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Characterizing the developable surfaces with curves whose position vectors lie in the planes spanned by Darboux frame

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Abstract

Curves on surface are called osculating Darboux curve and rectifying Darboux curve if its position vectors always lies in the osculating Darboux plane and rectifying Darboux plane which are spanned by Darboux frame. We study the case that osculating Darboux curve and rectifying Darboux curve are geodesic, line of curvature or asymptotic curve. In this case, we determine each curve is related to rectifying curve, spherical curve and planer curve. Using this result, we also give the classification of a special developable surface under the condition of the existence of osculating Darboux curve as a geodesic or a line of curvature. As a result, in this case, we determine these developable surfaces become conical surfaces.

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References

  1. Altunkaya, B., Kahraman, F., Kula, L., Aytekin, C.: On rectifying slant helices in Euclidean 3-space. Konuralp J. Math. 4, 17-24 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Camci, Ç., Kula, L., İlarslan, K.: Characterizations of the position vector of a surface curve in Euclidean 3-space. An. Şt. Univ. Ovidius Constanţa. 19(3), 59-70 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Chen, B.Y.: Rectifying curves and geodesics on a cone in the Euclidean 3-space. Tamkang J Math. 48, 209-214 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, B.Y.: When does the position vector of a space curve always lie in its rectifying plane? The American Mathematical Monthly. 110, 147-152 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, B.Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bull Inst Math Acad Sinica. 33, 77-90 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Deshmukh, S., Chen, B.Y., Alshammari, S.H.: On rectifying curves in Euclidean 3-space. Turk J Math. 42, 609-620 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hananoi, S., Izumiya, S.: Normal developable surfaces of surfaces along curves. Proc. Roy. Soc. Edinburgh Sect. A. 147(1), 177-203 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hananoi, S., Ito, N., Izumiya, S.: Spherical Darboux images of curves on surfaces. Beitr. Algebra Geom. 56, 575-585 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Izumiya, S.: Generating families of developable surfaces in \(\mathbb{R}^3\). Hokkaido University preprint series in mathematics. 512, 1-18 (2001)

  10. Izumiya, S., Otani, S.: Flat Approximations of surfaces along curves. Demonstr Math. 48(2), 217-241 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Izumiya, S., Takeuchi, N.: Generic properties of helices and Bertrand curves. J. Geom. 74, 97-109 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk J Math. 28, 153-163 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Izumiya, S., Takeuchi, N.: Special Curves And Ruled Surface. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry. 44, 203-212 (2003)

  14. Izumiya, S., Katsumi, H., Yamasaki, T.: The rectifying developable and the spherical Darboux image of a space curve. Geometry and topology of caustics-Caustics’98-Banach Center Publications. 50, 137-149 (1999)

  15. Izumiya, S., Saji, K., Takeuchi, N.: Flat surfaces along cuspidal edges. Journal of Singularities. 16, 73-100 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Shaikh, A.A., Ghosh, P.R.: Curves on a Smooth Surface with Position Vectors Lie in the Tangent Plane. Indian J Pure Appl Math. 51, 1097-1104 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Spivak, M.: A Comprehensive Introduction to Differential Geometry. vol.3 Second Edition, Publish Or Perish (1979)

  18. Takahashi, T.: Curves always lie in the plane spanned by Darboux frame. Rend. Circ. Mat. Palermo, II. Ser. 70, 1083-1098 (2021)

  19. Yayli, Y., Gok, I., Hacisalihoglu, HH.: Extended rectifying curves as new kind of modified Darboux vectors. TWMS J Pure Appl Math. 9, 18-31 (2018)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai Tsukuba, Ibaraki, 305-8571, Japan

    Takeshi Takahashi

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  1. Takeshi Takahashi
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Correspondence to Takeshi Takahashi.

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Communicated by Indranil Biswas.

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Takahashi, T. Characterizing the developable surfaces with curves whose position vectors lie in the planes spanned by Darboux frame. Indian J Pure Appl Math 54, 456–466 (2023). https://doi.org/10.1007/s13226-022-00267-0

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  • DOI: https://doi.org/10.1007/s13226-022-00267-0

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