Research article

$ C^* $-partner curves with modified adapted frame and their applications

  • Received: 05 April 2022 Revised: 26 July 2022 Accepted: 29 September 2022 Published: 19 October 2022
  • MSC : 53A04, 53A55

  • In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space $ E^3 $, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the "modified adapted frame" denoted by $ \{N^*, C^*, W^*\} $. Then, we have given some characterizations of this new frame. In addition to that, we have defined $ N^* $-slant helices and $ C^* $-slant helices according to $ \{N^*, C^*, W^*\} $. Moreover, we have studied $ C^* $-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between $ C^* $-partner curves and helices, slant helices.

    Citation: Sezai Kızıltuǧ, Tülay Erişir, Gökhan Mumcu, Yusuf Yaylı. $ C^* $-partner curves with modified adapted frame and their applications[J]. AIMS Mathematics, 2023, 8(1): 1345-1359. doi: 10.3934/math.2023067

    Related Papers:

  • In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space $ E^3 $, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the "modified adapted frame" denoted by $ \{N^*, C^*, W^*\} $. Then, we have given some characterizations of this new frame. In addition to that, we have defined $ N^* $-slant helices and $ C^* $-slant helices according to $ \{N^*, C^*, W^*\} $. Moreover, we have studied $ C^* $-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between $ C^* $-partner curves and helices, slant helices.



    加载中


    [1] A. Ali, R. Lopez, Slant helices in Minkowski space $E^3_1$, J. Korean Math. Soc., 480 (2011), 159–167.
    [2] A. Ali, M. Turgut, Some characterizations of slant helices in the Euclidean space $E^n$, Hacet. J. Math. Stat., 39 (2010), 327–336.
    [3] F. Ateş, E. Kocakuşaklı, İ. Gök, Y. Yaylı, A study of the tubular surfaces constructed by the spherical indicatrices in Euclidean 3-space, Turk. J. Math., 42 (2018), 1711–1725. https://doi.org/10.3906/mat-1610-101 doi: 10.3906/mat-1610-101
    [4] J. Burke, Bertrand curves associated with a pair of curves, Math. Mag., 34 (1960), 60–62. https://doi.org/10.1080/0025570X.1960.11975181 doi: 10.1080/0025570X.1960.11975181
    [5] S. Deshmukh, B. Chen, A. Alghanemi, Natural mates of Frenet curves in Euclidean 3-space, Turk.J. Math., 42 (2018), 2826–2840. https://doi.org/10.3906/mat-1712-34 doi: 10.3906/mat-1712-34
    [6] R. Farouki, C. Giannelli, M. L. Sampoli, A. Sestini, Rotation-minimizing osculating frames, Comput. Aided Geom. D., 31 (2014), 27–42. https://doi.org/10.1016/j.cagd.2013.11.003 doi: 10.1016/j.cagd.2013.11.003
    [7] S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math., 28 (2008), 153–163.
    [8] S. Kızıltuǧ, M. Önder, Ö Tarakcı, Bertrand and Mannheim partner D-curves on parallel surfaces, Bol. Soc. Parana. Mat., 35 (2017), 159–169. https://doi.org/10.5269/bspm.v35i2.24309 doi: 10.5269/bspm.v35i2.24309
    [9] L. Kula, N. Ekmekci, Y. Yaylı, K. İlarslan, Characterizations of slant helices in Euclidean 3-space, Turk. J. Math., 34 (2010), 261–273. https://doi.org/10.3906/mat-0809-17 doi: 10.3906/mat-0809-17
    [10] E. Özdamar, H. H. Hacısalihoǧlu, A characterization of inclined curves in Euclidean n-space, Commun. Fac. Sci. Univ. Ankara, 24 (1975) 15–22. https://doi.org/10.1501/Commua1_0000000262 doi: 10.1501/Commua1_0000000262
    [11] D. Struik, Lectures on classical differential geometry, Addison Wesley, NewYork, 1961.
    [12] İ. Gök, Ç. Camcı, H. H. Hacısalihoǧlu, $V_n$ slant helices in Euclidean $n$-space $E^n$, Math. Commun., 14 (2012), 317–329.
    [13] R. Hord, Torsion at an inflection point of a space curve, Am. Math. Mon., 79 (1972), 371–374. https://doi.org/10.2307/2978088 doi: 10.2307/2978088
    [14] S. Izumiya, N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom., 74 (2002), 97–109. https://doi.org/10.1007/PL00012543 doi: 10.1007/PL00012543
    [15] O. Kaya, M. Önder, C-partner curves and their applications, Differ. Geom. Dyn. Syst., 19 (2017), 64–74.
    [16] S. Kızıltuǧ, S. Yurttançıkmaz, Ö Tarakcı, The slant helices according to type-2 Bishop frame in Euclidean 3-space, Int. J. Pure Appl. Math., 85 (2013), 211–222. https://doi.org/10.12732/ijpam.v85i2.3 doi: 10.12732/ijpam.v85i2.3
    [17] L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput., 169 (2005), 600–607. https://doi.org/10.1016/j.amc.2004.09.078 doi: 10.1016/j.amc.2004.09.078
    [18] M. Lancret, Memoire surless courbes a double courbure, Memoirespresentes Inst., 1806,416–454.
    [19] M. Lone, H. Es, M. K. Karacan, B. Bükcü, On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. Technol., 43 (2019), 1905–1916. https://doi.org/10.1007/s40995-018-0661-2 doi: 10.1007/s40995-018-0661-2
    [20] P. Lucas, J. Ortega-Yagües, Slant helices in three dimensional sphere, J. Korean Math. Soc., 54 (2017), 1331–1343.
    [21] M. Önder, M. Kazaz, H. Kocayiǧit, O. Kılıç, $B_2$-slant helix in Euclidean 4-space $E^4$, Int. J. Contemp. Math. Sci., 3 (2008), 1433–1440.
    [22] T. Sasai, Geometry of analytic space curves with singularities and regular singularities of differential equations, Funkc. Ekvacioj, 30 (1987), 283–303.
    [23] T. Sasai, The fundamental theorem of analytic space curves and apperent singularities of fuchsian differential equations, Tohoku Math. J., 36 (1984), 17–24. https://doi.org/10.2748/tmj/1178228899 doi: 10.2748/tmj/1178228899
    [24] A. Şenol, E. Zıplar, Y. Yaylı, İ. Gök, A new approach on helices in Euclidean N-space, Math. Commun., 18 (2012), 241–256.
    [25] B. Uzunoǧlu, İ. Gök, Y. Yaylı, A new approach on curves of constant precession, Appl. Math. Comput., 275 (2016), 317–323. https://doi.org/10.1016/j.amc.2015.11.083 doi: 10.1016/j.amc.2015.11.083
    [26] Y. Yaylı, E. Zıplar, On slant helices and general helices in Euclidean n-spaces, Math. Aeterna, 1 (2011), 599–610.
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1446) PDF downloads(124) Cited by(0)

Article outline

Figures and Tables

Figures(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog