Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

Abstract

In this paper, we define the curve r λ = r + λd at a constant distance from the edge of regression on a curve r(s) with arc length parameter s in Galilean 3-space. Here, d is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve r(s) in 3-dimensional Galilean space. We build the Frenet frame {T λ , N λ , B λ } of the constructed curve r λ with respect to two types of the vector d and we indicate the properties related to the curvatures of the curve r λ . Also, for the curve r λ , we give the conditions to be a circular helix. Furthermore, we discuss ruled surfaces of type A generated via the curve r λ and the vector D which is defined as tangent of the curve r λ in 3-dimensional Galilean space. The constructed ruled surfaces also appear in two ways. The first is constructed with the curve r λ (s) = r(s) + λT(s) and the non-isotropic vector D. The second is formed by the curve r λ = r(s) + λ 2 N + λ 3 B and the non-isotropic vector D. We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable. Finally, we provide examples and visuals to back up our research.