The E. Study Mapping for Directed Lines in 3-Space1 Mina Rashidi¹, Mehdi Shahsavari² and Mehdi Jafari3*

The E. Study Mapping For Directed Lines in 3-Space1 Mina Rashidi¹, Mehdi Shahsavari² and Mehdi Jafari3*

1,2. Department of Mathematics, Islamic Azad university, Miyaneh branch, Miyaneh, Iran 3. Payame Noor university, Iran

*Corresponding author email: [email protected]

ABSTRACT: E. Study(Geometrie der Dynamen, Leibzig, 1903) showed that there exists a one-to-one 2 3 correspondence between the vectors of dual unit sphereSD and the directed lines of space of lines E. 3 In this paper, we introduce a generalized inner product and so obtain a new 3-space E. Furthermore, the generalized E. Study mapping is defined for the lines in this space. We show that this map corresponds with a dual orthogonal matrix. AMS Math. Sub. Class.: 47B37. Key Word: Generalized inner product, Generalized Study map

INTRODUCTION

Dual numbers are useful for analytical treatment in kinematics and dynamics of spatial mechanisms (Gu and 2 Luh, 1987). Clifford(1873) introduced dual numbers in the form of x  y with   0 to form biquaternions for studying the non-Euclidean geometry. Eduard Study(1903) defined dual numbers as dual angles to specify the relation between two lines in the Euclidean space. Also, he devoted special attention to the representation of directed line by dual unit vectors and defined the mapping which is called with his name; he proved that there 2 exists a one-to-one correspondence between the points of the dual unit sphere SD and the directed lines of 3 Euclidean 3-space E (E.Study's mapping). Subsequently, in (Uğurlu and Çalişkan, 1996 ) by taking the 3 3 Minkowski 3-space E1 instead of E, Uğurlu and Çalışkan gave a correspondence of E.Study mapping as follows: There exists a one-to-one correspondence between the dual spacelike unit vectors of dual Lorentzian unit sphere 2 3 3 S1 in the dual Lorentzian space D1 and the directed spacelike lines of the Minkowski 3-space E.1 In this paper, we consider a generalized inner product on a real 3-dimensional vector space while we ignore the fact that, up to a constant factor and up to a (diagonal) change of basis, this is the Euclidean bilinear form or 3 3 the Minkowski bilinear form, or a degenerate bilinear form. Furthermore, we take a 3-space E instead of E, and we generalize E.Study map in this space. Hence, there is a one-to-one correspondence between directed 3       lines of E and ordered pair of vectors (,)a a such that a, a  1and a, a  0.

Preliminaries In this section, we define a new inner product and give a brief summary of the dual numbers and a new dual vector space.   3 Definition 1. For the vectors x (,,) x1 x 2 x 3 and y ( y1 , y 2 , y 3 ), the generalized inner product on R is given by g(,), x y x1 y 1   x 2 y 2   x 3 y 3

1 This work has been supported by a grant from the Islamic Azad University, Miyaneh Branch with number: 52183910828005 Intl. Res. J. Appl. Basic. Sci. Vol., 5 (11), 1374-1379, 2013

  where  and  are positive numbers. If   0 and   0 then g(,) x y is called the generalized 3 Lorentzian inner product. The vector space on R equipped with the generalized inner product is called 3- 3 3 dimensional generalized space and denoted by E . The cross product in E is defined by      xy ()()(). xy2332  xyi   xy 3113  xyj  xy 1221  xyk Special Cases: 3 3 1. If    1, then E is an Euclidean 3-space E. 3 3 2. If 1,    1 , then E is a Minkowski 3-space E.1   3   Definition 2.Let u (,,) u u u and v (,,) v v v be in R. If ,R,   the generalized inner product of u and  1 2 3 1 2 3 v is defined by   g( u , v ) u1 v 1   u 2 v 2   u 3 v 3 , (1) It could be written  0 0    T   g( u , v ) u 0 0  v . 0 0     3 Also, if 0,   0, g(,) u v is called the generalized Lorentzian inner product. The vector space on R 3 equipped with the generalized inner product, is called 3-dimensional generalized space, and is denoted by E. 3 The vector product in E is defined by    i  j k   u v  u1 u 2 u 3

v1 v 2 v 3    ()()(),uv  uvi   uvuvj    uv  uvk  2332    3113    1221 where i j  k,, j  k  i k  i   j . Special cases: 3 3 1. If    1, then E is an Euclidean 3-space E. 3 3 2. If 1,    1, then E is a Minkowski 3-space E.1

Proposition 1. For ,,   the inner and vector products satisfy the following properties;     1.u v   v  u .          2.(guvw ,)  gvwu (  ,)  guwv (  ,).          3.u ( vw  )  guwv (  )  guvw ( , ) . 3 An oriented line L in the three-dimensional Euclidean space E can be determined by a point p  L   and a normalized direction vector a of L , i.e. a  1. To obtain components for L , one forms the moment vector    3    a p  a with respect to the origin point in E . If p is substituted by any point q p  a ,(R)   on L    then the above equation implies that a is independent of p on L. The two vectors a and a are unindependent, they satisfy the following relationships;     a, a  1, a, a  0.

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   The six components ai , ai i 1,2,3 of a and a are called the normalized Plücker coordinates of the line L.   Hence, the two vectors a and a determine the oriented line L.  Conversely, any six-tuple ai , ai i 1,2,3 with relations 2 2 2 * * * a1 a 2  a 3 1, a 1 a 1  a 2 a 2  a 3 a 3  0, , represents a line in the three-dimensional space E³ (Abdel-Baky, 2005).

Definition 3. Each element of the set D aˆ  a  a: a , a   and   0}, is called a dual number. Summation and multiplication of two dual numbers are defined similar to the complex 2 numbers with the condition   0 . D is a commutative group under addition, the associative law holds for multiplication, it contains a unit element with respect to multiplication, and it is distributive. Division, however, is not  defined for a pure dual number (i.e. aˆ   a ) in D. A complete discussion of these and other properties can be found in (Gu and Luh, 1987). A example of dual number is the dual angle between two skew lines in three- 3 * dimensional Euclidean space  defined as     where  is the projected angle between the lines and *  is the minimal distance between the lines long their common perpendicular line. 3 The set of dual numbers can be extended to vector spaces. If we have a vector space V in E we can define 3 3   the space D as the set of all pairs of vectors which formV. The elements of D are written as a  a   a where   3 a,. a   The operations of addition, multiplication by scalar (where the scalar is in D ), the scalar product, and 2 the vector product are all defined in the usual way remembering that   0. Let us give a generalized inner 3 product in D .     3 Definition 4.Let a  a   a and b  b   b be in D and ,   R. The generalized inner product of a and b is defined by      gab(,)   gab (,)  ((,) gab  gab (,)).  (2) 3 3   3 We put D ( D , g (  ,  )). The norm of dual vector a  a  a  D is a dual number given by    g(,) a a    a g(,), a a  a    2 a

  3 where a  0. For ,,    the set of all unit dual vector in D is said the dual elliptical unit sphere and is 2 3 denoted by ED. The vector product in D is      a  b  a  b  () a  b  a   b 3  Lemma 1.Let u, v   D and ,.    In this case, we have u v   u . v  sin .s  * where    is the dual angle subtended by the two axes, and s is the unit dual vector which is orthogonal to both u and v. 3 Theorem 1.(E.Study): The oriented lines in E are in one-to-one correspondence with the points of the dual unit 3 sphere in D (Guggenheimer, 1963).

Generalized E.Study Map

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In this section, we generalize the theorem which is attributed to Eduard Study (Study, 1903). 3 Theorem 2. The set of all oriented lines in E is in one-to-one correspondence with the set of the points of the 3 dual unit sphere in the dual 3-space D . 3      Proof: First let ,   0. In E, a directed line can be given by y x   a , where x and a are the    position vector and the direction vector of the line, respectively. The moment vector a x  a is not depending   on the chosen point on the line. For this reason, by the help of ordered pair of vectors ( a, a ), a directed line is 3 determined by one unique point in D and the following conditions are satisfied:     g( a , a ) 1, g ( a , a )  0.   3  Suppose that a  a   a is a dual unit vector in D . Taking b a in equation (2) we obtain:     g(,) a a   g (,)2((, a a  g a a ))1    where the dual unit vector a represents the unique directed line ( a, a ). Note that this theorem holds for   0 and   0.

Special cases 3 1) If   1, then we get E. Study mapping in Euclidean 3-space E . 2) If 1,    1, then we will have E.Study mapping for a space which is isomorphic to Minkowski 3-space 3 E1 (Uğurlu, Çalişkan,1996). 3 3 Example: In D , the unit dual vector corresponding to the line L=x y , z  2 E is

1  ⊂ u ((1,1,0)  2 (   ,  ,0)) , where P  (0,0,2) and u  (1,1,0) are the point of vector and the direction    *  vector of line L , respectively. The moment vector is u op  u 2(  ,  ,0).

 2 1 1 1 Let us give a point of E , for instance, u (0, , )   (0,  ,  ). It determines the line D 2    1  y  z   2 2   3 L=x  2 ,   E. 1 1  2 2  ⊂      Definition 5.The six components ai , ai i 1,2,3 of a and a are called Plucke- rian homogenous coordinates of the directed line. 2  2 Let EOD , and {,,,}o e1 e  2 e  3 denote the dual elliptic unit sphere, the center of ED and the dual orthonormal system at O respectively, where we have   ei e i  e i , 1  i  3 (3) e1 e  2  e  3, e  2  e  3  e  1 , e  3  e  1  e  2 (4) and         e1 e 2  e 3, e 2  e 3  e 1 , e 3  e 1  e 2 (5) for

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1  1  1 e( ,0,0), e  (0, ,0), e  (0,0, ). (6) 1 2  3     3 In this case, the orthonormal system {,,,}o e1 e 2 e 3 is the system of the space of lines in E. The moment  vectors ei can be written as *  ei MO  e i ,1  i  3, MO  (,,)3  2  1 (7) 3 Since these moment vectors are the vectors of E we may write 3  ei ij e j,  ij  , 1  i  3 (8) j1 Hence, the relations (7) and (8) give us *  e  0      1 1 2 e1       e*   0    e  2   1 3   2  *     e     0 e3  3  2 3     2 Hence the E. Study mapping can be given as a mapping from the dual orthogonal system, in ED , to the 3 real orthogonal system, in E. Using the above relations; we can express the E. Study mapping in the matrix form as follows: 1       e1 1 2  e 1     e   1    e  2  1 3   2     e3     1  e 3   2 3    which says the Study mapping corresponds with a dual orthogonal matrix. Since we know that the linear mappings are in one-to-one correspondence with the matrices, then we may give the following theorem. Theorem 3. The E.Study mapping is a linear isomorphism. 3 A ruled surface in 3-space E, is a differentiable one-parameter set of straight lines, which is generated by the motion of a straight line.  2  Let x and y denote any two different points at ED and  denote the dual elliptic angle x, y  .The elliptical angle   has a value which is a dual number, where  and  are the elliptic angle and the minimal distance     between directed lines x and y, respectively.  2   * Theorem 4.Let x, y   ED , then we have g( x ,y  ) cos  , where cos  cos  sin  .   proof: Moment vectors x and y are independent of choice of the points p and q on the directed lines L1 and L2 3 which correspond to x and y in E. Thus the points p and q can be thought of feet points of common perpendicular line of L and L. The unit vector of common perpendicular is   1 2  x y n     . x y  If we show the shortest distance between L and L by  , we get   1 2   x y p q      x y

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Now we consider following equations;         gxy( , ) gxqy ( ,  )   gqxy ( ,  ),         gxy( , ) gpxy (  , )  gpxy ( ,  ),       where x p  x and y q  y . By adding the last two equations,we got         gxy(,)(,)(,) gxy   gpqxy     x y   g(,)     x  y x y     x  y   sin  If we choose the minus (sign), we get g( x ,y  ) cos   sin  , and from the Taylor formula g( x ,y  ) cos  . Special cases: i) The condition g( x ,y  ) 0 means that the lines x and y cannot be orthogonal.  ii) If g(,) xy  pure real and   0 , then the lines x and y intersect each other.  iii) And if g( x ,y  )  1,   0 and   0 , then the line x and y are coincident.

CONCLUSION

3 3 In this paper, we took a new 3-space E instead of E , then E. Study map is generalized. In the next work, we   2 3 will study E. Study map of circle which lie on the dual elliptical unit sphere ED at the dual space D (Jafari M, Yayli Y. 2013).

ACKNOWLEDGMENT

The authors would like to thank Professor Yusuf Yayli for fruitful discussions.

REFERENCES

Abdel-Baky RA. 2005. One-parameter closed dual spherical motions and Holdich's theorem. Sitzungsber. Abt. II 214: 27-41. Clifford WK. 1873. Preliminary sketch of bi-quaternions, Proc. London Math. Soc. 4, 64: 361-395. Gu Y, Luh JYS.1987. Dual-number transformation and its applications to robotics. IEEE journal of robotics and automation, Vol. Ra-3, No. 6 : 615-623. Guggenheimer HW. 1963. Differential Geometry, McGraw-Hill book company, Inc. , New York,. 3 Jafari M, Yayli Y. 2012. Generalized quaternions and rotation in 3-space E . Arxiv.org/abs/1204.2476.  2 Jafari M, Yayli Y. 2013. The E.Study map of circle on dual elliptical unit sphere ED. Work in progress. Study E. 1903. Die Geometrie der Dynamen, VerlagTeubner, Leipzig, pp. 437. Uğurlu HH, Çalişkan A. 1996. The Study mapping for directed space-like and time-like in Minkowski 3-space ₁³. Mathematical & Computational

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