BEÜ Fen Bilimleri Dergisi BEU Journal of Science 4(1), 12-20, 2015 4(1), 12-20, 2015
Araştırma Makalesi / Research Article
ퟐ Dual Eliptik Birim Küre È푫 Üzerindeki Çemberin E.Study Dönüşümü
Mehdi JAFARI*1, Yusuf YAYLI2
1Department of Mathematics, University College of Science and Technology Elm o Fan, Urmia, Iran 2Department of Mathematics, Faculty of Sciences, Ankara University, 06100 Ankara, Turkey
Özet 3 Bu çalışmada, ilk olarak genelleştirilmiş E.Study dönüşümü üç boyutlu ℝ훼훽 uzayındaki doğrular için tanımlandı. 3 2 Sonra, 퐷훼훽 dual uzayındaki birim dual eliptik küre È퐷 üzerinde ki çemberin E.Study dönüşümü incelendi. Ayrıca, herbiri bir geometrik sonuç olan bazı özel durumlar araştırıldı.
Anahtar Kelimeler: Dual eliptik birim küre, genelleştirilmiş iç çarpım, eğimli kongruans
The E.Study Map of Circle on Dual Eliptic ퟐ Unit Sphere È푫
Abstract 3 In this paper, first the generalized E. Study mapping is defined for the lines in 3-space ℝ훼훽. Then, the E. Study 2 3 map of circle which lie on the dual elliptical unit sphere È퐷 at the dual space D is studied. Furthermore, some special case is examined, each of which is a geometrical result.
Keywords: Dual elliptical unit sphere, generalized inner product, inclined congruence
1. Introduction
Dual numbers were introduced in the 19th century by W.K. Clifford (1849-79), as a tool for his geometrical investigation. After him E. Study [5] and Kotelnikov [3] systematically applied the dual number and dual vector in their studies of line geometry and kinematics and independently discovered the transfer principle. Study devoted special attention to the representation of directed line by dual unit vectors and defined the mapping that is said with his name; he proved that there exists one-to-one 2 correspondence between the points of the dual unit sphere SD and the directed lines of Euclidean 3- 3 space ℝ훼훽(E. Study’s mapping). Hence, a differentiable curve on the sphere corresponds to a ruled surface in the line space [1]. Hacisalihoglu [2] showed that the E. Study map of a circle on a dual unit sphere is a family of hyperboloids of on sheet with two parameters. In [4], by taking the Minkowski 3 3 3-space ℝ1 instead of ℝ ; Ugurlu and Çaliskan gave a correspondence of E. Study mapping as follows: There exists one to one correspondence between the dual time-like and space-like unit vectors of dual 2 2 3 hyperbolic and Lorentzian unit sphere H0 and S1 at the dual Lorentzian space D1 and the directed time- 3 like and space-like lines of the Minkowski 3-space ℝ1. Subsequently, Yaylı and et. al [6] investigated the E. Study maps of circles on dual hyperbolic and Lorentzian unit spheres and at is the dual
*Corresponding author: [email protected]
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3 2 Lorentzian space D1 . By this mapping, a curve on a dual hyperbolic unit sphere H0 corresponds to a 3 timelike ruled surface in the Lorentzain line space ℝ1, that is, there exists a one-to-one correspondence 2 3 between the geometry of curves on H0 and the geometry of timelike ruled surfaces in ℝ1. This paper is organized as follows: In the first part, the authors consider a generalized inner product on a real 3- 3 3 dimensional vector space. In the second part, we take a 3-space ℝ훼훽 instead of ℝ , then, the E. Study 2 maps is generalized. In doing so, E. Study map of circle which lie on the dual elliptical unit sphere È퐷 at 3 the dual space D is studied.
2. Basic Concepts
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 x x x( , ,123 ) and y y y y( , ,123 ) , the generalized inner product on ℝ is given by
gxyxyxyxy(,), 112233 (1) where and are positive numbers. If 0 and 0 then g(,) x y is called the generalized Lorentzian inner product. The vector space on ℝ3 equipped with the generalized inner product is called 3 3-dimensional generalized space and denoted by ℝ훼훽. 3 The cross product in ℝ훼훽 is defined by
xyx yx y()()() ix2 33 yx . 23 yjx 11 yx 31 y 22 k 1 It is clear gxxy(,)0 and g( y , x y ) 0.
Special cases: 3 3 1. If 1, then ℝ훼훽 is a Euclidean 3-space ℝ . 3 3 2. If1,1 , then ℝ훼훽 is a semi-Euclidean 3-space ℝ2.
Proposition 1. For 훼, 훽 ∊ ℝ+ the inner and vector products satisfy the following properties; 1.. uvvu 2.(,g )(,uv wg )(, vw )det( ug , uw , vu ). w v 3.()()(uvwg , uw ) . vg u v w
Definition 2. Each element of the set
퐷 = {퐴 = 푎 + 휀푎∗: 푎, 푎∗ ∈ ℝ 푎푛푑 휀 ≠ 0, 휀2 = 0} = {퐴 = (푎, 푎∗): 푎, 푎∗ ∈ ℝ} is called a dual number. Summation and multiplication of two dual numbers are defined as similar to the complex numbers but it is must be forgotten that 2 0 . Thus, D is a commutative ring with a unit element. The set
3 Da ( A1 , A 2 , A 3 ) Ai D ,1 i 3 is a module over the ring D which called a D-module or dual space. The elements of D3 are called dual vectors. Thus a dual vector a can be written a a a
13 M. Jafari, Y. Yaylı / BEÜ Fen Bilimleri Dergisi 4(1), 12-20, 2015 where a and a are real vector at ℝ3.
Definition 3. Let a a a and b b b be in D3 and 훼, 훽 ∈ ℝ. The generalized inner product of a and b is defined by
ga(,)(,)((,)(,)).(2) bga bga bgab
33 3 We put D(D,(,)). g The norm of dual vector a a a D is a dual number given by gaa(,) agaaaa(,),0. a 2 + 3 For 훼, 훽 ∈ ℝ , the set of all unit dual vector in D is said the dual elliptical unit sphere and is denoted 2 3 by E.D The vector product in D is
abababab ()
3 + Lemma 1. Let uv,D and 훼, 훽 ∈ ℝ . In this case, we have
uvuvs .sin. 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.
Definition 4. An oriented line L in the three-dimensional Euclidean space E3 can be determined by a point pL and a normalized direction vector a of L , i.e. a 1. To obtain components for , one forms the moment vector apa with respect to the origin point in . If p is substituted by any point 푞⃗ = 푝⃗ + 휆푎⃗, (휆 ∈ ℝ) on then the above equation implies that a is independent of on L. The two vectors and a are unindependent, they satisfy the following relationships; aa,1 , aa,0.
The six components ai , ai i 1,2,3 of and a are called the normalized Plücker coordinates of the line L. Hence, the two vectors and determine the oriented line Conversely, any six-tuple , with relations
222*** aaaa1231 12 aa 23 aa 3 a1,0, represents a line in the three-dimensional space .
Theorem 1. (E. Study) The oriented lines in ℝ3 are in one-to-one correspondence with the points of the dual unit sphere in D3 [1].
3. Generalized E. Study Map
In this section, we generalize the theorem which is attributed to Eduard Study [5]. 3 Theorem 2. The set of all oriented lines in ℝ훼훽 is in one-to-one correspondence with the set of the 3 points of the dual unit sphere in the dual 3-space D . Proof: First let , 0.In ℝ3 a directed line can be given by y x a , where x and are the 훼훽 position vector and the direction vector of the line, respectively. The moment vector a x a is not
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depending on the chosen point on the line. For this reason, by the help of ordered pair of vectors ( aa, 3 ), a directed line is 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 ba in equation (2) we obtain: gaagaagaa(,)(,)2((,))1 where the dual unit vector a represents the unique directed line ( aa, ). Note that this theorem holds for 0 and 0. Special cases: 1) If 1, then we get E. Study mapping in Euclidean 3-space ℝ3. 2) If 1, 1, then we will have E. Study mapping for a space which is isomorphic to Minkowski 3 3-space ℝ1 [4].
3 3 Example 1. In D , the unit dual vector corresponding to the line L = , 2x y z ⊂ ℝ훼훽 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 uopu* 2(,,0).
2 111 Let us give a point of E,D for instance, u (0,,)(0,,). It determines the line 2 1 y z 3 22⊂ℝ L=2,x 훼훽 11 22 Definition 5. The six components ai , ai i 1,2,3 of a and a are called Pluckerian homogenous coordinates of the directed line.
Let E,2 O and {,,,}oeee denote the dual elliptic unit sphere, the center of E2 and the dual D 123 D orthonormal system at O respectively, where we have
eeeiiii ,13 ( 3 )
eeeeeeeee123231312,, (4) and
eeeeeeeee123231312 ,,(5) for 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 ℝ훼훽. The moment vectors ei can be written as
* eii MO e,13, i MO (,,)3 2 1 (7) 3 Since these moment vectors are the vectors of ℝ훼훽 we may write
∗ 3 푒푖 = ∑푗=1 휆푖푗푒푗, 휆푖푗 ∈ ℝ, 1 ≤ 푖 ≤ 3 (8)
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Hence, the relations (7) and (8) give us
* e1 0 12 e 1 ee* 0 2 1 3 2 * e 0 e3 3 23
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 ℝ훼훽. Using the above relations; we can express the E. Study mapping in the matrix form as follows: ee1 12 11 ee 1 2132 ee33 1 23 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 ℝ훼훽 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 xy, .The elliptical angle has a value which is a dual number, where and are the elliptic angle and the minimal distance between directed lines and y, respectively.
2 Theorem 4. Let xyE, D , then we have gxy(,)cos, where coscos * sin. Proof: Moment vectors x and y are independent of choice of the points p and q on the directed lines 3 L1 and L2 which correspond to x and y in ℝ훼훽. Thus the points and can be thought of feet points of common perpendicular line of and L.2 The unit vector of common perpendicular is xy n . xy If we show the shortest distance between and by , we get xy pq xy Now we consider following equations; gxygxq( , )( ,)( ,), ygqx y
gx( , ygp )(, )( xygpx ,), y where x p x and yqy . By adding the last two equations, we got
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gxygxygpqxy(,)(,)(,) xy gxy(,) xy xy sin If we choose the minus (sign), we get gxy(,)cossin, and from the Taylor formula gxy(,)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 and intersect each other. iii) And if g( x , y ) 1, 0 and , then the line and are coincident.
2 4. The E. Study mapping of a circle on ED
Let l be the straight line corresponding to the dual vector e3. If we choose the point P on then we have 230 and so the matrix (8) reduces to
ee101 11 ee 10 (9) 212 ee001 33 The inverse of this mapping ee101 11 ee 10. (10) 212 ee001 33 Let 12 E{E(DD3xg ,)cosconstant} x e
1 be the circle on the sphere and a point of ED be x .Thus the dual vector x can be expressed as 1 1 1 xsin cos e sin sin e cos e (11) 1 2 3 where and are the dual elliptic angle and dual angle, respectively. Since we have the relations xxx
sin sin cos , sin sin cos (12) cos cos sin , cos cos sin .
(9) and (11) give us the vector x and x in the matrix form:
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1 sincos 1 xeee 123 sinsin 1 cos 1 coscossinsin1 1 xeee 1231 sincossincos 1 sin
On the other hand the point x is on the circle whose center is a point of the axis e3 .Thus we may write gx(,)cos(cossin)constant e 3 which means that
c1( c o n s t a n t) and c2 (c o n s t a n t) The above equations permit us to write the following relation: g(,)1, x x g(,)0, x x
g(,)cos0, x e3 g(,)(,)sin0. x eg33 xe
3 The last equations have only two parameters and . So it represents a line Congruence in ℝ훼훽. Now we may calculate the equations of this congruence in Plucker coordinates. Let y denote a point of this congruence then we have
yxxv(,)(,)(,) x
If the coordinates of are (,,),yyy123 then this gives us
111 yv11 sinsin cos cossin cos
111 yv21 cossin cos sinsin sin 1 yv sinsin2222 ( sincos )cos . 31
Case 1: if 1, we have
yv11 sin sin cos cos sin cos yv21cos sin cos sin sin sin 2 yv31( )sin cos In the case, we have a hyperbolic of one sheet
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2 22 yy y31 12 1 [2]. cc22 2 22 cc21cot Case 2: if , we have
11 yv11 sinsincoscossincos 11 (18) yv21 cossincossinsinsin 1 yv()sincos 2 31
If yY11 , yY22 and Yy33 then; 2 22 YY Y31 12 1 (19) cc22 2 22 cc21cot
Which has two parameters and 1 . So it represents a line congruence with degree 2. The lines of this congruence are located so that (a) The shortest distance of these lines and line l is c2 .
(b) The angle of these lines and the line l is c1 .
Hence we can say that the lines of this congruence intersect the generators of a cylinder whose radius is =constant, and the axis is l , under the angle =constant.
Definition 6. If all the lines of a line congruence have a constant angle with a definite line then the congruence is called an inclined congruence.
According to this definition, (19) represents an inclined congruence. Hence we may give the following theorem.
1 2 Theorem 5. Let ED be a circle with two parameter on the dual elliptic unit sphere ED . The Study map of is an inclined congruence with degree two.
On the other hand we know that the shortest distance of the axis l of the cylinder and the lines of the congruence is c2 .Therefore, this cylinder is the envelope of the lines of the congruence.
Special cases: (i) The case that 0 and . 2 In this case the lines of the congruence (19) orthogonally intersect the generators of the cylinder whose axis is l and the radius is . Indeed, in the case (19) reduces to
YYcv2222 122 Y31 (ii) The case that and 0. In this case the lines of the congruence coincide with the generators of the cylinder which is the envelope of the lines congruence. This means that the Study map of reduces to the cylinder whose equations, form (19), are
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Y Y222 c 122
Yv3 (iii) The case that 0 and 0. In this case, all of the lines congruence intersect the axis l under the constant elliptic angle . We can say that the lines of the congruence are the common lines of two linear line complexes. Form (19) the equation of congruence is
2 Y31 22 YY12 2 1 cot c1 (iv)The case that 0 and . 2 1 2 In this case ED is a great circle on ED .Then all of the lines of congruence orthogonally intersect the axis l. This means that the inclined congruence reduces to a linear line complex whose axis is l.(19) gives us that equation of congruence is YYv222 12 Y31()
(v) The case that 0 and 0. In the case, all of the lines of the congruence are coincide with the line l. Indeed, (19) reduces to the line l : YY220 12
Yv3 . Definition 7. If all the lines of a line congruence orthogonally intersect a constant line then the congruence is called a recticongruence.
2 Theorem 6. Let be a great circle on ED , that is, 12 E{E,0}DD3xx e Then the E. Study mapping of is a recticongruence.
References
1. Guggenheimer H.W. 1963. Differential Geometry, McGraw-Hill book company, New York. 2. Hacısalihoğlu H.H. 1977. Study map of a circle, Journal of the Faculty of science of Karadeniz Technical university, 1 (7): 69-80. 3. Kotel’nikov A.P.1895. Vintovoe Schislenie i Nikotoriya Prilozheniye evo k geometrie i mechaniki, Kazan. 4. Uğurlu H.H., Çalışkan A. 1996. The Study Mapping for Directed Space-Like and Time-Like in Minkowski 3-space ℝ₁³. Mathematical & Computational App., 1: 142-148. 5. Study E. 1903. Die Geometrie der Dynamen, Verlag Teubner, Leipzig. 6. Yaylı Y., Çalışkan A., Uğurlu H.H. 2002. The E. Study map of circles on dual hyperbolic and 2 2 Lorentzian unit sphere H0 and S1 , Math. Proceedings of the Royal Irish Academy, 102 (1): 37-47.
Geliş Tarihi: 25/01/2015 Kabul Tarihi: 25/03/2015
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