The relation of framed curves with Frenet curves in four-dimensional Euclidean space is given and Bishop-type frame of framed curves is introduced with the help of Euler angles. In addition, by using Bishop-type framed curves in four-dimensional Euclidean space, framed rectifying curves, framed osculating curves and framed normal curves are introduced. Also, some characterizations depending on framed curvatures are obtained.

On the Framed Normal Curves in Euclidean 4-space

In this paper, we introduce the adapted frame of framed curves and we give the relations between the adapted frame and Frenet type frame of the framed curve in four-dimensional Euclidean space. Moreover, we define the framed normal curves in four-dimensional Euclidean space. We obtain some characterizations of framed normal curves in terms of their framed curvature functions. Furthermore, we give the necessary and sufficient condition for a framed curve to be a framed normal curve.