In this paper, Cyclic surfaces are introduced using the foliation of circles of curvature of a space curve. The conditions on a space curve such that these cyclic surfaces are of type Weingarten surfaces or HK-quadric surfaces are obtained. Finally, some examples are given and plotted.

In this paper, we study the surfaces foliated by ellipses in three dimensional Euclidean space E-3. We prove the following results: (1) The surface foliated by an ellipse have constant Gaussian curvature K if and only if the surface is flat, i.e. K = 0. (2) The surface foliated by an ellipse is a flat if and only if it is a part of generalized cylinder or part of generalized cone.

In this paper, motion of Darboux vector on two different space curves in Euclidean 3-space is investigated. The structure of the motion is based on ruled surfaces generated by Darboux vector ~ ω. According to this, developability of the considered ruled surfaces are studied. An extensive comparison between these developable ruled surfaces is performed. Some relations between the corresponding curves on the developable ruled surfaces are performed. Finally, special correspondence is given and plotted.

In this article, we study skew ruled surfaces by using the geodesic Frenet trihedron of its generator. We obtained some conditions on this surface to ensure that this ruled surface is flat, II-flat, minimal, II-minimal and Weingarten surface. Moreover, the parametric equations of asymptotic and geodesic lines on this ruled surface are determined and illustrated through example using the program of mathematica.

In this paper, a three dimensional surface using equiform motion of a surface of revolution in Euclidean 3-space E3 is generated.The main results obtained in this paper are that the surface foliated by equiform motion of sphere has a zero scaler curvature if the motion of sphere are in parallel planes.Also, the surface foliated by equiform motion of tours has a zero scaler curvature if the motion of torus are in parallel planes.Finally, for some special cases, new examples are constructed and plotted.

In this paper we consider the equiform motion of a sphere in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant. Under this assumption, we prove that $|\mathbf{K}|<2$.

In this paper we present a local study of a cyclic surface in E5 generated by equiform motions of circles. We describe partially such surfaces with some assumption on its curvature. In general, we shall consider the situation that the scalar curvature is locally constant.

In this paper, we study cyclic surfaces in E 5 generated by equiform motions of a circle. The propertiesof this cyclic surfaces up to the first order are discussed. We prove the following new result: A cyclic 2-surfaces in E 5 in general are contained in canal hypersurfaces. Finally we give an example.

In this article, a new type of ruled surfaces in a Lorentz 3-space R13 is obtained by a strictly connected timelike oriented line moving with Frenet’s frame along a spacelike curve. These surfaces are classified into timelike and spacelike surfaces. The well-known theorems due to Bonnet and Chasles in the 3-dimensional Euclidean space are proved for a timelike ruled surface. For developable and maximal timelike ruled surfaces [Minimal Surfaces of the 3-Dimensional Minkowski Space, World Scientific Publishing, Singapore, 1990, p. 344], some results are obtained. The relation between geodesic curvature, normal curvature and the curvature of a general curve on a timelike ruled surface is derived.

In this paper, we investigate motions of the 7-parameter group of equiform transformations with the property that three points move on three cir- cles with axes in one plane. We give an algorithm to flnd the corresponding one-parametric motion. It can be displayed as a curve in the space of motion parameters. As in general there seems to be no global parametrization of this curve, we give a local one up to the second order. An example demonstrates the eficiencyofthepresentedmethod.

In this paper, we present a differential geometric local study of two-parameter spatial motions: we look for points, which (up to the second order) instantaneously move on one-dimensional point paths, further we locally characterise these motions, which move ∞1 points in these way.