On parallel p-equidistant ruled surfaces by using mofied orthogonal frame with curvature in E³

In this paper, it is investigated Ruled surfaces according to modified orthogonal frame with curvature in 3-dimensional Euclidean space. Firstly, we give apex angle, pitch and drall of closed ruled surface in E³. Then, it characterized the relationship between these invariant of parallel p-equidistant ruled surfaces.

A new characterization of null scrolls as parallel surfaces

In this paper, we present a theorem which explains the necessary onditions for parallel surfaces of null scrolls to be surfaces again. Then, we give some properties of parallel ruled surfaces such as the frame fields in terms of the original base curve’s frame vectors, the cases of cross products of these frames’ vector fields. Also, we show the parallel ruled surfaces to be developable from a different point of view. Additionally, we obtain a relation between Gauss curvatures and mean curvatures. Finally, we characterize the striction curve of the parallel ruled surface. In the example, we draw a graph for a particular null scroll and its parallel surface.

Characteristic Properties of Type-2 Smarandache Ruled Surfaces According to the Type-2 Bishop Frame in E 3

In this paper, we define and investigate a special kind of ruled surfaces called type-2 Smarandache ruled surfaces related to the type-2 Bishop frame in E 3 . From this point and depending on the type-2 Bishop curvature, we provide the necessary and sufficient conditions that allow these surfaces to be developable in a minimal amount of time. Furthermore, an example is given to clear the results.