CONSTRUCTION OF CHEN CURVES IN EUCLIDEAN SPACES
In this paper, first we study dilation of tangent, normal and bi-normal spherical images of a unit speed Frenet curve in the Euclidean space by a positive dilation factor and find necessary and sufficient conditions on the dilation factor so that these dilated spherical images are Chen curves (rectifying curves). It turns out that dilation of tangent spherical image and binormal spherical images by a positive dilation factor are Chen curves for any unit speed Frenet curve in However, for a dilated normal spherical image of a unit speed Frenet curve to be a Chen curve requires that Frenet curve should be nonhelical. Similarly, we find necessary and sufficient conditions for a dilation factor for the dilation of tangent, normal, first binormal and second binormal spherical images of a unit speed Frenet curve in the Euclidean space to be a Chen curve. It turns out that they all are Chen curves for any unit speed Frenet curve in the Euclidean space and as a consequence, we conclude that Chen curves in are too many.
Chen curves, dilation of a curve, spherical images, curvatures.