The Equivalence Problem Of Dual Parametric Curves

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 operationin the algebra D . For an element *1 A  a  a D and a transformation 2 2 S :  where S A Sa a    , we define the sets *1 *0, 0, , AaID S a a aa a              and*1 *0 1 0, 0, ,0 1aID a a aa a              . Let us denote 1 1 1 ID ID ID  . Moreover, we denote the setℳ 1ID  ℳ 1ID  ℳ 1ID whereℳ   2 2 21 1 : , ( ) , , , A ID F F B S B C A D B C        andℳ 2 2 21 11 0: , ( ) ( ) , , , ,0 1 A ID F F B S W B C A D B C W               . Let ( , ) T a b  be an openinterval 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 - equivalentif the equality ( ) ( ) t Ft  is satisfied for an element FG and all tT . Then, it is denoted by ( ) ( )GttThis work is devoted to the solutions of problems of G-equivalence of parametric curves in Euclidean space2 for the groups G  ℳ 1ID  , ℳ 1ID .

Özet

reel sayılar cismi ve   * * * 2 ( , ) , , , 0 D a a a a a a        dual cebir olsun. D nin