Legendrian dualities between spherical indicatrixes of curves and surfaces according to Bishop frame

Volume 9, Issue 5, pp 2875--2887 http://dx.doi.org/10.22436/jnsa.009.05.82

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Authors

Haiming Liu - School of Mathematics, Mudanjiang Normal University, 157011 Mudanjiang, P. R. China. Donghe Pei - School of Mathematics and Statistics, Northeast Normal University, 130024 Changchun, P. R. China.

Abstract

Legendrian dualities between spherical indicatrixes of curves in Euclidean 3-space are investigated by using the theory of Legendrian duality. Moreover, the singularities of the ruled surfaces according to Bishop frame which are deeply related to space curves are classified from the viewpoints of wave fronts. We also give some more detail descriptions on the conditions of those singularities.

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Keywords

• Ruled surfaces
• Bishop frame
• Legendrian dualities
• singularity theory.

MSC

•  32S25
•  53A04
•  53A05.

References

• [1] R. L. Bishop, There is more than one way to frame a curve , Amer. Math. Monthly, 82 (1975), 246-251.


• [2] J. W. Bruce, P. J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge (1992)


• [3] B. Bükcü, M. K. Karacan, Special Bishop motion and Bishop Darboux rotation axis of the space curve, J. Dyn. Syst. Geom. Theor., 6 (2008), 27-34.


• [4] L. Chen, Q. Han, D. Pei, W. Sun , The singularities of null surfaces in anti de Sitter 3-space , J. Math. Anal. Appl., 366 (2010), 256-265.


• [5] N. Clauvelin, W. K. Olson, I. Tobias, Characterization of the geometry and topology of DNA pictured as a discrete collection of atoms, J. Chem. Theory Comput., 8 (2012), 1092-1107.


• [6] C. Y. Han , Nonexistence of rational rotation-minimizing frames on cubic curves , Comput. Aided Geom. Design, 25 (2008), 298-304.


• [7] A. J. Hanson, H. Ma, Parallel transport approach to curve framing, Indiana Univ., 11 (1995), 3-7.


• [8] M. K. Karacan, B. Bükcü, An alternative moving frame for tubular surfaces around timelike curves in the Minkowski 3-space , Balkan J. Geom. Appl., 12 (2007), 73-80.


• [9] Ş. Kiliçouğlu, H. Hacaısalihoğlu, On the ruled surfaces whose frame is the Bishop frame in the Euclidean 3-space , Int. Electron. J. Geom., 6 (2013), 110-117.


• [10] T. Körpinar, E. Turhan, On characterization of B-canal surfaces in terms of biharmonic B-slant helices according to Bishop frame in Heisenberg group Heis3, J. Math. Anal. Appl., 382 (2011), 57-65.


• [11] H. Liu, D. Pei , Singularities of a space curve according to the relatively parallel adapted frame and its visualization, Math. Probl. Eng., 2013 (2013), 12 pages.


• [12] H. Liu, D. Pei, Cusps of Bishop spherical indicatrixes and their visualizations , Math. Probl. Eng., 2013 (2013), 11 pages.


• [13] H. Liu, D. Pei , Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter 3-space, J. Nonlinear Sci. Appl., 9 (2016), 2563-2576.


• [14] T. Nagai , The Gauss map of a hyper surface in Euclidean sphere and the spherical Legendrian duality, Topology Appl., 159 (2012), 545-554.


• [15] D. Pei, T. Sano, The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-space, Tokyo J. Math., 23 (2000), 211-225.


• [16] K. Shoeemake, Animating rotation with quaternion curves, Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques, 19 (1985), 245-254.


• [17] D. Ünal, I. Kisi, M. Tosun, Spinor Bishop equations of curves in Euclidean 3-space, Adv. Appl. Clifford Algebr., 23 (2013), 757-765.


• [18] G. Wassermann, Stability of caustics , Math. Ann., 216 (1975), 43-50.


• [19] S. Yilmaz, M. Turgut , A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371 (2010), 764-776.


• [20] N. Yüksel , The ruled surfaces according to Bishop frame in Minkowski 3-space, Abstr. Appl. Anal., 2013 (2013), 5 pages.