Research Article
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A New Method to Obtain PH-Helical Curves in E^(n+1)

Year 2021, Issue: 37, 45 - 57, 31.12.2021
https://doi.org/10.53570/jnt.1027564

Abstract

Helical curves are constructed by the property that their unit tangents make a constant angle with a chosen constant direction. There are relations between polynomial planar curves, helices and Pythagorean-hodograph or shortly PH-curves. The aim of this paper is to obtain a method which generate PH-curves and PH-helical curves from a planar curve in Euclidean Space E^(n+1). Furthermore, some examples are given in E^4 and E^5 to explain the method neatly.

Project Number

FHD-2020-3452

References

  • M. A. Lancret, Mémoire sur la théorie des courbes à double courbure, Mémoires présentés ‘a l’Institut des Sciences, Letters et arts par divers savants, Tome 1(1802) 416–454.
  • W. Kuhnel, Differential Geometry: Curves-Surfaces-Manifolds, Braunchweig, Friedr. Vieweg & Sohn, 1999.
  • R. T. Farouki, T. Sakkalis, Pythagorean Hodographs, IBM Journal of Research and Development 34(5) (1999) 736–752.
  • R. T. Farouki, T. Sakkalis, Pythagorean-Hodograph Space Curves, Advances in Computational Mathematics 2(1) (1994) 41–66.
  • R. T. Farouki, C. Y. Han., C. Manni, A. Sestini, Characterization and Construction of Helical Polynomial Space Curves, Journal of Computational and Applied Mathematics 162(2) (2004) 365–392.
  • R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Springer, Berlin, Heidelberg, 2008.
  • R. T. Farouki, C. Giannelli, A. Sestini, Helical Polynomial Curves and Double Pythagorean Hodographs I Quaternion and Hopf map representations, Journal of Symbolic Computation 44(2) (2009) 161–179.
  • R. T. Farouki, C. Giannelli, A. Sestini, Helical Polynomial Curves and Double Pythagorean Hodographs II. Enumeration of Low-Degree Durves, Journal of Symbolic Computation 44(4) (2009) 307–332.
  • R. T. Farouki, M. Al-Kandari, T. Sakkalis, Structural Invariance of Spatial Pythagorean Hodographs, Computer Aided Geometric Design 19(6) (2002) 395–407.
  • R. T. Farouki, Z. Šír, Rational Pythagorean-Hodograph Space Curves, Computer Aided Geometric Design 28(2) (2011) 78–88.
  • H. Pottman, Curve Design with Rational Pythagorean-Hodograph Curves, Advances in Computational Mathematics 3 (1995) 147–170.
  • S. Izumiya, N. Takeuchi, Generic Properties of Helices and Bertrand Curves, Journal of Geometry 74 (2002) 97–109.
  • Ç. Camcı K. İlarslan, A New Method for Construction of PH-Helical Curves in E^3, Comptes Rendus De L Academıe Bulgare Des Scıences, 72(3) (2019) 301–308.
  • H. Gluck, Higher Curvatures of Curves in Euclidean Space, The American Mathematical Monthly 37 (1966) 699–704.
  • M. C. Romero-Fuster, E. Sanabria-Codesal, Generalized Helices Twistings and Flattenings of Curves in n-Space, Matematica Contemporanea 17 (1999) 267–280.
  • E. Ödamar, H. H. Hacısalihoğlu, A Characterization of Inclined Curves in Euclidean n-Space, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics 24 (1975) 15–23.
  • Ç. Camcı K. İlarslan, L. Kula, H. H. Hacısalihoğlu, Harmonic Curvature and General Helices, Chaos Solitons Fractals 40 (2009) 2590–2596.
  • K. K. Kubota, Pythagorean Triples in Unique Factorization Domains, The American Mathematical Monthly 79 (1972) 503–505.
Year 2021, Issue: 37, 45 - 57, 31.12.2021
https://doi.org/10.53570/jnt.1027564

Abstract

Supporting Institution

Çanakkale Onsekiz Mart Üniversitesi Bilimsel Araştırma Projeleri Koordinasyonu

Project Number

FHD-2020-3452

References

  • M. A. Lancret, Mémoire sur la théorie des courbes à double courbure, Mémoires présentés ‘a l’Institut des Sciences, Letters et arts par divers savants, Tome 1(1802) 416–454.
  • W. Kuhnel, Differential Geometry: Curves-Surfaces-Manifolds, Braunchweig, Friedr. Vieweg & Sohn, 1999.
  • R. T. Farouki, T. Sakkalis, Pythagorean Hodographs, IBM Journal of Research and Development 34(5) (1999) 736–752.
  • R. T. Farouki, T. Sakkalis, Pythagorean-Hodograph Space Curves, Advances in Computational Mathematics 2(1) (1994) 41–66.
  • R. T. Farouki, C. Y. Han., C. Manni, A. Sestini, Characterization and Construction of Helical Polynomial Space Curves, Journal of Computational and Applied Mathematics 162(2) (2004) 365–392.
  • R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Springer, Berlin, Heidelberg, 2008.
  • R. T. Farouki, C. Giannelli, A. Sestini, Helical Polynomial Curves and Double Pythagorean Hodographs I Quaternion and Hopf map representations, Journal of Symbolic Computation 44(2) (2009) 161–179.
  • R. T. Farouki, C. Giannelli, A. Sestini, Helical Polynomial Curves and Double Pythagorean Hodographs II. Enumeration of Low-Degree Durves, Journal of Symbolic Computation 44(4) (2009) 307–332.
  • R. T. Farouki, M. Al-Kandari, T. Sakkalis, Structural Invariance of Spatial Pythagorean Hodographs, Computer Aided Geometric Design 19(6) (2002) 395–407.
  • R. T. Farouki, Z. Šír, Rational Pythagorean-Hodograph Space Curves, Computer Aided Geometric Design 28(2) (2011) 78–88.
  • H. Pottman, Curve Design with Rational Pythagorean-Hodograph Curves, Advances in Computational Mathematics 3 (1995) 147–170.
  • S. Izumiya, N. Takeuchi, Generic Properties of Helices and Bertrand Curves, Journal of Geometry 74 (2002) 97–109.
  • Ç. Camcı K. İlarslan, A New Method for Construction of PH-Helical Curves in E^3, Comptes Rendus De L Academıe Bulgare Des Scıences, 72(3) (2019) 301–308.
  • H. Gluck, Higher Curvatures of Curves in Euclidean Space, The American Mathematical Monthly 37 (1966) 699–704.
  • M. C. Romero-Fuster, E. Sanabria-Codesal, Generalized Helices Twistings and Flattenings of Curves in n-Space, Matematica Contemporanea 17 (1999) 267–280.
  • E. Ödamar, H. H. Hacısalihoğlu, A Characterization of Inclined Curves in Euclidean n-Space, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics 24 (1975) 15–23.
  • Ç. Camcı K. İlarslan, L. Kula, H. H. Hacısalihoğlu, Harmonic Curvature and General Helices, Chaos Solitons Fractals 40 (2009) 2590–2596.
  • K. K. Kubota, Pythagorean Triples in Unique Factorization Domains, The American Mathematical Monthly 79 (1972) 503–505.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Ahmet Mollaoğulları 0000-0002-5769-9831

Mehmet Gümüş 0000-0003-4608-2446

Kazım İlarslan 0000-0003-1708-280X

Çetin Camcı 0000-0002-0122-559X

Project Number FHD-2020-3452
Publication Date December 31, 2021
Submission Date November 23, 2021
Published in Issue Year 2021 Issue: 37

Cite

APA Mollaoğulları, A., Gümüş, M., İlarslan, K., Camcı, Ç. (2021). A New Method to Obtain PH-Helical Curves in E^(n+1). Journal of New Theory(37), 45-57. https://doi.org/10.53570/jnt.1027564
AMA Mollaoğulları A, Gümüş M, İlarslan K, Camcı Ç. A New Method to Obtain PH-Helical Curves in E^(n+1). JNT. December 2021;(37):45-57. doi:10.53570/jnt.1027564
Chicago Mollaoğulları, Ahmet, Mehmet Gümüş, Kazım İlarslan, and Çetin Camcı. “A New Method to Obtain PH-Helical Curves in E^(n+1)”. Journal of New Theory, no. 37 (December 2021): 45-57. https://doi.org/10.53570/jnt.1027564.
EndNote Mollaoğulları A, Gümüş M, İlarslan K, Camcı Ç (December 1, 2021) A New Method to Obtain PH-Helical Curves in E^(n+1). Journal of New Theory 37 45–57.
IEEE A. Mollaoğulları, M. Gümüş, K. İlarslan, and Ç. Camcı, “A New Method to Obtain PH-Helical Curves in E^(n+1)”, JNT, no. 37, pp. 45–57, December 2021, doi: 10.53570/jnt.1027564.
ISNAD Mollaoğulları, Ahmet et al. “A New Method to Obtain PH-Helical Curves in E^(n+1)”. Journal of New Theory 37 (December 2021), 45-57. https://doi.org/10.53570/jnt.1027564.
JAMA Mollaoğulları A, Gümüş M, İlarslan K, Camcı Ç. A New Method to Obtain PH-Helical Curves in E^(n+1). JNT. 2021;:45–57.
MLA Mollaoğulları, Ahmet et al. “A New Method to Obtain PH-Helical Curves in E^(n+1)”. Journal of New Theory, no. 37, 2021, pp. 45-57, doi:10.53570/jnt.1027564.
Vancouver Mollaoğulları A, Gümüş M, İlarslan K, Camcı Ç. A New Method to Obtain PH-Helical Curves in E^(n+1). JNT. 2021(37):45-57.


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