Slant Helices Research Articles

A New Link to Helices in Euclidean $3$-Space

In this paper, we introduce a novel approach for obtaining the parametric expression and description of a general helix, slant helix, and Darboux helix. The new method involves projecting $\alpha $ onto a plane passing through $\alpha \left( 0\right) $ and orthogonal to the unit axis vector $U$ in order to determine the position vector of the general helix $\alpha $. The position vector of the helix with the plane curve $\gamma $ and its axis $U$ is then established. Additionally, a relation between the curvatures of $\alpha $ and $\gamma $ is presented. The proposed technique is then applied to derive the parametric representation of a slant helix and Darboux helix, followed by the provision of various examples obtained through the application of this methodology.

The Notes on Slant Helices According to Equiform Frame on Symplectic Space

In this this paper, we have dealed with the properties of k-type. We express some characterizations for k-type slant helix according to equiform frame geometry in symplectic regular curves.

Salkowski curves and spherical epicycloids

The relationship between Salkowski curves, a family of slant helices with constant curvature and non-constant torsion, and the family of spherical epicycloid curves is studied. It is shown that, for some values of the parameter defining the Salkowski curve, the curve is the image by a shear transformation along the z-axis of a spherical epicycle. Therefore, the projection of both curves on the xy-plane is the same. This result can be extended to the whole family of Salkowski curves if some parameter defining the spherical epicycle is allowed to be a complex imaginary number.

Exploring helices in Minkowski $3$-space $\mathbb{E}_{1}^{3}$

This study focuses on the geometric exploration of helices in Minkowski $3-$ space. For this purpose, we study the problem of constructing a spacelike general helix, slant helix, or Darboux helix with a timelike principal from a given plane curve, fixed vector, and constant angle. We obtain a parametric representation for the helices whose projection is onto the plane $P$ perpendicular to the fixed vector $U$ share the same fixed vector. In addition, we furnish various illustrative examples showcasing the geometric characteristics of these helices.

Cα-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}_\\alpha -$$\\end{document}helices and Cα-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}_\\alpha -$$\\end{document} slant helices in fractional differential geometry

In this study, the theory of curves is reconstructed with fractional calculus. The condition of a naturally parametrized curve is described, and the orthonormal conformable frame of the naturally parametrized curve at any point is defined. Conformable helix and conformable slant helix curves are defined with the help of conformable frame elements at any point of the conformable curve. The characterizations of these curves are obtained in parallel with the conformable analysis Finally, examples are given for a better understanding of the theories and their drawings are given with the help of Mathematics.

Spacelike Ac-Slant Curves with Non-Null Principal Normal in Minkowski 3-Space

In this paper, we define a spacelike ac-slant curve whose scalar product of its acceleration vector and a unit non-null fixed direction is a constant in Minkowski 3-space. Furthermore, we give a characterization depending on the curvatures of the spacelike ac-slant curve. After that, we get the relationship between a spacelike ac-slant curve and several distinct types of curves, such as spacelike Lorentzian spherical curves, spacelike helices, spacelike slant helices, and spacelike Salkowski curves, enhancing our understanding of its geometric properties in Minkowski 3-space. Finally, we used Mathematica, a symbolic computation software, to support the notions of an ac-slant curve with attractive images.

CHARACTERIZATION OF BI-NULL SLANT(BNS) HELICES OF (k,m )-TYPE IN R^3_1 AND R^5_ 2

The present study discusses bi-null slant helices of $(k, m)$ type in $R_2^5$ and give the characterization for a curve to be certain $(k,m)$ type bi null slant helix (BNS helix). The discussion includes the proofs for the non existence cases of $(k,m)$ type bi null slant helices in $R_2^5$. Moreover certain characterizations and non existence have also been obtained for bi null slant helix to be $(k,m)$ type using modified orthogonal frame.

Osculating mate of a Frenet curve in the Euclidean 3-space

A new kind of partner curves called osculating mate of a Frenet curve is introduced. Some characterizations for osculating mate are obtained and using the obtained results some special curves such as slant helix, spherical helix, C-slant helix and rectifying curve are constructed.

An Alternative Method for Determination of the Position Vector of a Slant Helix

In this paper, we provide an alternative method to determine the position vector of a slant helix with the help of an alternative moving frame. We then construct a vector differential equation in terms of the principal normal vector of a slant helix using an alternative moving frame. By solving this vector differential equation, we determine the position vector of the slant helix. Afterward, we obtain parametric representations of some examples of slant helices for chosen curvature and torsion functions as an application of the proposed method. Finally, we discuss the method and whether further research should be conducted or not.

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.

$ C^* $-partner curves with modified adapted frame and their applications

<abstract><p>In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space $ E^3 $, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the "modified adapted frame" denoted by $ \{N^*, C^*, W^*\} $. Then, we have given some characterizations of this new frame. In addition to that, we have defined $ N^* $-slant helices and $ C^* $-slant helices according to $ \{N^*, C^*, W^*\} $. Moreover, we have studied $ C^* $-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between $ C^* $-partner curves and helices, slant helices.</p></abstract>

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.

K-TYPE BI-NULL CARTAN SLANT HELICES IN R6 2

In the present paper, we give the notion of k-type bi-null Cartan slant helices in R6 2, where k ∈ {1,2,3,4,5,6}. We give the necessary and sufficient conditions for bi-null Cartan curves to be k-type slant helices in terms of their curvature functions.

Framed isophote curves on the framed surface

The isophote curve on surfaces is an important consequence of Lambert reflect law in optical research. In this paper, we define the framed isophote curve along a surface which original curve and surface may be singular. We give a condition of the framed isophote curve, and the expression of the light vector with different conditions. The relationship between the framed isophote curve and the framed helix is expounded. Moreover, we get the relationship between the framed isophote curve and the framed slant helix. Finally, we give the theorem for singularities recognition of framed isophote curves and two examples.

Characterization of proper curves and proper helix lying on $S_{1}^2(r)$

In this paper, we analyse the proper curve $\gamma(s)$ lying on the pseudo-sphere. We develop an orthogonal frame $\lbrace V_{1}, V_{2}, V_{3} \rbrace$ along the proper curve, lying on pseudosphere. Next, we find the condition for $\gamma(s)$ to become $V_{k} -$ slant helix in Minkowski space. Moreover, we find another curve $\beta(\bar{s})$ lying on pseudosphere or hyperbolic plane heaving $V_{2} = \bar{V_{2}}$ for which $\lbrace \bar{V_{1}},\bar{V_{2}},\bar{V_{3}} \rbrace$, an orthogonal frame along $\beta(\bar{s})$. Finally, we find the condition for curve $\gamma(s)$ to lie in a plane.

SOME INTEGRAL CHARACTERIZATIONS OF TIMELIKE HELICES IN R2^4

In this study, we examine timelike helices in R2^4 and some integral characterizations of these curves in terms of Frenet frame. In addition, we study timelike B2 slant helices in R2^4 and present the differential equations for vector positions.

On the Characterisations of Curves with Modified Orthogonal Frame in $\mathbb{E}^{3}$

This study analyses (k,m)-type slant helices in compliance with the modified orthogonal frame in 3-dimensional Euclidean space ($\mathbb{E}^{3}$). Furthermore, we perform some characterisations of curves with modified orthogonal frames in $\mathbb{E}^{3}$.

A unified method for constructing developable surface pencils interpolating characteristic curves

Developable surface is a simple and common surface in surface modeling. Geodesic, line of curvature, asymptotic curve, and D-type curve are important characteristic curves on the surfaces. This study gives a unified method for constructing developable surface pencils interpolating these four kinds of characteristic curves. Given a regular space curve R(r), we derive a new condition that a surface pencil P (r, t) interpolating R(r) is developable. The result shows that the condition completely depends on a univalent function λ and an angle θ. By choosing different λ and θ, we can not only control the shape of P (r, t), but also make R(r) become any kind of characteristic curve on P (r, t). Furthermore, we take natural and conjugate curve pairs as those characteristic curves to construct developable surface pairs. Finally, an example of a slant helix shows that the proposed unified method is more general than other methods, and has good interactivity and convenience.

A New Approach For Smarandache Curves

In this paper, we introduce new adjoint curves which are associated curves in Euclidean space of three dimension. They are generated with the help of integral curves of special Smarandache curves. We attain some connections between Frenet apparatus of these new adjoint curves and main curve. We characterize these curves in which conditions they are general helix and slant helix. Finally, we exemplify them with figures.

ON DUAL SLANT HELICES IN $D^3$

In this paper, by using the method in [6] we study dual tangent indicatrix and dual binormal indicatrix of a dual slant helix. Moreover we obtain the relationship between the dual slant helices and dual general helices in $\mathbb{D}^{3}.$ We get some characterizations of a dual slant helix in $\mathbb{D}^{3}$.