An Examination on $\ NP^{\ast }$ Curves in $E^3$

Year 2020, Volume: 12 Issue: 1, 26 - 30, 29.06.2020

Süleyman Şenyurt , Şeyda Kılıçoğlu

Abstract

The evolute and involute curves, Mannheim curves or Bertrand curves are the famous examples of the associated curve pairs. In the view of such information we have defined $ NP^{\ast }$ curve pairs where the principal normal vector of the first curve and the vector $P^{\ast }$  lying on the normal plane of the second curve are linearly dependent. We have called these curve pairs $NP^{\ast }-$ curves. Second curve is named $NP^{\ast }-$ partner curve. Also, while the examination of $NP^{\ast }-$ curves we obtain some relations for the curvatures and Frenet apparatus of the second curve based on the Frenet apparatus of the first curve.

Keywords

Darboux vector , Offset curves , Mannheim curves , Bertrand pairs

References

• Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 205, 1997.
• Hacısalihog˘lu, H.H., Diferensiyel Geometri, Cilt 1, İnönü Üniversitesi Yayinlari, Malatya, 1994.
• Lipschutz, M.M., Diferential Geometry, Schaum’s Outlines.
• Liu, H., Wang F., Mannheim partner curves in 3-space, Journal of Geometry, 88(1)(2008), 120-126.
• Schief, W.K., On the integrability of Bertrand curves and Razzaboni surfaces, Journal of Geometry and Physics, 45(1–2)(2003), 130–150.