Spherical Indicatrix Curves of Spatial Quaternionic Curves

In this paper, the spherical indicatrix curves drawn by quaternionic frenet vectors are computed. Also the quaternionic geodesic curvatures of the spherical indicatrix curves to E3 and S2 are found. Mathematics Subject Classification: 11R52, 53A04


Introduction
Quaternions were discovered, for the first time in 1843, by the Irish mathematician Sir William R. Hamilton [2].Hamilton wanted to generalize complex numbers in order to be used in geometric optics.In 1987, Bharathi and Nagaraj defined the quaternionic curves in E 3 and E 4 , they studied the differential geometry of space curves and introduced Frenet frames and formulae by using quaternions [4].About a decade later, quaternionic inclined curves have been defined and harmonic curvatures studied by Karadag and Sivridag [6].Tuna and C ¸öken have studied quaternion valued functions and quaternionic inclined curves in the semi-Euclidean space E 4  2 [1].They have given the Serret-Frenet formulae and they have defined quaternionic inclined curves and harmonic curvatures for the quaternionic curves in the semi-Euclidean space.Quaternionic rectifying curves have been studied by Güngör and Tosun, [5].S ¸enyurt and C ¸alışkan have founded the Darboux vector of the spatial quaternionic curve according to the Frenet frame.Then, they calculated the curvature and torsion of the spatial quaternionic Smarandache curve formed by the unit Darboux vector with the normal vector, [7].

Preliminaries
In this section, we give the basic elements of the theory of quaternions and quaternionic curves.A more complete elementary treatment of quaternions and quaternionic curves can be found in [3] and [4], respectively.A real quaternion q is an expression of the form where a, b, c ∈ R and e i , 1 ≤ i ≤ 3, are quaternionic units which satisfy the non-commutative multiplication rules The algebra of the quaternions is denoted by Q and its natural basis is given by {e 1 , e 2 , e 3 }.A general quaternion can be given by the form where S q = d is the scalar part and V q = ae 1 + be 2 + ce 3 is the vector part of q.The Hamilton conjugate of q = S q + V q is defined by q = S q − V q .Summation of two quaternions q 1 = S q 1 + V q 1 and q 2 = S q 2 + V q 2 is defined as . Multiplication of a quaternion q = S q + V q with a scalar λ ∈ R is identified as λ q = λS q + λV q .If q = 1, then q is called unit quaternion.Let c 2 e 3 be two quaternions in, then the quaternion product of q 1 and q 2 is given by where , and ∧ denote the inner product and vector product in Euclidean 3-space.
In this space it is defined a symmetric real-valued, non-degenerate, bilinear form as follows: which is called the quaternion inner product.As a result the norm of q is (2.6) The element q is called a spatial quaternion whenever q + q = 0 and called a temporal quaternion whenever q − q = 0.A general quaternion q can be given as q = 1 2 (q + q) + 1 2 (q − q).The three-dimensional Euclidean space is identified with the space of spatial quaternions [4].The space Q H = {q ∈ Q | q + q = 0} is defined in an obvious manner.Let I = [0, 1] be an interval in the real line R and s ∈ I be the arc-length parameter along the smooth curve (2.7) The tangent vector γ (s) = t(s) has unit length t(s) = 1 for all s.
Let γ : [0, 1] → Q H be a differentiable spatial quaternions curve with arclength parameter s and {t(s), n 1 (s), n 2 (s)} be the Frenet frame of γ at the point γ(s), where and if the curve γ(s) is a non unit speed curve then we say that Let {t(s), n 1 (s), n 2 (s)} be the Frenet frame of γ(s).Then Frenet formula, curvature and the torsion are given by and (2.11) where t(s), n 1 (s), n 2 (s) are, respectively, the unit tangent, the unit principal normal and the unit binormal vector of a quaternionic curve [4].The functions k, r are called the principal curvature and the torsion, respectively.Considering the spatial quaternions curve γ : [0, 1] → Q H , the moving frame {t(s), n 1 (s), n 2 (s)} moves with a certain angular velocity around each axis for any s ∈ [0, 1].This axis is called instantaneous rotation axis of the spatial quaternionic curve.The Darboux axis vector in the direction indicated by D in defined as follows: (2.12) Let D be the instantaneos Pfaff vector of curve γ.Let denote the angle between D and n 2 with ϕ, The unit vector of quaternionic Darboux vector [7] is indicated by w: 3 Spherical Indicatrix Curves of Spatial Quaternionic Curves In this section we find the arc lenght of (t), (n 1 ), (n 2 ) and (w) for the Spherical Indicatrix Curves, we compute the geodesic curvatures in E 3 and S 2 .We indicate the arc length for (t) with s t , that is If the arc length for (n 1 ), (n 2 ) and (w) are s n 1 , s n 2 and s w , we have: Let α t (s t ) = t(s) be a unit speed regular spherical curves.We denote s t as the arc-lenght parameter of tangents indicatrix (t) Let us consider the tangent vector t t of curve (t).Supposing that the tangent vector of the (t) geodeseic curvature is Λ t , we get Computing the derivatives and after some algebra we get, By substituting (3.5), we obtain or from the equation (2.13), Likewise, let us suppose that the tangent vector of the curve (n 1 ) is t n 1 .Supposing that Λ n 1 is the geodesic curvature of (n 1 ) at E 3 , we get (3.10) Let us define the tangent vector of the curve (n 2 ) as t n 2 .Supposing that Λ n 2 is the geodesic curvature of (n 2 ) at E 3 , we get Following a similar approach, we consider the tangent vector is t w of the curve (w).Supposing that Λ w is the geodesic curvature of (w) at E 3 , we get

0 N
(ϕ (cos ϕt − sin ϕn 2 ))ds = Because of the Gauss equation and S(t t ) = t t , we haveD tt t t = D tt t t + S(t t ), t t | Q t, D tt t t = D tt t t + t, D tt t t = r k n 2 = tan φn 2 .Supposing that the geodesic curvature for (n 1 ) is Γ n 1 , it is Γ n 1 = N (D tn 1 t n 1 ).Because S(t n 1 ), t n 1 | Q = 1 at the statement D tn 1 t n 1 = D tn 1 t n 1 + S(t n 1 ), t n 1 | Q t, D tn 1 t n 1 = ϕ N(w)(sin ϕt + cos ϕn 2 ) Likewise, supposing that the geodesic curvatures for (n 2 ) are (w) are Γ n 2 and Γ w , these are Γ n 2 = N (D tn 2 t n 2 ) and Γ w = N (D tw t w ).Then, Γ n 2 = cot ϕ, .12) Let find the geodesic curvatures of (t), (n 1 ), (n 2 ) and (w) to S 2 .Supposing that the geodesic curvature for (t) is Γ t , it is Γ t = N (D tt t t ).