Salkowski Curves Research Articles

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.

Bishop Frames of Salkowski Curves in E3

In this study, alternative, type-1 Bishop, type-2 Bishop and N-Bishop frames of Salkowski curves in E3 are calculated. Moreover, curvatures, Darboux and pol vectors of these frames are found. Also, relationships between the Bishop frames, Darboux vectors and pole vectors are given.

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.

On Determining the Equation of a Salkowski Curve Satisfying tau/kappa=1/s

In this paper, we determine the equation of a Salkowski curve whose ratio of torsion to curvature is given by 1/s, where s is the arc length of the curve. The Frenet-Serret equations provide the third-order vector differential equation for the unit tangent vector T(s) and the general (series) solution was obtained. In the end, the series solution is entirely determined by the given initial conditions.

On Darboux Frames of Indicatrices of Spacelike Salkowski Curve with Spacelike Binormal in E13

The aim of this study is to examine Darboux frames and some other geometric properties (geodesic curvatures, geodesic torsions, normal curvatures, Darboux derivative formulas, Darboux vectors, angles, etc.) of the spherical indicatrices on Lorentzian unit sphere S_1^2 and hyperbolic unit sphere H_0^2 of the spacelike Salkowski curve with spacelike binormal in Lorentzian 3-space E_1^3. In this context, new and interesting results have been obtained for this curve. Thus, relationships between the newly obtained curvatures and torsions and the curvature and torsion of the original curve are given. Moreover, the matrix relationship between the Darboux and Frenet frames of these indicatrices is shown. Finally, the Darboux vectors belong to the Darboux frames and the Darboux vectors belong to the Frenet frames of these curves are compared.

On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion

<abstract><p>If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.</p></abstract>

On the pole indicatrix curve of the spacelike Salkowski curve with timelike principal normal in Lorentzian 3-space

Bu çalışmada 3-boyutlu Lorentz uzayında timelike asli normalli spacelike Salkowski eğrisinin pol gösterge eğrisinin Frenet çatısı, eğrilikleri, Frenet türev formülleri, Darboux vektörü, yay uzunluğu, e ve (Lorentzian küre) ye göre geodezik eğrilikleri hesaplanmıştır ve bu pol gösterge eğrisi Lorentzian küre üzerinde gösterilmiştir.

Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$

In this study, we examine some properties of Salkowski curves in $\mathbb{E}^{3}$. We then make sense of the angle $(nt)$ in the parametric equation of the Salkowski curves. We provide the relationship between this angle and the angle between the binormal vector and the Darboux vector of the Salkowski curves. Through this angle, we obtain the unit vector in the direction of the Darboux vector of the curve. Finally, we calculate the modified orthogonal frames with both the curvature and the torsion and give the relationships between the Frenet frame and the modified orthogonal frames of the curve.

Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$

In this present paper, rectifying curves are re-characterized in a shorter and simpler way using harmonic curvatures and some relations between rectifying curves and focal curves are found in terms of their harmonic curvature functions in $n-$dimensional Euclidean space. Then, a rectifying Salkowski curve, which is the focal curve of a given space curve is investigated. Finally, some figures related to the theory are given in the case $n=3$.

Some Smarandache Curves Constructed from a Spacelike Salkowski Curve with Timelike Principal Normal

In this article, we investigate the regular Smarandache curves constructed from the Frenet vectors of spacelike Salkowski curve with a timelike principal normal. In the first part of the study, literature research was conducted. In the second part, general information about the curve and spacelike Salkowski curve in Minkowski space are given. In the last part, the Frenet apparatus of the Smarandache curves are calculated. We draw a graphic of the obtained Smarandache curves and some related results about Smarandache curves are given.

On a surface pencil with a common new type of special surface curve in Galilean space G 3

In this study, we investigate a new type of surface curve called a D-type special curve. We also show that this special curve is more general than a geodesic curve or an asymptotic curve. Then, we give the necessary and sufficient conditions for a curve to be the new D-type special curve using Frenet frame in Galilean space. Moreover, we investigate some results by taking account of a D-type special curve as a helix, a Salkowski curve and an anti-Salkowski curve. After then, for the sake of visualizing this study, we plot some examples for this surface pencil (i.e. surface family).

Space Curves Related by a Transformation of Combescure

In this paper, the curves associated with the Combescure transform are discussed. With the help of the fact that these curves have a common Frenet frame, an equivalence relation is defined. The equivalence classes obtained by this equivalence relation have been examined for some special curves and it has been obtained that all curves in the equivalence class of a helix curve are also helix curves. This is also true for k-slant helix curves. The important part of this paper consists of the useful construction method to obtain a curve from the given curve α with the help of Combescure transformation. With this method, some special curves such as Bertrand, Mannheim, Salkowski, anti-Salkowski or spherical curve can be obtained from any curve related by a Combescure transform. For example, we obtain an example of Mannheim curves explicitly obtained from an anti-Salkowski curve. It is not easy to find an example of Mannheim curves except circular helix in the literature. In general, the condition for being Bertrand anti-Salkowski or spherical curve of the curve β obtained from the given curve α with the help of Combescure transformation were obtained.

Polynomial Parametric Equations of Rectifying Salkowski Curves

The aim of the paper is to find polynomial parametric equations of rectifying Salkowski curves in Minkowski 3-space, via a serial approach. These curves are characterized by according to their curvature; in particular those curves with constant curvature functions and linear harmonic curvature functions are fully characterized. Then, the equations of the rectifying Salkowski curves are obtained as serial solutions of differential equations with third-order polynomial coefficients.

Equiform spacelike Smarandache curves of anti-eqiform Salkowski curve according to equiform frame

Equiform spacelike Smarandache curves of anti-eqiform Salkowski curve according to equiform frame

3 Boyutlu Uzaylarda Torsiyon İle Oluşturulan Modifiye dik çatıya göre Adjoint Eğriler

The main purpose in this study is to create the adjoint curve of a curve according to the modified orthogonal frame created by torsion in Euclidean and Minkowski 3-space. Besides, it is another purpose to reveal the relationships between the modified orthogonal frame of these curves.In this paper, firstly, the adjoint curve of a space curve in 3D Euclidean space with respect to the modified orthogonal frame formed by torsion is defined and the modified orthogonal frame of the adjoint curve is given by a theorem in terms of the modified orthogonal frame of the main curve.With the help of this theorem, results about properties such as being a general helix and a Salkowski curve between main curve and adjoint curve were given. Similar approaches are discussed for non-null curves in 3D Minkowski space and the adjoint curve of a space curve is defined.

Smarandache Curves According to Sabban Frame Generated by the Spherical Indicatrix Curves of the Unit Darboux Vector of Salkowski Curve

Bu çalışmada ilk olarak Salkowski eğrisine ait birim Darboux vektörünün birim küre yüzeyi üzerinde çizdiği küresel eğrinin Sabban çatısı oluşturuldu. Daha sonra bu çatı konum vektörü olarak alınarak Smarandache eğrileri tanımlandı. Son olarak da her bir Smarandache eğrisinin geodezik eğrilikleri hesaplanarak esas eğrinin Frenet aparatlarına bağlı ifadeleri elde edildi. Maple programı ile çizimleri yapıldı. Bu çalışmada ilk olarak Salkowski eğrisine ait birim Darboux vektörünün birim küre yüzeyi üzerinde çizdiği küresel eğrinin Sabban çatısı oluşturuldu. Daha sonra bu çatı konum vektörü olarak alınarak Smarandache eğrileri tanımlandı. Son olarak da her bir Smarandache eğrisinin geodezik eğrilikleri hesaplanarak esas eğrinin Frenet aparatlarına bağlı ifadeleri elde edildi. Maple programı ile çizimleri yapıldı.

Smarandache curves according to the Sabban frame belonging to spherical indicatrix curve of the Salkowski curve

In this study, Smarandache curves are defined according to the Sabban frame by means of the spherical indicatrix curve of the Salkowski curve. Then using these curves we calculated geodesic curvatures. Finally each curve is drawn with maple program.

The Bertrand curve associated to a Salkowski curve

After a review of two methods of construction of Bertrand curves, we characterize Bertrand curves defined from Salkowski curves as the only family of Bertrand curves which are slant helices.

Smarandache Curves of Spacelike Salkowski Curve with a Spacelike Principal Normal According to Frenet Frame

Bu çalışmada ilk olarak spacelike normalli spacelike Salkowski eğrisinin Frenet vektörlerinden elde edilen regüler Smarandache eğrileri tanımlandı. Daha sonra her bir Smarandache eğrisinin Frenet vektörleri, eğrilik ve torsiyonu hesaplandı. Son olarak elde edilen eğrilerin Frenet elemanları spacelike Salkowski eğrisinin Frenet elemanları cinsinden yazılarak grafikleri çizildi.

Slant Helices that Constructed from Hyperspherical Curves in the n-dimensional Euclidean Space

In this work, we study slant helices in the n-dimensional Euclidean space. We give methods to determine the position vectors of slant helices from arclength parameterized curves that lie on the unit hypersphere. By means of these methods, first we characterize slant helices and Salkowski curves which lie on 2n-dimensional hyperboloid. After that, we characterize rectifying slant helices which are geodesics of 2n-dimensional cone.