EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 1, 2014, 86-96 ISSN 1307-5543 – www.ejpam.com Osculating Spheres of a Semi Real Quaternionic Curve in E4 2 Ö. Bektaş1,∗, N. Bayrak Gürses 2, S. Yüce 1 1 Department of Mathematics, Faculty of Arts And Sciences, Yıldız Technical University, Istanbul, Turkey 2 Department of Mathematics, Graduate School of Natural and Applied Sciences, Yıldız Technical University, Istanbul, Turkey Abstract. In this study, we define the osculating spheres of a semi real quaternionic curve in semi- Euclidean spaces E3 1 and E4 2. We give the equation of the osculating spheres with respect to Frenet frames ¦ t0,n10 ,n20 © and ¦ T0,N10 ,N20 ,N30 © . 2010 Mathematics Subject Classifications: 14Q05, 53A17, 53C50. Key Words and Phrases: Osculating sphere, semi real quaternionic curve, semi-Euclidean space. 1. Introduction Quaternion algebra was introduced by Hamilton in 1843. Important precursors to this work included Euler’s four square identity and Olinde Rodrigues parametrization of general rotations by four parameters, but neither of these writers treated the four parameter rota- tions as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the complex numbers could be in- terpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. While walking along the towpath of the Royal Canal with ∗Corresponding author. Email addresses: obektas@yildiz.edu.tr (Ö. Bektaş), nbayrak@yildiz.edu.tr (N. Gürses), sayuce@yildiz.edu.tr (S. Yüce) http://www.ejpam.com 86 c© 2014 EJPAM All rights reserved. Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 87 his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, e1 2 = e2 2 = e3 2 = e1e2e3 =−1 into the stone of Brougham Bridge as he paused on it. Hamilton called a quadruple with these rules of multiplication a quaternion, and he de- voted most of the remainder of his life to studying and teaching them. Hamilton’s treatment is more geometric than the modern approach, which emphasizes quaternions’ algebraic proper- ties. He founded a school of ”quaternionists“, and he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long; it was published shortly after his death. After Hamilton’s death, his student Peter Tait continued pro- moting quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinemat- ics in space and Maxwell’s equations, were described entirely in terms of quaternions. There was even a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems. From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vec- tor analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature of quaternions. However, vector analysis was con- ceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side effect of this transition is that Hamilton’s work is difficult to comprehend for many modern readers. Hamilton’s original definitions are un- familiar and his writing style was wordy and difficult to understand. However, quaternions have had a revival since the late 20th Century, primarily due to their utility in describing spa- tial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles they are not susceptible to gimbal lock. For this reason, quaternions are used in computer graphics, computer vision, robotics, control theory, signal processing, attitude control, physics, bioin- formatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude-control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms [1]. The set of quaternions is denoted byH . While the quaternions are not commutative, they are associative, and they form a group known as the quaternion group. By analogy with the complex numbers being representable as a sum of real and imaginary parts, a01 + a1e1, a quaternion can also be written as a linear combination a01+ a1e1+ a2e2+ a3e3. As a set, the quaternions H are equal to E4 , a four dimensional vector space over the real numbers. H has three operations: addition, scalar multiplication and quaternion multiplica- tion. The sum of two elements of H is defined to be their sum as elements of E4. Similarly Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 88 the product of an element of H by a real number is defined to be same as the product in E4.To define the product of two elements inH requires a choice of basis for E4. The elements of this basis are customarily denoted as 1, i, j, and k. Every element of H can be uniquely written as a linear combination of these basis elements, that is, as a01+ a1e1 + a2e2 + a3e3, where a0, a1, a2, and a3 are real numbers. The basis element 1 will be the identity element ofH , meaning that multiplication by 1 does nothing, and for this reason, elements ofH are usually written a01+ a1e1 + a2e2 + a3e3, suppressing the basis element 1. Given this basis, associative quaternion multiplication is defined by first defining the products of basis elements and then defining all other products using the distributive law, [2–4]. In [5], Baharathi and Nagaraj studied the differential geometry of a smooth quaternionic curve in E3 and E4. Elements of E4 were identified with quaternions in a natural way. The Serret-Frenet formulae for a quaternionic curves in E3 and E4 were given by them. Then Serret Frenet formulae for quaternionic curves in semi-Euclidean Space are given by [6]. By using of these formulas, new definitions and new characterizations of quaternionic curves are studied in [2, 3]. In Euclidean space E3, there is a unique sphere for a curve α which contacts α at the third order at α(0). The intersection of the sphere with the osculating plane is a circle which contacts at the second order at α(0), [7–9]. This concept studied in [10] in terms of quater- nionic curves in E4. In [11], the osculating sphere and the osculating circle of the curve are investigated in semi-Euclidean spaces E3 1, E4 1, and E4 2. In this study, we give definition of the osculating spheres for semi quaternionic curve in semi-Euclidean spaces E3 1 and E4 2 with respect to Frenet frames ¦ t0,n10 ,n20 © and ¦ T0,N10 ,N20 ,N30 © , respectively. 2. Preliminaries A semi real quaternion is defined by q = ae1+be2+ce3+ de4 such that ei × ei =− εei , 1≤ i ≤ 3 ei × e j =εei εe j ek, in E3 1 ei × e j =− εei εe j ek, in E4 2, where (i jk) is an even permutation of (123). Notice here that we denote the set of all spatial semi real quaternions byHp and all semi real quaternions byH . It is defined by H = ¦ q = ae1+ be2+ ce3+ de4 : a, b, c, d ∈ R, e1, e2, e3 in E3 1, ei , ei � = εei , 1≤ i ≤ 3 © where index = 1,2. If ei is a spacelike or timelike vector, then εei = +1 or −1, respectively. For p = Sp + Vp and q = Sq + Vq, the multiplication of two semi real quaternions p and q is defined as follows: p× q = SpSq − ¬ Vp,Vq ¶ + SpVq + SqVp +Vp ∧Vq, ∀p, q ∈H , Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 89 where we have used the scalar and cross products in E3 1. Let q denotes the conjugate of a quaternion, q = −ae1 − be2 − ce3 + d for every q ∈ H . This helps to define the symmetric, non-degenerate, bilinear form as follows: 〈 , 〉 :H ×H →R, (p, q)→ p, q � p = 1 2 � εpεq � p× q � + εqεp � q× p � � for E3 1 (p, q)→ p, q � =− 1 2 � εpεq � p× q � + εqεp � q× p � � for E4 2. The norm of semi real quaternion q is denoted by q = � � q, q � � �= � �p � q× q � � �= � �−a2− b2+ c2+ d2 � � for p, q ∈ H . If p, q � = 0, then p and q are called orthogonal. q is called a spatial semi quaternion whenever q + q = 0 [6]. The Serret Frenet formulae for semi-real quaternionic curves in E3 1 and E4 2 are as follows: The three-dimensional semi-Euclidean space E3 1 is identified with the space of spatial quaternions Hp = ¦ γ ∈H � �γ+ γ= 0 © in an obvious manner. Let I = [0,1] be an inter- val in the real line R and γ : I ⊂ R→Hp s→ γ (s) = 3 ∑ i=1 γi (s) ei , (1≤ i ≤ 3) be an arc length curve with nonzero curvatures {k, r}. Let � t,n1,n2 denote the Frenet frame of the curve γ. Then Frenet formulae are given by t ′ (s) =εn1 kn1 (s) n ′ 1 (s) =− εtkt (s) + εn1 rn2 (s) n ′ 2 (s) =− εn2 rn1 (s) , (1) where 〈t, t〉p = εt, n1,n1 � p = εn1 , n2,n2 � p = εn2 [6]. The four dimensional semi-Euclidean space E4 2 is identified with the space of unit quater- nionsH . Let I = [0,1] be an unit interval in the real line R and β : I ⊂ R→H s→ β (s) = 4 ∑ i=1 γi (s) ei , e4 =+1 Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 90 be a smooth curve in E4 2 with nonzero curvatures {K , k, r − K}. Let � T,N1,N2,N3 be Frenet Frame of β . Then Frenet formulae are given by T ′ (s) =εN1 KN1 (s) N ′ 1 (s) =− εN1 εtKT (s) + εn1 kN2 (s) N ′ 2 (s) =− εtkN1 (s) + εn1 � r − KεTεtεN1 � N3 (s) N ′ 3 (s) =− εn2 � r − KεTεtεN1 � N2 (s) , (2) where 〈T,T〉= εT, N1,N1 � = εN1 , N2,N2 � = εN2 and K = εN1 T ′ (s) , [6]. 3. Spatial Semi Quaternionic Osculating Spheres of a Spatial Semi Quaternionic Curve in E3 1 Definition 1. Let γ : I ⊂ R→Hp s→ γ (s) = 3 ∑ i=1 γi (s) ei , be a spatial semi quaternionic curve in E3 1 identified withHp in an obvious manner. Let I = [0,1] be an interval in the real line R and s be an arc-length parameter. So we say that γ ′ (s) p = ‖t (s)‖p = 1. We assume that l = � l1, l2, l3 � be a rectangular coordinate system of E3 1. We take a sphere 〈l −m, l −m〉p = r2 with origin m and radius r. We define a function f (s) = γ (s)−m,γ (s)−m � p − r2 holds the following equations f (0) = f ′ (0) = f ′′ (0) = f ′′′ (0) = 0, f (4) 6= 0. Then we called that the sphere contacts at third order to the curve γ at γ (0). The sphere is called spatial semi real quaternionic osculating sphere for spatial semi quaternionic curves in E3 1. Theorem 1. Let γ : I ⊂ R→Hp be a spatial semi quaternionic curve with nonzero curvatures k (0) and r (0) at γ (0). Then there exists a sphere which contacts at the third order to the curve γ at γ (0) and the equation of the spatial semi quaternionic osculating sphere according to the Frenet frame ¦ t0,n10 ,n20 © as follows: εt0 x2 1 + εn10 � x2− εt0 ρ0 �2 + εn20 x3− εt0 εn20 ρ′0σ0 !2 = εn10 ρ2 0 + εn20 � ρ′0σ0 �2 where ρ0 = 1 k0 and σ0 = 1 r0 Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 91 Proof. If f (0) = 0 then γ (0)−m,γ (0)−m � p = r2. By differentiating this equation, we get f ′ (0) = 0 and f ′ = 2 γ′,γ−m � p = 0. Because of this, we can write t0,γ (0)−m � p = 0. (3) In a similar way we can get f ′′ = 2 � γ′′,γ−m � p + γ′,γ′ � p � = 0 and f ′′ (0) = 0. If we use the equation (1) we have D εn10 k0n10 (s) ,γ (0)−m E p + t0, t0 � p = 0 and ¬ n10 (s) ,γ (0)−m ¶ p =− εt0 εn10 k0 =− εt0 εn10 ρ0 (4) where 1 k0 = ρ0. By considering f ′′′ = 2 � γ′′′,γ−m � p + 3 γ′′,γ′ � p � = 0 and f ′′′ (0) = 0 and using the equations (3) and (4), we have ¬ n20 (s) ,γ (0)−m ¶ p =−εt0 k′0 k2 0 r0 =−εt0 ρ′0σ0. (5) When all is said and done, presently we study to find the numbers ϑ1, ϑ2, and ϑ3 such that γ (0)−m= ϑ1t0+ ϑ2n10 + ϑ3n20 . (6) From t0,γ (0)−m � p = ϑ1 and by using the equation (3) we obtain ϑ1 = 0. In the same vein, by using the equations (4) and (5), we get ϑ2 = −εt0 ρ0 and ϑ3 = − εt0 εn20 ρ′0σ0. Also the origin of the sphere that contacts at the third order to the curve at the point γ (0) is m= γ (0)− ϑ1t0− ϑ2n10 − ϑ3n20 . Let Q be a spatial semi quaternionic variable on spatial semi quaternionic osculating sphere, assume Q = γ (0) + x1t0+ x2n10 + x3n20 . From the above discussions, we have the following statement Q−m= x1t0+ � x2− εt0 ρ0 � n10 + x3− εt0 εn20 ρ′0σ0 ! n20 Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 92 and 〈Q−m,Q−m〉p = εt0 x2 1 + εn10 � x2− εt0 ρ0 �2 + εn20 x3− εt0 εn20 ρ′0σ0 !2 . Finally, with the aim of the equation (6) we get r2 = γ (0)−m,γ (0)−m � p = εn10 ρ2 0 + εn20 � ρ′0σ0 �2 which is the equation of the spatial semi quaternionic osculating sphere in E3 1. This completes proof. 4. Semi Quaternionic Osculating Spheres of a Semi Quaternionic Curve in E4 2 Definition 2. β : I ⊂ R→H s→ β (s) = 3 ∑ i=0 γi (s) ei , be a semi quaternionic curve in E4 2 identified with the space of semi quaternionsH . Let I = [0, 1] be an interval in the real line R and s be an arc-length parameter. In this position we say that β ′ (s) = ‖T (s)‖ = 1. We assume that L = � L1, L2, L3, L4 � be a rectangular coordinate system of E4 2. We take a sphere〈L−M , L−M〉 = R2 with origin M and radius R. We define a function g (s) = β (s)−M ,β (s)−M � − R2 satisfies the following equations g (0) = g ′ (0) = g ′′ (0) = g ′′′ (0) = g(4) = 0, g(5) 6= 0. Then we called that the sphere contacts at fourth order to the curve β at β (0). The sphere is called semi real quaternionic osculating sphere for semi quaternionic curves in E4 2. Theorem 2. Let β : I ⊂ R → H be a semi quaternionic curve with nonzero curvatures K (0), r (0), and (r − K) (0) at β (0). Then there exists a sphere which contacts at the fourth order to the curve β at β (0) and the equation of the semi quaternionic osculating sphere according to the Frenet frame ¦ T0,N10 ,N20 ,N30 © such that εT0 X 2 1 + εN10 X2− εT0 εN10 ς0 !2 + εN20 X3− εT0 εN10 εn10 ς′0ρ0 !2 + εN30 � X4−Ω0 � εT0 εN10 � � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0 − 4 � ρ0 ς0 � � + 3 εT0 εN10 � ρ0 ς0 ����2 Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 93 =εN10 ς2 0+ εN20 � ς′0ρ0 �2 + εN30 � Ω2 0 � � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0 − 4 � ρ0 ς0 � � + 3 εT0 εN10 � ρ0 ς0 ��� where ς0 = 1 K0 , ρ0 = 1 k0 , and Ω0 = 1 r0−εt0 εT0εN10 K0 . Proof. If g (0) = 0 then β (0)−M ,β (0)−M � = R2. By differentiating this equation, we get g ′ (0) = 0 and g ′ = 2 � β ′ (0) ,β (0)−M �� = 0. Because of this, we can write T0,β (0)−M � = 0. (7) Similarly we can get g ′′ (0) = 2 � β ′′ (0) ,β (0)−M � + β ′ (0) ,β ′ (0) �� = 0 and g ′′ (0) . By taking into account the equations (2) and using the last equation we have D εN10 K0N10 ,β (0)−M E + T0,T0 � = 0 ¬ N10 ,β (0)−M ¶ =− εT0 εN10 K0 =− εT0 εN10 ς0 (8) By considering g ′′′ (0) = 2 � β ′′′ (0) ,β (0)−M � + 3 β ′′ (0) ,β ′ (0) �� = 0 and g ′′′ (0) = 0 and from equations (7) and (8), we get ¬ N20 (s) ,β (0)−M ¶ = εT0 K ′0 εN10 εn10 k0K2 0 =− εT0 εN10 εn10 ς′0ρ0. (9) Additionally, we obtain g(4) (0) = 2 �¬ β (4) (0) ,β (0)−M ¶ + 4 ¬ β ′′′ (0) ,β ′ (0) ¶ + 3 β ′′ (0) ,β ′′ (0) � � = 0 and g(4) (0) = 0 where ¬ β ′′′ (0) ,β ′ (0) ¶ =−εt0 εT0 K2 0 , β ′′ (0) ,β ′′ (0) � = εN10 K2 0 . So we have ¬ N30 (s) ,β (0)−M ¶ =−Ω0    − εT0 εN10    K ′′0 ς 2 0ρ0− εt0 εn10 � ς0 ρ0 � − εt0 εN10 ρ0 +2K ′0ς ′ 0ρ0ς0+ k′0ς ′ 0ρ 2 0 + 4εt0 � ρ0 ς0 �    + 3 ρ0 ς0    . Ö. Bektaş, N. Gürses, S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 86-96 94 The following two statements are found such that ¬ N30 (s) ,β (0)−M ¶ =−Ω0 � − εT0 εN10    2 � ς′0 �2 ρ0 ς0 − ς′′0ρ0− εt0 εn10 � ς0 ρ0 � − εt0 εN10 ρ0 −2 � ς′0 �2 ρ0 ς0 − ς′0ρ ′ 0+ 4εt0 � ρ0 ς0 �    + 3 ρ0 ς0 � and ¬ N30 (s) ,β (0)−M ¶ =−Ω0 � εT0 εN10 � � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0− 4 � ρ0 ς0 �� + 3 εT0 εN10 � ρ0 ς0 ��� (10) Furthermore, let take the numbers ω1, ω2, ω3 and ω4 such that β (0)−M =ω1T0+ω2N10 +ω3N20 +ω4N30 . From T0,β (0)−M � =ω1 and by using the equation (7) we obtain ω1 = 0. Similarly, using the equations (8), (9), and (10) we get ω2 =− εT0 εN10 ς0, ω3 =− εT0 εN10 εn10 ς′0ρ0, and ω4 = −Ω0 � εT0 εN10 � � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0− 4 � ρ0 ς0 �� + 3 εT0 εN10 � ρ0 ς0 � �� . Also the origin of the sphere that contacts at the fourth order to the curve at the point β (0) is M = β (0)−ω1T0−ω2N10 −ω3N20 −ω4N30 . Let P be a semi quaternionic variable on semi quaternionic osculating sphere, assume P = β (0) + X1T0+ X2N10 + X3N20 + X4N30 . Then we can write P −M =X1T0+ X2− εT0 εN10 ς0 ! N10 + X3− εT0 εN10 εn10 ς′0ρ0 ! N20 + � X4−Ω0 � εT0 εN10 � � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0 − 4 � ρ0 ς0 � � + 3 εT0 εN10 � ρ0 ς0 ���� N30 . From the last equation, we get 〈P −M , P −M〉=εT0 X 2 1 + εN10 X2− εT0 εN10 ς0 !2 + εN20 X3− εT0 εN10 εn10 ς′0ρ0 !2 REFERENCES 95 + εN30 � X4−Ω0 � εT0 εN10 � � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0 − 4 � ρ0 ς0 � � + 3 εT0 εN10 � ρ0 ς0 ����2 and from the equation (10) we finally calculate the equation of the semi real quaternionic osculating sphere in E4 2 such that R2 = β (0)−M ,β (0)−M � = εN10 ς2 0+ εN20 � ς′0ρ0 �2 + εN30 ( Ω2 0   � ς′0ρ0 �′ + εt0 � εn10 � ς0 ρ0 � − εN10 ρ0− 4 � ρ0 ς0 �� + 3 εT0 εN10 � ρ0 ς0 �   ) . This completes proof. ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal successful. References [1] W. K. Clifford. Preliminary sketch of biquaternions, Proceedings of the London Mathe- matical Society, 4, 361-395, 1873. [2] A. C. Çöken and A. Tuna. On the quaternionic inclined curves in the semi-Euclidean space E4 2, Appl. Math. Comput. 155, 373–389, 2004. [3] F. Kahraman, İ. Gök, and H. H. 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