Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean

Abstract

A Bertrand curve is a space curve whose principal normal line is the same as the principal normal line ofanother curve. On the other hand, a Mannheim curve is a space curve whose principal normal line is the same as thebinormal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of theprincipal normal vector. Even if that is the regular case, the existence conditions of the Bertrand and Mannheim curvesseem to be wrong in some previous research. Moreover, parallel curves may have singular points. As smooth curveswith singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand andMannheim curves of framed curves. We clarify that the Bertrand and Mannheim curves of framed curves are dependenton the moving frame.