A Characterization of Involutes and Evolutes of a Given Curve in E^n
0
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
The orthogonal trajectories of the rst tangents of the curve are called theinvolutes of x. The hyperspheres which have higher order contact with a curve x areknown osculating hyperspheres of x. The centers of osculating hyperspheres form a curvewhich is called generalized evolute of the given curve x in n-dimensional Euclidean space E^n. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space E^n. Further, weobtain some results on these type of curves in E^3 and E^4, respectively.
Abstract
The orthogonal trajectories of the rst tangents of the curve are called theinvolutes of x. The hyperspheres which have higher order contact with a curve x areknown osculating hyperspheres of x. The centers of osculating hyperspheres form a curvewhich is called generalized evolute of the given curve x in n-dimensional Euclidean space E^n. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space E^n. Further, weobtain some results on these type of curves in E^3 and E^4, respectively.
참고문헌 (18)
- J. W. Bruce, "Curves and singularities: a geometrical introduction to singularity theory" (1992), Cambridge University Press
- B. Divjak, "Involutes and evolutes in n-dimensional simply isotropic space <TEX>${\mathds{I}}_n^{(1)}$<", TEX>, 1999, pp.23(1), 71~79
- T. Fukunaga, "Evolutes of fronts in the Euclidean plane", J. Singul, 10, pp.92~107 (2014)
- T. Fukunaga, "Involutes of fronts in the Euclidean plane", Beitr. Algebra Geom, 57, pp.637~653 (2016)[DOI]
- H. Gluck, "Higher curvatures of curves in Euclidean space", Amer. Math. Monthly, 73, pp.699~704 (1966)[DOI]
- G. P. Henderson, "Parallel curves", Canadian J. Math, 6, pp.99~107 (1954)[DOI]
- C. Huygens, "Horologium oscillatorium sive de motu pendulorum ad horologia aptato, Demonstrationes Geometricae"
- B. Kilic, "Tangentially cubic curves in Euclidean spaces", Differ. Geom. Dyn. Syst, 10, pp.186~196 (2008)
- F. Klein, "Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren linearen Transformationen in sich ubergehen", Math. Ann, 4, pp.50~84 (1871)[DOI]
- Z. Milin-Sipus, "Curves in n-dimensional k-isotropic space", Glas. Mat. Ser. III, 33(53), pp.267~286 (1998)
- J. Monterde, "Curves with constant curvature ratios", Bol. Soc. Mat. Mexicana (3), 13(1), pp.177~186 (2007)
- G. Ozturk, "A characterization of ccr-curves in <TEX>${\mathbb{R}}^m$<", TEX>, 2008, pp.57(4), 217~224
- E. Ozyimaz, "Involute-Evolute curve couples in the Euclidean 4-space", Int. J. Open Probl. Comput. Sci. Math, 2(2), pp.168~174 (2009)
- M. C. Romero-Fuster, "Generalized evolutes, vertices and conformal invariants of curves in <TEX>${\mathbb{R}}^{n+1}$<", TEX>, 1999, pp.10, 297~305
- E. Salkowski, "Zur transformation von raumkurven", Math. Ann, 66(4), pp.517~557 (1909)[DOI]
- M. Turgut, "Some characterizations of special curves in the Euclidean space <TEX>${\mathbb{E}}^4$<", TEX>, 2010, pp.2(1), 111~122
- R. Uribe-Vargas, "On singularites, "perestroikas" and differential geometry of space curves", Enseign. Math, 50, pp.69~101 (2004)
- R. Uribe-Vargas, "On vertices, focal curvatures and differential geometry of space curves", Bull Braz. Math. Soc, 36, pp.285~307 (2005)[DOI]