Galilean Space Research Articles

Involute-evolute curves with modified orthogonal frame in Galilean space G 3

UDC 514 We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space G 3 . The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

Quasi-position vector curves in Galilean 4-space

The Frenet frame is not suitable for describing the behavior of the curve in the Galilean space since it is not defined everywhere. In this study, an alternative frame, the so-called quasi-frame, is investigated in Galilean 4-space. Furthermore, the quasi-formulas in Galilean 4-space are deduced and quasi-curvatures are obtained in terms of the quasi-frame and its derivatives. Quasi-rectifying, quasi-normal, and quasi-osculating curves are studied in Galilean 4-space. We prove that there is no quasi-normal and accordingly normal curve in Galilean 4-space.

Surface family mate with Bertrand mate as mutual curvature lines in Galilean 3-space G3

Surface family mate with Bertrand mate as mutual curvature lines in Galilean 3-space G3

Surfaces with a common asymptotic curve in the 3D Galilean space 𝔾3

Surfaces with a common asymptotic curve in the 3D Galilean space 𝔾₃

Investigate and Designing Translation surface with a 2-Bishop frame in the Galilean space.

Investigate and Designing Translation surface with a 2-Bishop frame in the Galilean space.

Development and Isometry of Surfaces Galilean Space G3

Development and Isometry of Surfaces Galilean Space G3

Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space G3. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreover, the necessary and sufficient condition for the given Bertrand couple to satisfy the principal curves and the geodesic requirements are also analyzed. As implementations of our main consequences, we expound upon some models to confirm the method.

Equiform rectifying curves in Galilean space [formula omitted

This research paper presents the equiform rectifying curves in Galilean space G4, and establish the relation between equiform curves and their equiform curvature functions. The proposed methodology involves studying the necessary and sufficient conditions for the curve α(σ) with non zero curvatures K1(σ),K2(σ), and K3(σ) to be congruent to an equiform rectifying curve. By employing this approach we derive various characterizations of these curves, also deduce that there are no equiform rectifying curves in G4 with zero constant equiform curvature functions K2(σ), and K3(σ). The findings contribute to a deeper understanding of the geometric properties and behaviour of equiform rectifying curves in Galilean space G4. Thereby offering potential applications in fields such as mathematical physics and differential geometry.

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.

A Surface Family with a Mutual Geodesic Curve in Galilean 3-Space G3

This article gives an approach for establishing a surface family with a mutual geodesic curve in Galilean 3-space G3. Given a smooth space curve, we derive the sufficient and necessary condition for the given curve to be geodesic on it. Furthermore, we resolve the conditions when the surface is a ruled surface. Consequently, its ability to develop with the mutual geodesic of the elements of the surface family is researched. Meanwhile, we explain this approach by submitting considerable examples.

Surface Family Pair with Bertrand Pair as Common Geodesic Curves in Galilean 3-Space 𝔾3

This paper is about deriving the necessary and sufficient conditions of a surface family pair with a Bertrand pair as common geodesic curves in Galilean 3-space G3. Thereafter, the consequence for the ruled surface family pair is also deduced. Meanwhile, some examples are provided to show the surfaces family with common Bertrand geodesic curves.

The Physical Concepts on Special Tube Surfaces Generated by Normal Curves in Galilean 3-Space

In this study, we examine the tube surfaces formed by normal curves in Galilean 3-space, and we give Clairaut’s theorem on the tube surfaces using geodesic normal curves. Also, we attempted to explain why the specific kinetic energy and angular momentum of particles may be on tube surfaces

Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

In this paper, we define the curve at a constant distance from the edge of regression on a curve r(s) with arc length parameter s in Galilean 3-space. Here, d is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve r(s) in 3-dimensional Galilean space. We build the Frenet frame of the constructed curve with respect to two types of the vector d and we indicate the properties related to the curvatures of the curve . Also, for the curve , we give the conditions to be a circular helix. Furthermore, we discuss ruled surfaces of type A generated via the curve and the vector D which is defined as tangent of the curve in 3-dimensional Galilean space. The constructed ruled surfaces also appear in two ways. The first is constructed with the curve and the non-isotropic vector D. The second is formed by the curve and the non-isotropic vector D. We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable. Finally, we provide examples and visuals to back up our research.

On dual surfaces in Galilean 3-space

<abstract><p>In the present paper, we describe dual translation surfaces in the Galilean 3-space having the constant Gaussian and mean curvatures as well as Weingarten and linear Weingarten dual translation surfaces. We also study dual translation surfaces in $ \mathbb{G}_{3} $ under the condition $ \Delta ^{II}r = \lambda _{i}r_{i} $, where $ \lambda _{i}\in R $ and $ \Delta ^{II} $ denotes the Laplacian operator respect to the second fundamental form.</p></abstract>

3-Boyutlu Galilean Uzayında Tüp Yüzeyler Üzerine Bir Araştırma

In this study, the tube surfaces generated by the curve defined in Galilean 3-space are examined and some certain results of describing the geodesics on the surfaces are also given. Furthermore, the conditions of being geodesic on the tubular surface are obtained with the help of Clairaut’s theorem, which allows us to constitute the specific energy. The physical meaning of the specific energy and the angular momentum is of course related with the physical meaning itself. Our results show that the specific energy and the angular momentum obtained on tubular surfaces can be expressed using arbitrary geodesic curve in Galilean space. In addition, some characterizations are given for these surfaces, with the obtained mean and Gaussian curvatures.

On the Geometry of Equiform Normal Curves in the Galilean Space G4

On the Geometry of Equiform Normal Curves in the Galilean Space G4

A mathematical interpretation on special tube surfaces in Galilean 3-space

In this paper, we study the special tube surfaces generated by rectifying curves with respect to the Darboux frame in terms of the geodesic curvature, the normal curvature and the geodesic torsion in Galilean 3-space. During this study we establish some definite results of geodesics on specific tube surfaces with the help of Clairaut’s theorem in detail and we compute the Gaussian curvature and the mean curvature of the special tube surfaces with respect to the Darboux frame. After that, considering the geodesic conditions and the curvatures of the special tube surface, we give some theorems for the rectifying curves with $v$-parameter (and $w$-parameter) being a geodesic curve and an asymptotic curve, respectively.

THA-surfaces of finite type in the Galilean space 𝔾3

Abstract In this article, we study THA-surfaces in the Galilean space 𝔾3, which has finite type immersion.

CHARACTERIZATIONS OF NORMAL AND BINORMAL SURFACES IN G3

In this paper, our aim is to give surfaces in the Galilean 3-space G3 with the property that there exist four geodesics through each point such that every surface built with the normal lines and the binormal lines along these geodesics is a surface with a minimal surface and a constant negative Gaussian curvature. We show that should be an isoparametric surface in G3: A plane or a circular hyperboloid.

Hypersurfaces with a common asymptotic curve in the 4D Galilean space 𝔾4

In this study, we investigate how to construct hypersurfaces using an asymptotic curve in the 4D Galilean space [Formula: see text]. Then, by utilizing the Serret–Frenet frame, we give the conditions that a curve becomes asymptotic on a hypersurface when the marching-scale functions are three types of functions. After then, an example connected to our method is given for the aim of visibility.