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