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

Year 2021, Issue: 37, 45 - 57, 31.12.2021

Ahmet Mollaoğulları , Mehmet Gümüş , Kazım İlarslan , Çetin Camcı

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.

Keywords

PH-curve, helical curve, PH-helical curve, planar curve, polynomial curve

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.