open access

https://doi.org/10.14943/89248

https://hdl.handle.net/2115/74203

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

英語

2019-05-20

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1126

1-18

A Bertrand curve is a special class of space curves that the principal normal line of the curve and the principal normal line of another curve are the same. On the other hand, a Mannheim curve is also a special class of space curves that the principal normal line of the curve and the bi-normal line of another curve are the same. By definitions, the other curves are parallel curves to the direction of the principal normal vector. Even if regular cases, the existence conditions of the Bertrand and Mannheim curves seem to be wrong. Moreover, parallel curves may have singular points. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define Bertrand and Mannheim curves of framed curves. Moreover, we clarify the Bertrand and Mannheim curves are depend of the moving frame.

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