Frenet Curves in Euclidean 4-Space

Abstract

In this paper, we study rectifying curves arising through the dilation of unit speed curves on the unit sphere S-3 and conclude that arcs of great circles on S-3 do not dilate to rectifying curves, which develope previously obtained results for rectifying curves in Eucidean spaces. This fact allows us to prove that there exists an associated rectifying curve for each Frenet curve in the Euclidean space E-4 and a result of the fact rectifying curves in the Euclidean space E-4 are ample, indeed as an appication, we present an ordinary differential equation satisfied by the distance function of a Frenet curve in E-4 which alows us to characterize the spherical curves and rectifying curves in E-4. Furthermore, we study ccr-curves in the Euclidean space E-4 which are generalizations of helices in E-3 and show that the property of a helix that its tangent vector field makes a constant angel with a fixed vector (axis of helix) does not go through for a ccr-curve.