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

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

Keywords:

Bertrand curve, Mannheim curve framed curve, singularity,

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[1] Aminov Y. Differential Geometry and the Topology of Curves. Translated from the Russian by V. Gorkavy. Amsterdam, the Netherlands: Gordon and Breach Science Publishers, 2000.
[2] Banchoff Y, Lovett S. Differential Geometry of Curves and Surfaces. Natick, MA, USA: AK Peters, Ltd., 2010.
[3] Berger M, Gostiaux B. Differential Geometry: Manifolds, Curves, and Surfaces. Translated from the French by Silvio Levy. Graduate Texts in Mathematics, 115. New York, NY, USA 1988.
[4] Bertrand J. Mémoire sur la théorie des courbes à double courbure. J. de methématiques pures et appliquées 1850; (15): 332-350 (in French).
[5] Bishop RL. There is more than one way to frame a curve. American Mathematical Monthly 1975; (82): 246-251.
[6] Bruce JW, Giblin PJ. Curves and Singularities. A Geometrical Introduction to Singularity Theory. 2nd ed. Cambridge, UK: Cambridge University Press, 1992.
[7] Do Carmo MP. Differential Geometry of Curves and Surfaces. Translated from the Portuguese. Englewood Cliffs, NJ, USA: Prentice-Hall, Inc., 1976.
[8] Choi JH, Kang TH, Kim YH. Bertrand curves in 3-dimensional space forms. Applied Mathematics and Computation 2012; (219): 1040-1046.
[9] Fukunaga T, Takahashi M. Evolutes of fronts in the Euclidean plane. Journal of Singularities 2014; (10): 92-107.
[10] Fukunaga T, Takahashi M. Evolutes and involutes of frontals in the Euclidean plane. Demonstratio Mathematica 2015; (48): 147-166. doi: 10.1515/dema-2015-0015
[11] Fukunaga T, Takahashi M. Existence conditions of framed curves for smooth curves. Journal of Geometry 2017; (108): 763-774. doi: 10.1007/s00022-017-0371-5
[12] Honda S, Takahashi M. Framed curves in the Euclidean space. Advances in Geometry 2016; (16): 265-276. doi: 10.1515/advgeom-2015-0035
[13] Honda S, Takahashi M. Evolutes and focal surfaces of framed immersions in the Euclidean space. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 2020; (150): 497-516. doi: 10.1017/prm.2018.84
[14] Huang J, Chen L, Izumiya S, Pei D. Geometry of special curves and surfaces in 3-space form. Journal of Geometry and Physics 2019; (136): 31-38.
[15] Izumiya S, Takeuchi N. Generic properties of helices and Bertrand curves. Journal of Geometry 2002; (74): 97-109. [16] Kühnel W. Differential geometry. Curves-surfaces-manifolds. Translated from the 1999 German original by Bruce Hunt. Student Mathematical Library, 16. Providence, RI, USA: American Mathematical Society, 2002.
[17] Liu H, Wang F. Mannheim partner curves in 3-space. Journal of Geometry 2008; (88): 120-126.
[18] Lucas P, Ortega-Yagües JA. A variational characterization and geometric integration for Bertrand curves. Journal of Mathematical Physics 2013; (54): 043508, 12 pp.
[19] Papaioannou SG, Kiritsis D. An application of Bertrand curves and surfaces to CADCAM. Computer-Aided Design 1985; (17): 348-352.
[20] Struik DJ. Lectures on classical differential geometry. Reprint of the second edition. New York, NY, USA: Dover Publications, Inc., 1988.
[21] Takahashi M. Legendre curves in the unit spherical bundle over the unit sphere and evolutes. Contemporary Mathematics 2016; (675): 337-355. doi: 10.1090/conm/675/13600