ISSN: 1304-7981 Number: 10, Year:2016, Pages: 60-68 http://jnrs.gop.edu.tr Received: 18.05.2016 Editors-in-Chief : Bilge Hilal C»ad³rc³ Accepted: 06.06.2016 Area Editor: Serkan Demiriz Spherical Curves with Modi¯ed Orthogonal Frame Bahaddin BUKCUa;1 ([email protected]) Murat Kemal KARACANb ([email protected]) aGazi Osman Pasa University, Faculty of Sciences and Arts, Department of Mathematics,60250,Tokat-TURKEY bUsak University, Faculty of Sciences and Arts, Department of Mathematics,1 Eylul Campus, 64200,Usak-TURKEY Abstract - In [2-4,8-9], the authors have characterized the spherical curves in di®erent spaces.In this paper, we shall charac- Keywords - Spherical curves, terize the spherical curves according to modi¯ed orthogonal frame Modi¯ed orthogonal frame in Euclidean 3-space. 1 Introduction In the Euclidean space E3 a spherical unit speed curves and their characterizations are given in [8,9]. In [2-4,7], the authors have characterized the Lorentzian and Dual 3 spherical curves in the Minkowski 3-space E1 . In this paper, we shall characterize the spherical curves according to modi¯ed orthogonal frame in the Euclidean 3-space. 2 Preliminaries We ¯rst recall the classical fundamental theorem of space curves, i.e., curves in Euclid- ean 3-space E3. Let ®(s) be a curve of class C3, where s is the arc-length parameter. Moreover we assume that its curvature ·(s) does not vanish anywhere. Then there exists an orthonormal frame ft; n; bg which satis¯es the Frenet-Serret equation 2 3 2 3 2 3 t0(s) 0 · 0 t(s) 4 n0(s) 5 = 4 ¡· 0 ¿ 5 4 n(s) 5 (1) b0(s) 0 ¡¿ 0 b(s) 1Corresponding Author Journal of New Results in Science 6 (2016) 60-68 61 where t; n and b are the tangent, principal normal and binormal unit vectors, respec- tively, and ¿(s) is the torsion. Given a function ·(s) of class C1 and a continuous func- tion ¿(s), there exists a curve of class C3 which admits an orthonormal frame ft; n; bg satisfying the Eq.(1) with given · and ¿ as its curvature and torsion, respectively. Such a curve is uniquely determined by a motion of E3: 3 Let ®i (i = 1; 2; 3) be coordinates in E : Let ®(s) an analytic curve , where s runs through some interval and ®(s) be analytic in s. We assume that ® is non-singular, i.e., X3 µ ¶2 d®i ds i=1 is nowhere zero. Therefore we can parametrize ® by its arc length s. In the rest of this paper, we only consider ® in the following form: ® = ®(s) = (®1; ®2; ®3) ; s 2 I where ®(s) is analytic in s and I is a non-empty open interval. We assume that the curvature ·(s) of ® is not identically zero. Now we de¯ne an orthogonal frame fT; N; Bg as follows: d® dT T = ;N = ;B = T £ N; ds ds where T £ N is the vector product of T and N. The relations between those and the classical Frenet frame ft; n; bg at non-zero points of · are 8 < T = t N = ·n (2) : B = ·b: Thus N(s0) = B(s0) = 0 when ·(s0) = 0 and squares of the length of N and B vary analytically in s. By the de¯nition of fT; N; Bg or Eq.(2), a simple calculation shows that 2 3 2 3 2 3 T 0(s) 0 1 0 T (s) 4 0 5 4 2 ·0 5 4 5 N (s) = ¡· · ¿ N(s) (3) B0(s) ·0 B(s) 0 ¡¿ · where a dash denotes the di®erentiation with respect to the arc length s and det (®0; ®00; ®000) ¿ = ¿(s) = ·2 is the torsion of ®. From the Frenet-Serret equation, we know that any zero point of ·2 is a removable singularity of ¿. The Eq.(3) corresponds to the Frenet-Serret equation in the classical case. Moreover, fT; N; Bgsatis¯es: ½ hT;T i = 1; hN; Ni = hB; Bi = ·2; (4) hT;Ni = hT;Bi = hN; Bi = 0 where h; i denotes the inner product of E3. We note that the essential quantities in Eqs.(3) and (4) are ·2(s) and ¿(s) which are analytic in s [7]. Journal of New Results in Science 6 (2016) 60-68 62 3 Spherical Curves with Modi¯ed Orthogonal Frame De¯nition 3.1. Let ® be in E3given with coordinate neighborhood (I; ®). If ® ½ E3 then ® is called a spherical curve of E3: De¯nition 3.2. The sphere having su±ciently close common four points at m 2 ® with the curve ® ½ E3 is called the osculating sphere or curvature sphere of the curve ® at the point m 2 ®: Now let us calculate the geometric locus of the sphere having su±ciently close common three points with curve ® ½ E3 at the point m 2 ®: Theorem 3.3. Let ® be in E3given with coordinate neighborhood (I; ®) : The geometric locus of the centers of the spherical curves having su±ciently close common three points with the curve ® providing the modi¯ed orthogonal frame vectors fT; N; Bgat the point ®(s); s 2 I is a(s) = ®(s) + m2(s)N(s) + m3(s)B(s); where p ¡2 ¡2 2 2 m2 : I ! R; m2(s) = · ; m3(s) = §· r · ¡ 1: Proof. Let (I; ®) be a coordinate neighborhood, and s 2 I be parameter for the curve ®. Let also a be the center and r be the radius of the sphere having su±ciently close common three points with ®. In accordance to this, let us consider f : I ! R (5) s ! f(s) = ha ¡ ®(s); a ¡ ®(s)i ¡ r2: Since f(s) = f 0(s) = f 00(s) = 0 (6) at the point ®(s), then the sphere © ª S2 = x 2 E3 : hx ¡ a; x ¡ ai = r2 ; (x generic point of the sphere) with the curve ® at this point passes su±ciently close three points. So, considering Eqs.(5) and (6) together f 0(s) = ¡2 hT; a ¡ ®(s)i = 0: is obtained. From this, since f 00(s) = 0, we get hT;T i + ha ¡ ®(s); ¡Ni = 0: Considering Eq.(3) with this, we have ha ¡ ®(s);Ni = 1: On the other hand, for the base fT; N; Bg ; a ¡ ®(s) = m1(s)T (s) + m2(s)N(s) + m3(s)B(s); m1(s); m2(s); m3(s) 2 R (7) Journal of New Results in Science 6 (2016) 60-68 63 is obtained. However, by using Eq.(6), we have m1(s) = ha ¡ ®(s);T (s)i ) m1(s) = 0; (8) and 1 ·2m = ha ¡ ®(s);N(s)i ) m = ; (9) 2 2 ·2 By using f(s) = 0, we get 2 2 2 2 2 2 2 ha ¡ ®(s); a ¡ ®(s)i = r ) m1(s) + m2(s)· (s) + m3(s)· (s) = r : (10) Considering Eq.(8) in Eq.(10), we have £ 2 2 ¤ 2 2 m2(s) + m3(s) · (s) = r : (11) Considering also Eq.(9) in Eq.(11) then p ¡2 2 2 m3(s) = §· r · ¡ 1 = ¸: (12) Therefore, subtituting Eqs.(8),(9),(12) in Eq.(3) 1 p a(s) = ®(s) + N(s) § ·¡2 r2·2 ¡ 1B(s): ·2 Here a and r change when the spheres change. Hence, m3(s) = ¸ 2 R is a parameter. This completes the proof of the theorem. Corollary 3.4. Let the curve ® in E3 be given with neighborhood coordinate (I; ®): Then the centers of the spheres which pass su±ciently close common three points with ® at the points ®(s) 2 ® are located on a straight line. Proof. By Theorem 3.3, we have 1 a(s) = ®(s) + N(s) + ¸B(s) ·2(s) The equation with ¸ parameter denotes a line which pass through the point C(s) = 1 ®(s) + ·2(s) N(s) and is parallel to the B: 1 De¯nition 3.5. The line a(s) = ®(s) + ·2 N(s) + ¸B(s) is the geometric locus of the centers of the spheres which have su±ciently close common three points with the curve ® ½ E3 at the point m 2 ® is called curvature the axis at the point m 2 ® of curve ® ½ E3: The point 1 C(s0) = ®(s0) + 2 N(s0) · (s0) on curvature the axis is called curvature the center at the point m = ®(s0) of curve ® ½ E3. Theorem 3.6. Let curve ® ½ E3 be given by (I; ®) coordinate neighborhood. If a(s) = ®(s) + m2(s)N(s) + m3(s)B(s) is the center of the osculating sphere at the point ®(s) 2 ®, then (·¡2)0 m0 m (s) = ·¡2(s); m (s) = or m (s) = 2 . 2 3 2¿ 3 2¿ Journal of New Results in Science 6 (2016) 60-68 64 Proof. The proof of the theorem is similar to the proof of Theorem3.3. The osculating sphere with the curve ® have su±ciently close common four points. Therefore, since f 00(s) = 0 in (6) thus f 000(s) = 0. Then we have ·0 h¡T;Ni ¡ ·2 ha ¡ ®(s);T i + ha ¡ ®(s);Ni + ¿ ha ¡ ®(s);Bi = 0 · Considering Eqns.(1), (8) and (9) in the last equality, we obtain ·0 ha ¡ ®(s);Bi = ¡ : ¿· or ·0 (·¡2)0 m (s) = ¡ or m = 3 ¿·3 3 2¿ Using Eq.(4) in the last equation yields m0 m = 2 : 3 2¿ Corollary 3.7. Let curve ® in E3 is given by (I; ®) neighbouring coordinate. If r is the radius of the osculating sphere at ®(s) 2 ®, then s q · 0 ¸2 1 (·¡2) r = [m2(s) + m2(s)] ·2 = + : 2 3 ·2 2¿ Proof. If the center of the osculating sphere at ®(s) is a, then by Theorem 3.3, a = ®(s) + m2(s)N(s) + m3(s)B(s): Thus we have s q · 0 ¸2 1 (·¡2) r = ka ¡ ®(s)k = ·2(s)m2(s) + ·2(s)m2(s) = + : 2 3 ·2 2¿ 2 2 Theorem 3.8.