In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.
In this paper, some special developable ruled surfaces with point-wise Gauss map are studied. Three different special developable ruled surfaces called rectifying ruled surface, generalized normal ruled surface and osculating type ruled surface are considered. The conditions for such surfaces to have 1-type Gauss map and also to be minimal are introduced. Finally, some examples are given for the obtained results.
In this study, we give definitions and characterizations of eikonal slant helix curves, eikonal Darboux helices and non-normed eikonal Darboux helices in three dimensional Riemannian manifold 3 M . We show that every eikonal slant helix is also an eikonal Darboux helix. , where κ and τ are curvature and torsion of α , respectively. MSC: 53A04, 14H45.
In this study, first we investigate the Fibonacci vectors, Lucas vectors and their vector products considering two Fibonacci vectors, two Lucas vectors and one of each vector. We give some theorems for the mentioned vector products and then we give the conditions for such vectors to be perpendicular or parallel. We also introduce the area formulas for the parallelograms constructed by Fibonacci and Lucas vectors with respect to Fibonacci and Lucas numbers. Moreover, we determine some formulas for the cosine and sine functions of the angles between two Fibonacci vectors, two Lucas vectors and lastly a Fibonacci vector and a Lucas vector. Finally, we investigate the Fibonacci quaternions and Lucas quaternions. We give some corollaries regarding the quaternion products of two Fibonacci quaternions, two Lucas quaternions and one of each quaternion. We conclude with the result that the quaternion product of such quaternions is neither a Fibonacci quaternion nor a Lucas quaternion.
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