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The Equivalence Problem Of Dual Parametric Curves

Year 2020, Volume: 13 Issue: ÖZEL SAYI I, 18 - 32, 28.02.2020
https://doi.org/10.18185/erzifbed.598364

Abstract

Let be the field of real numbers and   * * * 2 D  (a,a )  a  a ,a,a  ,  0 be the algebra of dual numbers.

The subset   * *

1 ( , ), 0, , D a a a a a    of D is an abelian group with respect to the multiplication operation

in the algebra D . For an element *

1 A  a  a D and a transformation 2 2 S :  where S A S

a a

 

  

 

, we define the sets *

1 *

0

, 0, , A

a

ID S a a a

a a

    

       

   

and

*

1 *

0 1 0

, 0, ,

0 1

a

ID a a a

a a

    

      

    

. Let us denote 1 1 1 ID ID ID  . Moreover, we denote the set

ℳ 1ID  ℳ 1ID  ℳ 1ID where

ℳ   2 2 2

1 1 : , ( ) , , , A ID F F B S B C A D B C        and

ℳ 2 2 2

1 1

1 0

: , ( ) ( ) , , , ,

0 1 A ID F F B S W B C A D B C W   

         

   

. Let ( , ) T a b  be an open

interval of . A (2) C -function 2 :T for tT where, ( ) ( ( ), ( )) t x t y t  is called a parametrized curve

(path) on the plane. Let G be a group. Two parametric curves (paths) ()t  and ()t  are called G - equivalent

if the equality ( ) ( ) t Ft  is satisfied for an element FG and all tT . Then, it is denoted by ( ) ( )

G

tt

This work is devoted to the solutions of problems of G-equivalence of parametric curves in Euclidean space

2 for the groups G  ℳ 1ID  , ℳ 1ID .

References

  • [1] Khadijev D., Oren I., Peksen O. 2018. ‘Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space’, International Journal of Geometric Methods in Modern Physics Vol. 15.
  • [2] Khadjiev D.,1988, ‘’Application of the Invariant Theory to the Differential Geometry of Curves’’ Fan Publisher, Tashkent.
  • [3] Klein, F. 1872. “Vergleichende Betrachtungen Über Neuere Geometrische Forschungen”,Erlangen: Verlag.
  • [4] TOMAR, M. 2012. ‘Applications of dual numbers and dual numbers to two-dimensional dual geometry’, Master’s Thesis, Science Institute, Trabzon.

Dual Parametrik Eğrilerin Denklik Problemi

Year 2020, Volume: 13 Issue: ÖZEL SAYI I, 18 - 32, 28.02.2020
https://doi.org/10.18185/erzifbed.598364

Abstract

reel sayılar cismi ve   * * * 2 ( , ) , , , 0 D a a a a a a        dual cebir olsun. D nin
  * *
1 D  (a,a ),a  0,a,a  alt kümesi çarpma işlemine göre değişmeli bir grup oluşturur. Bir
*
1 A a a D     elemanı ve 2 2 : S  dönüşümü için *
0
( ) A
a
S A S
a a
 
  
 
olmak üzere;
*
1 *
0
, 0, , A
a
ID S a a a
a a
    
       
   
ve *
1 *
0 1 0
, 0, ,
0 1
a
ID a a a
a a
    
      
    
kümelerini
tanımlayalım. 1 1 1 ID ID ID     olsun. Ayrıca;
ℳ   2 2 2
1 1 : , ( ) , , , A ID F F B S B C A D B C        ve
ℳ 2 2 2
1 1
1 0
: , ( ) ( ) , , , ,
0 1 A ID F F B S W B C A D B C W   
         
   
olmak üzere;
ℳ 1 ID  ℳ 1 ID  ℳ 1ID  şeklinde tanımlayalım. T  (a,b) ’de bir açık aralık olsun. Bir 2 :T   , tT
için (t)  (x(t), y(t)) şeklindeki (2) C -fonksiyonuna düzlemde bir parametrik eğri (yol) denir. G bir grup
olsun. tT ve bir FG için ( ) ( ) t F t    eşitliği sağlanıyorsa ()t  ve ()t  iki parametrik eğriye (yollara)
G -denk eğriler denir. ( ) ( )
G
 t  t ile gösterilir. Bu çalışma 2 Öklid uzayındaki parametrik eğriler (yollar) için
G  ℳ 1ID  , ℳ 1ID gruplarına göre G -denklik probleminin çözümünü bulmaya yönelik bir çalışmadır.

References

  • [1] Khadijev D., Oren I., Peksen O. 2018. ‘Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space’, International Journal of Geometric Methods in Modern Physics Vol. 15.
  • [2] Khadjiev D.,1988, ‘’Application of the Invariant Theory to the Differential Geometry of Curves’’ Fan Publisher, Tashkent.
  • [3] Klein, F. 1872. “Vergleichende Betrachtungen Über Neuere Geometrische Forschungen”,Erlangen: Verlag.
  • [4] TOMAR, M. 2012. ‘Applications of dual numbers and dual numbers to two-dimensional dual geometry’, Master’s Thesis, Science Institute, Trabzon.
There are 4 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Makaleler
Authors

Nurcan Demircan Bekar 0000-0001-5670-8655

Publication Date February 28, 2020
Published in Issue Year 2020 Volume: 13 Issue: ÖZEL SAYI I

Cite

APA Demircan Bekar, N. (2020). Dual Parametrik Eğrilerin Denklik Problemi. Erzincan University Journal of Science and Technology, 13(ÖZEL SAYI I), 18-32. https://doi.org/10.18185/erzifbed.598364