TBILISI - MATHEMATICS (eBook)

Tbilisi Mathematical Journal Collection of Special Issues, 1

Inassaridze Hvedri (Herausgeber)

eBook Download: EPUB

2020
145 Seiten
De Gruyter Poland (Verlag)
978-83-957938-9-9 (ISBN)

Infos

The Special Issues of Tbilisi Mathematical Journal are fully refereed international publications, publishing original research papers in all areas of pure and applied mathematics. The editors are well known experts in the field, particularly from leading universities of USA and Europe. Papers should satisfy high standards and only works of high quality are recommended for publication. They constitute a collection around selected themes related to mathematical sciences, or coming from a specific group of mathematicians or event, or coming from a workshop, symposia and international mathematical conferences.

ABSTRACTING & INDEXING

TBILISI - MATHEMATICS is covered by the following services:

Baidu Scholar
CNKI Scholar (China National Knowledge Infrastructure)
Dimensions
ExLibris
Google Books
Google Scholar
Naviga
ReadCube
Semantic Scholar
TDOne (TDNet)
WorldCat (OCLC)
X-MOL

P>Hvedri Inassaridze, Tbilisi State University, Georgia

TBILISI - MATHEMATICS

Tbilisi Mathematical Journal

Special Issues, 1

Editor Hvedri Inassaridze

DOI 10.2478/9788395793882-fm

Special issue on 8th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2019), 27 – 30 August 2019, Baku, Azerbaijan

Lead Guest Editor: Murat Tosun (Sakarya University, Turkey) Guest Editors: Cristina Flaut (Ovidius University, Romania), Etibar S. Penahov (Baku State University, Azerbaijan), H. Hidayet Kosal (Sakarya University, Turkey),

Lyudmila Romakina (Saratov State University),

Mehmet Ali Gungor (Sakarya University, Turkey),

Soley Ersoy (Sakarya University, Turkey),

Wolfgang Sproessig (Technical University Freiburg, Germany).

DOI 10.2478/9788395793882-001

New versions of q-surface pencil in Euclidean 3-space Aziz Yazla1 and Muhammed T. Sariaydin2

1,2Selcuk University, Faculty of Science, Department of Mathematics, 42130, Konya, Turkey E-mail: [e-mail]1, [e-mail]2

Abstract

In this paper, the q-surface pencil is studied in Euclidean 3-space. By using q-frame in Euclidean space, Firstly, we define q-surface pencil. Then, we give the necessary and sufficient condition for a curve to be a geodesic curve and to be an asymptotic curve on a q-surface pencil. Then, we study this subject for an offset q-surface pencil.

2010 Mathematics Subject Classification. 53A05 53A04

Keywords. Geodesic Curve, Asymptotic Curve, q-Surface Pencil, Offset Surface.

1

Introduction

Geodesic on a surface corresponds to the shortest path between any two points on the surface.

When one flattens a developable surface into a planar shape without distortion, any geodesic on the surface will be mapped to a straight line in the planar shape. Therefore, a good algorithm which is needed to flatten a non-developable surface with as little distortion as possible must preserve the geodesic curvatures on the surface, [16].

A geodesic on a surface is a parametric curve on the surface such that the acceleration vector on every point of the curve is orthogonal to the surface. It is well-known that the tangent vector field of a parametric curve on a surface is parallel in Levi-Civita sense if and only if this curve is a geodesic on the surface. A geodesic on a surface can also be defined as a curve on the surface with zero geodesic curvature, [10]. Additionally, there are many works related to surface pencil [1-3,8].

Geodesic curves play an important role in many industrial applications, such as tent manufacturing and textile manufacturing. Most existing work on geodesics is finding geodesics on a given surface. However, the reverse problem that is how to characterize the surfaces that possess a given curve as a common geodesic is also studied. Shoe design can be given as an engineering application of this problem. The characteristic curve of a woman shoe which is called the girth is wanted to be a geodesic on the shoe surface, [16].

One important concept in differential geometry of curves and surfaces is also asymptotic curves.

An asymptotic curve on a surface is a curve such that its tangent vector field is an asymptotic direction. Along an asymptotic direction, the surface never makes twists from its tangent plane.

Furthermore, the Gaussian curvature of the surface is never positive along an asymptotic direction.

Moreover, the acceleration vector of the asymptotic curve on the surface is tangent to the surface,

[4]. Additionally, there are many works related to this surfaces [11-14].

In this paper, we study q-surface pencil and its offset. Given a space curve with the q-frame, we obtain the conditions for this curve to be a common geodesic and to be a common asymptotic curve on a q-surface pencil. Then we give answers to the same problem for the offset q-surface pencil.

Tbilisi Mathematical Journal Special Issue (IECMSA-2019), pp. 1–10.

Tbilisi Centre for Mathematical Sciences.

Received by the editors: 20 September 2019.

Accepted for publication: 27 December 2019.

2

Aziz Yazla, Muhammed T. Sariaydin

2

Preliminaries

In this section, we present the Frenet frame and the q-frame along a space curve. Also, we give some geometric properties for these frames.

Let α(s) be a space curve parameterized with arc-length in 3

R . The Frenet frame of α(s) is

{T(s), N(s), B(s)}, where the vector fields are given as T(s)

=

α0(s),

α00(s)

N(s)

=

,

kα00(s)k

B(s)

=

T(s)×N(s).

Here, T(s) is called the tangent vector field, N(s) is called the principal normal vector field and B(s) is called the binormal vector field of the curve α(s).

The curvature κ(s) and the torsion τ (s) of the curve α(s) are given by κ(s)

=

kα00(s)k ,

det(α0(s), α00(s), α000(s))

τ (s)

=

.

kα00(s)k2

The q-frame of a space curve α(s) which is parameterized with arc-length is tq(s), n (s), b (s) , q

q

where the vector fields are given as

tq(s)

=

T(s),

−

→

T(s) × k

nq(s)

=

,

−

→

T(s) × k

bq(s)

=

T(s) × nq(s).

−

→

−

→

−

→

Here, k is the projection vector which can be chosen as k = (1, 0, 0) or k = (0, 1, 0) or

−

→

−

→

k = (0, 0, 1). In this paper, we choose the projection vector k = (0, 0, 1). nq(s) and bq(s) are called the quasi normal vector field and the quasi binormal vector field of the curve α(s), respectively, [5].

Let θ(s) be the angle between the principal normal vector field N(s) and the quasi normal vector field nq(s). The quasi formulas are given by



t





0

k

 

t



d

q (s)

1(s)

k2(s)

q (s)

n

=

−k

n

,



q (s) 



1(s)

0

k3(s)  

q (s) 

ds

bq(s)

−k2(s) −k3(s)

0

bq(s)

where ki(s) are called the quasi curvatures (1 ≤ i ≤ 3) which are given by k1(s)

=

κ(s)cosθ(s) = t0 (s), n

q

q (s) ,

k2(s)

=

−κ(s)sinθ(s) = t0 (s), b

q

q (s) ,

k3(s)

=

θ0(s) + τ (s)

= − nq(s), b0 (s) .

q

New versions of q-surface pencil

3

The relationship between the Frenet frame and the q-frame is given by, [6]. Additionally, there are many works related to q-frame, [7,15].



T(s) 



1

0

0

 

t



q (s)

N(s)

=

0

cos θ(s)

− sin θ(s)

n

.







 

q (s) 

B(s)

0

sin θ(s)

cos θ(s)

bq(s)

3

q-Surface pencil with a common geodesic curve

We can define q-surface pencil similar to the definition of surface pencil in [9], as follows: Definition 3.1. q-surface pencil possessing α(s) as a common curve is given by S(s, t) = α(s) + x1(s, t)tq(s) + x2(s, t)nq(s) + x3(s, t)bq(s), (3.1)

0 ≤ s ≤ L, 0 ≤ t ≤ T, where x1(s, t), x2(s, t) and x3(s, t) are C2 functions. The curve α(s) is a common parametric curve on the q-surface pencil S(s, t) if x1(s, t0) = x2(s, t0) = x3(s, t0) = 0.

Theorem 3.2. Let α(s) be a parametric curve on the q-surface pencil S(s, t) which is given as (3.1). Then, α(s) is a geodesic curve on the surface S(s, t) if and only if µ1(s, t0) = 0, µ2(s, t0) = k cos θ(s), µ3(s, t0) = −k sin θ(s), (3.2)

where m(s, t0) = µ1(s, t0)tq(s) + µ2(s, t0)nq(s) + µ3(s, t0)bq(s) is the normal vector field of S(s, t) along the curve α(s), θ(s) is the angle between the principal normal vector field N(s) and the quasi normal vector field nq(s) of the curve α(s) and k ∈ R, k 6= 0.

Proof. The normal vector field of S(s, t) is

∂S

∂S

m(s, t)

=

(s, t) ×

(s, t)

∂s

∂t

=

µ1(s, t)tq(s) + µ2(s, t)nq(s) + µ3(s, t)bq(s),

where

∂x3

∂x2

µ1(s, t)

=

(

(s, t)k1(s) −

(s, t)k2(s))x1(s, t)

∂t

∂t

∂x

∂x

−

3

2

(

(s, t)x3(s, t) +

(s, t)x2(s, t))k3(s)

∂t

∂t

∂x3

∂x2

∂x2

∂x3

+

(s, t)

(s, t) −

(s, t)

(s, t),

∂t

∂s

∂t

∂s

∂x1

∂x3

µ2(s, t)

=

(

(s, t)k3(s) +

(s, t)k1(s))x2(s, t)

∂t

∂t

∂x3

∂x1

+(

(s, t)x3(s, t) +

(s, t)x1(s, t))k2(s)

∂t

∂t

∂x1

∂x3

∂x3

∂x1

∂x3

+

(s, t)

(s, t) −

(s, t)

(s, t) −

(s, t),

∂t

∂s

∂t

∂s

∂t

4

Aziz Yazla, Muhammed T. Sariaydin

∂x1

∂x2

µ3(s, t)

=

(

(s, t)k3(s) −

(s, t)k2(s))x3(s, t)

∂t

∂t

∂x

∂x

−

2

1

(

(s, t)x2(s, t) +

(s, t)x1(s,...

• Erscheint lt. Verlag: 16.7.2020
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• ISBN-10: 83-957938-9-8 / 8395793898
• ISBN-13: 978-83-957938-9-9 / 9788395793899