Bertrand types of regular curves and Bertrand framed curves in the Euclidean 3-space

Abstract

A Bertrand (respectively, Mannheim) curve is a space curve whose principal normal line is the same as the principal normal (respectively, bi-normal) line of another curve. By definition, another curve is a parallel curve with respect to the direction of the principal normal vector. In this paper, we consider the other cases, that is, a space curve whose tangent (or, principal normal, bi-normal) line is the same as the tangent (or, principal normal, bi-normal) line of another curve, respectively. We say that a Bertrand type curve if there exists such another curve. We clarify that the existence conditions of Bertrand type curves in all cases. There are times when the Bertrand type curve does not exist. On the other hand, since the another curve may have singular points, we also consider curves with singular points. As smooth curves with singular points, it is useful to use the framed curves in the Euclidean space. Then we define and investigate Bertrand framed curves. We also clarify that the existence conditions of the Bertrand framed curves in all cases.