Bertrand 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.

A Generalization of Curve Mates: Normal Mate of a Curve

This a paper, a new curve pair is defined that generalizes some pairs of curves well known as Mannheim and Bertrand curve pairs. A normal curve pair is defined in such a way that a vector u obtained by overlapping the normal planes of the G and G* curves makes the same angle as the binormals of these curves. The relationship between torsions and curvatures of curve pairs was analyzed. Moreover, The unit quaternion q corresponding to the rotation matrix between the Frenet vectors of the curves was defined. In the conclusion, it is expressed express which famous pairs of curves will be obtained in which particular case.

Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space

Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space

Timelike surfaces with Bertrand geodesic curves in Minkowski 3–space

Geodesic curves on a surface play an essential role in reasonable implementation. A curve on a surface is a geodesic curve if its principal normal vector is aligned with the surface normal. Using the Serret–Frenet frame, the timelike (TL) surfaces can be specified as linear combinations of the components of the local frames in Minkowski 3–space E13. With these parametric representations, we obtained the indispensable and required events for the specified Bertrand (B) curves to be the geodesic curves on these surfaces. Afterword, the conclusion regarding the TL ruled surface is also made. Finally, the models are declared to assure that the suggested methods are effective in outcome manufacturing by modifying the styles of the surface pair.

Partner Curves in Three Dimensional Degenerated Space Form

Using elementary and effective methods we study cone curves and their associated curves or partner curves in the three dimensional lightlike cone $\Q^3$, which is called three dimensional degenerated space form of the four dimensional Minkowski space $\E^4_1$. We define the associated curve of the cone curve and also partner curves of some special curves, such as a Bertrand curve and a Mannheim curve. We consider the properties and relations of a curve and its associated curve or partner curve. Some geometric characterizations of these curves are also given.

Extending Natural Mates in Euclidean 3-space and Applications to Bertrand Pairs

In Euclidean 3-space, a family of curves, the co-successor, is motivated and then introduced in relation to the natural mate. A complete characterization of co-successors is proved, followed by an application of the co-successor towards describing Bertrand curves and their mates.

About Bertrand curves and type-1 Bishop frame in three-dimensional Weyl space W3

About Bertrand curves and type-1 Bishop frame in three-dimensional Weyl space W3

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

In the theory of curves in Euclidean $3$-space, it is well known that a curve $\beta $ is said to be a Bertrand curve if for another curve $\beta^{\star}$ there exists a one-to-one correspondence between $\beta $ and $\beta^{\star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Minkowski $3$-space. Related examples that meet these conditions are given. Moreover, thanks to this new approach, timelike, spacelike and Cartan null Bertrand mates of a timelike general helix have been obtained.

A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space

The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.

Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space

The primary objective of this work is to discuss a surface family with the similarity of Bertrand curves in 3D Galilean space. Subsequently, by applying the Serret–Frenet frame, we estimate the sufficient and necessary statuses of a surface family with Bertrand curves as joint asymptotic curves. The dilation to ruled surfaces is also summarized. Meanwhile, the epitomes are illustrated to provide an explanation of the theoretical results.

Surfaces Family with Bertrand Curves as Common Asymptotic Curves in Euclidean 3–Space E3

The main result of this paper is constructing a surfaces family with the similarity of Bertrand curves in Euclidean 3–space E3. Then, by utilizing the Serret–Frenet frame, we conclude the sufficient and necessary conditions of surfaces family interpolating Bertrand curves as common asymptotic curves. Consequently, the expansion to the ruled surfaces family is also depicted. As implementations of our main results, we demonstrate some examples to confirm the method.

The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature.

Position Vectors of the Natural Mate and Conjugate of a Space Curve

The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in E 3 . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterization of a general helix is introduced.

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.

The spinor representations of framed Bertrand curves

In this study, we intend to examine the framed Bertrand curves in three-dimensional Euclidean space E3 by using the spinors, which have a fundamental place and importance in different disciplines from mathematics to physics. For this purpose, we investigate the spinor representations of framed Bertrand mates in E3. Additionally, we present some geometric results and interpretations. Then, we construct numerical examples with illustrative figures in order to support the given materials.

Forming coupled dispersionless equations of families of Bertrand curves

In this study, we establish a link of the coupled dispersionless (CD) equations system with the motion of Bertrand curve pairs. Moreover, we find the Lax equations that provide the integrability of these equations. By taking an appropriate choice of variables we show the link of the short pulse (SP) equation with the motion of Bertrand curve pairs via the reciprocal (hodograph) transformation. Finally, we prove that the conserved quantity of the corresponding coupled dispersionless equations obtained from each of these curve pairs is constant.

On Quaternionic Bertrand Curves in Euclidean $3$-Space

In this article, spatial quaternionic Bertrand curve pairs in the 3-dimensional Euclidean space are examined. Algebraic properties of quaternions, basic definitions and theorems are given. Later, some characterizations of spatial quaternionic Bertrand curve pairs are obtained in the 3-dimensional Euclidean space.

Characterization of spherical and plane curves using rotation minimizing frames

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves as curves whose so-called natural mates are spherical.

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

In this paper, we give some characterization of quaternionic Bertrand curves whose the torsion is non-zero but bitorsion is zero in $\mathbb{R}^{4}$ according to Type 2-Quaternionic Frame. One of the most important points in working on quaternionic curves is that given a curve in $\mathbb{R}^{4}$, the curve in $\mathbb{R}^{3}$ associated with this curve is determined individually. So, we obtain some relationships between quaternionic Bertrand curve $\alpha^{(4)}$ in $\mathbb{R}^{4}$ and its associated spatial quaternionic curve $\alpha$ in $\mathbb{R}^{3}$. Also, we support some theorems in the paper by means of an example.

Bertrand Curves and B-Lift Curves in Lorentzian 3-Space

In this article, based on Thorpe's definition, we define a new curve called the B-lift curve in Lorentzian 3-space and examine the Frenet vectors of the B-lift curve. Furthermore, we examine the relationship between the Frenet vectors of the Bertrand curve and the Frenet vectors of the natural lift curve. Finally, we give an example on these results.