Öz
In this paper, tangent, principal normal and binormal wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal and rectifying plane of its partner, respectively. For each associated curve, a new moving frame and the corresponding curvatures are formulated in terms of Frenet frame vectors. In addition to this, the possible solutions for distance functions between the curve and its associated mate are discussed. In particular, it is seen that the involute curves belong to the family of tangent associated curves in general and the Bertrand and the Mannheim curves belong to the principal normal associated curves. Finally, as an application, we present some examples and map a given curve together with its partner and its corresponding moving frame.
Anahtar Kelimeler
Curve theory special curves associated curves Frenet frame Euclidean space
Destekleyen Kurum
None
Proje Numarası
None
Teşekkür
Thank you in advance for your valuable time spent on our manuscript.
Kaynakça
- Bertrand, J., Memoire sur la th´eorie des courbes `a double courbure, Journal de Mathematiques Pures et Appliquees 15 (1850), 332–350.
- Mannheim, A., De l’emploi de la courbe repr´esentative de la surface des normales principales d’une courbe gauche pour la d´emonstration de propri´et´es relatives `a cette courbure, C.R. Comptes Rendus des S´eances de l’Acad´emie des Sciences, 86 (1878), 1254-1256.
- O’Neill, B., Elementary differential geometry, Academic Press Inc., New York, 1966.
- Liu, H., Wang, F., Mannheim partner curves in 3-space, Journal of Geometry 88(1-2) (2008), 120-126. https://doi.org/10.1007/s00022-007-1949-0
- Menninger, T., Characterization of the slant helix as successor curve of the general helix, International Electronic Journal of Geometry, 7(2) (2014), 84-91. https://doi.org/10.36890/iejg.593986
- Kazaz, M., Uğurlu, H. H., Önder, M., Oral, S., Bertrand partner D- curves in the Euclidean 3-space $E^3$, Afyon Kocatepe University Journal of Science and Engineering, 16(1) (2016), 76-83. https://doi.org/10.5578/fmbd.25270
- Kaya, O., Önder, M., New partner curves in the Euclidean 3-space, International Journal of Geometry 6(2) (2017), 41-50.
- Kaya, O., Önder, M., C-partner curves and their applications, Differential Geometry-Dynamical Systems 19 (2017), 64-74.
- Körpınar, T., Sarıaydın, M. T., Turhan, E., Associated curves according to Bishop frame in Euclidean 3 space, Advanced Modeling and Optimization, 15(3) (2013), 713-717.
- Masal, M., Azak, A. Z., Mannheim B-curves in the Euclidean 3-space $E^3$, Kuwait Journal of Science, 44(1) (2017), 36-41.
- Yılmaz, B., Has, A., Alternative partner curves in the Euclidean 3-space, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1) (2020), 900-909. https://doi.org/10.31801/cfsuasmas.538177
- Choi, J. H., Kim, Y. H., Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, 218(18) (2012), 9116-9124. https://doi.org/10.1016/j.amc.2012.02.064
- Şahiner, B., Direction curves of principal normal indicatrix of a curve, Journal of Technical Sciences, 8(2) (2018), 46-54.
- Şahiner, B., Direction curves of tangent indicatrix of a curve, Applied Mathematics and Computation, 343 (2019), 273-284. https://doi.org/10.1016/j.amc.2018.09.021
Öz
Proje Numarası
None
Kaynakça
- Bertrand, J., Memoire sur la th´eorie des courbes `a double courbure, Journal de Mathematiques Pures et Appliquees 15 (1850), 332–350.
- Mannheim, A., De l’emploi de la courbe repr´esentative de la surface des normales principales d’une courbe gauche pour la d´emonstration de propri´et´es relatives `a cette courbure, C.R. Comptes Rendus des S´eances de l’Acad´emie des Sciences, 86 (1878), 1254-1256.
- O’Neill, B., Elementary differential geometry, Academic Press Inc., New York, 1966.
- Liu, H., Wang, F., Mannheim partner curves in 3-space, Journal of Geometry 88(1-2) (2008), 120-126. https://doi.org/10.1007/s00022-007-1949-0
- Menninger, T., Characterization of the slant helix as successor curve of the general helix, International Electronic Journal of Geometry, 7(2) (2014), 84-91. https://doi.org/10.36890/iejg.593986
- Kazaz, M., Uğurlu, H. H., Önder, M., Oral, S., Bertrand partner D- curves in the Euclidean 3-space $E^3$, Afyon Kocatepe University Journal of Science and Engineering, 16(1) (2016), 76-83. https://doi.org/10.5578/fmbd.25270
- Kaya, O., Önder, M., New partner curves in the Euclidean 3-space, International Journal of Geometry 6(2) (2017), 41-50.
- Kaya, O., Önder, M., C-partner curves and their applications, Differential Geometry-Dynamical Systems 19 (2017), 64-74.
- Körpınar, T., Sarıaydın, M. T., Turhan, E., Associated curves according to Bishop frame in Euclidean 3 space, Advanced Modeling and Optimization, 15(3) (2013), 713-717.
- Masal, M., Azak, A. Z., Mannheim B-curves in the Euclidean 3-space $E^3$, Kuwait Journal of Science, 44(1) (2017), 36-41.
- Yılmaz, B., Has, A., Alternative partner curves in the Euclidean 3-space, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1) (2020), 900-909. https://doi.org/10.31801/cfsuasmas.538177
- Choi, J. H., Kim, Y. H., Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, 218(18) (2012), 9116-9124. https://doi.org/10.1016/j.amc.2012.02.064
- Şahiner, B., Direction curves of principal normal indicatrix of a curve, Journal of Technical Sciences, 8(2) (2018), 46-54.
- Şahiner, B., Direction curves of tangent indicatrix of a curve, Applied Mathematics and Computation, 343 (2019), 273-284. https://doi.org/10.1016/j.amc.2018.09.021
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Research Article |
| Yazarlar | |
| Proje Numarası | None |
| Yayımlanma Tarihi | 30 Eylül 2022 |
| Gönderilme Tarihi | 20 Kasım 2021 |
| Kabul Tarihi | 2 Nisan 2022 |
| Yayımlandığı Sayı | Yıl 2022 Cilt: 71 Sayı: 3 |
Kaynak Göster
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.
