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On the Construction of a Surface Family with Common Geodesic in Galilean Space G3

  • Zühal Küçükarslan Yüzbaşı EMAIL logo and Mehmet Bektaş
Published/Copyright: December 30, 2016
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Open Physics
From the journal Open Physics Volume 14 Issue 1

Abstract

In this paper, we investigate the parametric representation for a family of surfaces through a given geodesic curve G3. We provide necessary and sufficient conditions for this curve to be an isogeodesic curve on the parametric surfaces using Frenet frame in Galilean space. Also, for the sake of visualizing of this study, we plot an example for this surfaces family.

Keywords: Geodesic curve; Isoparametric curve; Parametric surface; Galilean space
PACS: 02.40.-k

1 Introduction

Geodesics have a very important role in surface theory and physics. One of the primary reasons why they are so important for physics is that any mass point is not acted on by any forces but is constrained to remain on a fixed surfaces moves on a geodesic line of the surface. Also geodesics play a role in the Lagrange equations of the first kind. Geodesics are so important is that they generalize (locally) the shortest path between points in the space. Geodesics have been commonly studied in Riemannian geometry, more generally metric geometry and general relativity. More precisely, a curve in a surface is said to be geodesic if its geodesic curvature is equally zero. In other words the normal vector of a curve is everywhere parallel to the normal of the surface. Geodesics are helpful in many areas, for example; computer vision, industrial applications and image processing. Moreover, surface with common geodesic is one of the most important research topics in Differential Geometry. Some more studies and results about surfaces in G3 have been given in[1-4]

There are nine related plane geometries including Euclidean geometry, hyperbolic geometry and elliptic geometry. The Galilean geometry is one of these geometries whose motions are the Galilean transformations of classical kinematics[5]. There has been a lot of studying about the Differential geometry of the Galilean space G3 in[6-9.

The purpose of this paper is to introduce the parametric representation of surface through a given isogeodesic curve in Galilean space G3. We derive the necessary and sufficient conditions for the given curve as the geodesic and isoparametric on the parametric surface. Also, we define the family of parametric surfaces with common geodesic curve in Galilean space G3. Finally, for the sake of visualizing of this study, we demonstrate an example for this family of surfaces.

We present this paper from the Galilean point of view. The results can be easily transferred to the Pseudo-Galilean geometry with minor changes.

2 Preliminaries

The Galilean space G3 is a Cayley-Klein space equipped with the projective metric of signature (0,0,+,+), given in[10]. The absolute figure of the Galilean space consists of an ordered triple{ω, f, I} in which ω is the ideal (absolute) plane, f is the line (absolute line) in ω and I is the fixed elliptic involution of points of f.

Definition 2.1

A vector x = (x1, x2, x3) is called a non-isotropic if x1 ≠ 0. All unit isotropic vectors are of the form x = (1,x2, x3). For isotropic vectors x1 = 0 holds.

Definition 2.2

The Galilean scalar product between two vectors x = (x1, x2, x3) and y = (y1, y2, y3) vectors in G3, is given by

x,y=x1y1,if x1≠0 or y1≠0x2y2+x3y3,if x1=0 and y1=0,

in[11].

Definition 2.3

Let x = (x1, x2, x3) and y = (y1, y2, y3) be vectors in G3, the cross product of the vectors x and y is defined as follows

x∧y=0e2e3x1x2x3y1y2y3=(0,x3y1−x1y3,x1y2−x2y1),

in[11].

An admissible curve r of the class Cr (r > 3) in G3, and parametrized by the invariant parameter u, is given by

ru=u,fu,gu.

For an admissible curve, the associated invariant moving trihedron satisfies the following equation

tu=r′u=1,f′u,g′u,nu=r′′uκu=1κu0,f′′u,g′′u,bu=1κu0,−g′′u,f′′u,

where t, n and b are called the vectors of the tangent, principal normal and binormal of r(u), respectively, and the curvature κ(u) and the torsion τ (u) of the curve r can be given by, respectively,

κu=f′′u2+g′′u2,τu=detr′u,r′′u,r′′′uκ2u,

Frenet formulas may be written as

t′=κn,n′=τb,b′=−τn,

in[12].

Definition 2.4

Let M is a surface in G3, the equation of a surface in G3 can be expressed as the parametrization

ϕu,υ=ϕ1u,υ,ϕ2u,υ,ϕ3u,υ, u,υ∈R,

where ϕ1(u, υ), ϕ2(u, υ) and ϕ3(u, υ) ∈ C3, in\cite{13}.

Also, the isotropic normal vector field is given by

ηu,v=ϕu×ϕv,

where ϕu and ϕv are partial differentiations with respect to u and v, respectively.

3 Surfaces with Common Geodesic Curve in Galilean Space G3

Let ϕ = ϕ (u, v) be a parametric surface on the arc-length parametrized curve r(u) in G3. The surface is defined by

ϕu,v=r(u)+[xu,vtu+yu,vnu+zu,vbu],(1)
L1≤u≤L2 and T1≤v≤T2,(2)

where x(u, v), y(u, v) and z(u, v) are C1 functions {t(u), n(u), b(u)} is the frame associated with the curve r(u) in G3.

The normal η(u, v) of the surface is given by

ηu,v=ϕu×ϕv,(3)

from (1)

ϕu=1+xut+kx+yu−τzn+τy+zub,ϕv=xvt+yvn+zvb,

Taking account (3), the normal vector η(u, v) can be expressed as

ηu,v=−1+xuzv+τy+zuxvn+1+xuyv−kx+yu−τzxvb,

Let r(u) be a curve on a surface ϕ(u, v) in G3. If r(u) is isoparametric curve on this surface, then there exists a parameter v = v0 such that r(u) = ϕ(u, v0), that is

xu,v0=yu,v0=zu,v0=0.(4)

From (4), we get

ηu,v0=−1+xuzv+zuxvn+1+xuyv−yuxvb.

According to[14], the curve r(u) on the surface ϕ(u, v) is geodesic if and only if the normal vector n(u) of the curve r(u) is everywhere parallel to the normal vector η(u, v0) of the surface ϕ(u, v). Then, n(u) ‖ η(u, v0) if and only if

−1+xuzv+zuxv≠0,1+xuyv−yuxv=0.(5)

Thus, the necessary and sufficient conditions for the surface ϕ to have the curve r(u) in G3 as an isoparametric and geodesic can be given with the following theorem.

Theorem 3.1

Let ϕ be a surface having a curve r(u) in the 3-dimensional Galilean space with parametrization (1). The curve r(u) is isogeodesic on a surface ϕ(u, v) if and only if the following conditions are satisfied:

xu,v0=yu,v0=zu,v0=0,−1+xuzv+zuxv≠0,1+xuyv−yuxv=0.

We call the set of surfaces given by (1) and satisfying (4) and (5) the family of surfaces with common isogeodesic in G3. Any surface ϕ(u, v) defined by(1) and satisfying (4) and (5) is a member of the family.

The functions x(u, v), y(u, v) and z(u, v) can be chosen in two different forms:

Case 1

If we take

xu,v=∑i=1pa1iluixvi,yu,v=∑i=1pa2imuiyvi,zu,v=∑i=1pa3inuizvi,(6)

then, the sufficient condition for which the curve r(u) is an isogeodesic curve on the surface ϕ(u, v) can be given as

xv0=yv0=zv0=0,a21=0 or mu=0 or dyv0dv=0 ,a31≠0 and nu≠0,and dzv0dv≠0,(7)

where l(u), m(u), n(u), x(v), y(v) and z(v) are C1 functions, aij∈R (i = 1, 2, 3;j = 1, 2,…,p) and l(u), m(u) and n(u) are not identically zero.

For the case when the functions x(u, v), y(u, v) and z(u, v) depend only on the parameter v, the family of surfaces with common geodesic becomes

ϕu,v=r(u)+xvtu+yvnu+zvbu.
Case 2

If we take

xu,v=f(luxv),yu,v=g(muyv),zu,v=h(nuzv),(8)

then, the sufficient condition for which the curve r(u) is an isogeodesic curve on the surface ϕ(u, v) can be expressed as

xv0=yv0=zv0=0 and f0=g0=h0=0,mu=0 or g′0=0 or dyv0dv=0,nu≠0 and h′0≠0 and dzv0dv≠0,

where l(u), m(u), n(u), x(v), y(v), z(v), f, g and h are C1 functions and l(u), m(u) and n(u) are not identically zero.

So, we get the functions in (6) and (8) which are general for expressing surfaces with a given curve as an isogeodesic curve in G3. Also, different types of these functions can be chosen according to Theorem 3.1.

Example 3.1

Let r be a parametrized by

ru=u,sin⁡u,cos⁡u.

It is easy to calculate that

t=1,cos⁡u,−sin⁡u,n=0,−sin⁡u,−cos⁡u,b=0,cos⁡u,−sin⁡u,

where κ = 1 is the curvature and τ = 1 is the torsion of the curve in G3.

Then, we obtain the surfaces family with the common isogeodesic. If we take

xu,v=0 ,yu,v=1−cos⁡v and zu,v=sin⁡v,

and v0 = 0 such that Equation (7) is satisfied. Thus, a member of this family is obtained by

ϕu,v=[u,sin⁡u−(1−cos⁡v)sin⁡u+sin⁡vcos⁡u,cos⁡u−(1−cos⁡v)cos⁡u−sin⁡vsin⁡u].
Figure 1 The representation of the curve and a member of surfaces.
Figure 1

The representation of the curve and a member of surfaces.

4 Conclusions

We showed the parametric representation for a family of surfaces through a given geodesic curve G3. We gave a theorem related to the curve r(u) is isogeodesic on a surface ϕ(u, v). Consequently, an example for this surfaces family was plotted.

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Received: 2016-1-4
Accepted: 2016-7-21
Published Online: 2016-12-30
Published in Print: 2016-1-1

© 2016 Z. K. Yüzbaşı and M. Bektaş, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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Keywords for this article
Geodesic curve; Isoparametric curve; Parametric surface; Galilean space
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  83. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  84. Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
  85. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  86. On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
  87. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  88. On soliton solutions of the Wu-Zhang system
  89. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  90. Comparison between the (G’/G) - expansion method and the modified extended tanh method
  91. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  92. On the union of graded prime ideals
  93. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  94. Oscillation criteria for nonlinear fractional differential equation with damping term
  95. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  96. A new method for computing the reliability of consecutive k-out-of-n:F systems
  97. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  98. A time-delay equation: well-posedness to optimal control
  99. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  100. Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
  101. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  102. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
  103. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  104. The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
  105. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  106. Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
  107. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  108. A certain sequence of functions involving the Aleph function
  109. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  110. On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
  111. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  112. Solitary and compacton solutions of fractional KdV-like equations
  113. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  114. Regarding on the exact solutions for the nonlinear fractional differential equations
  115. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  116. Non-local Integrals and Derivatives on Fractal Sets with Applications
  117. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  118. On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
  119. Special issue on Information Technology and Computational Physics
  120. On uninorms and nullnorms on direct product of bounded lattices
  121. Special issue on Information Technology and Computational Physics
  122. Phase-space description of the coherent state dynamics in a small one-dimensional system
  123. Special issue on Information Technology and Computational Physics
  124. Automated Program Design – an Example Solving a Weather Forecasting Problem
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  126. Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
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  128. Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
  129. Special issue on Information Technology and Computational Physics
  130. Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
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  132. Intelligent Chatter Bot for Regulation Search
  133. Special issue on Information Technology and Computational Physics
  134. Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
  135. Special issue on Information Technology and Computational Physics
  136. Resource management for server virtualization under the limitations of recovery time objective
  137. Special issue on Information Technology and Computational Physics
  138. MODY – calculation of ordered structures by symmetry-adapted functions
  139. Special issue on Information Technology and Computational Physics
  140. Survey of Object-Based Data Reduction Techniques in Observational Astronomy
  141. Special issue on Information Technology and Computational Physics
  142. Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
  143. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  144. Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
  145. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  146. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
  147. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  148. Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
  149. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  150. Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
  151. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  152. Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
  153. Special Issue on Research Frontier on Molecular Reaction Dynamics
  154. Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
  155. Special Issue on Research Frontier on Molecular Reaction Dynamics
  156. Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
  157. Special Issue on Research Frontier on Molecular Reaction Dynamics
  158. The Interaction between Graphene and Oxygen Atom
  159. Special Issue on Research Frontier on Molecular Reaction Dynamics
  160. Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
  161. Special Issue: Functional Advanced and Nanomaterials
  162. Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
  163. Special Issue: Functional Advanced and Nanomaterials
  164. Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
  165. Special Issue: Functional Advanced and Nanomaterials
  166. Quasi-planar elemental clusters in pair interactions approximation
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