Abstract
Bonnet yüzeyin, asli eğrilikleri
koruyan en az bir ihmal edilebilir izometriyi kabul edebilen bir yüzey olduğunu
biliyoruz. Bu çalışmada, öncelikle, Öklidyen
üç boyutlu uzayında özel bir süpürme yüzeyi olarak adlandırılan kanal
yüzeylerin hangi şartlarda Bonnet yüzey olabileceğini inceledik. Sonrasında,
Öklidyen üç boyutlu uzayında, Bonnet
kanal yüzeylerini tanımladık ve Bonnet kanal yüzeyler için bazı özel sonuçlar
elde ettik. Kanal yüzeyi oluşturan eğrinin özel bir eğeri olması durumlarında
oluşacak olan Bonnet kanal yüzeyleri inceledik. Ayrıca elde edilen şartları göz
önüne alarak Bonnet kanal yüzey için örnek verdik.
References
- Bonnet, O., 1867. Memoire Sur la Theorie des Surfaces Applicables Sur une Surface Donee, Journal de l’Ecole Polytechnique, Paris, XLII, pp. 72-92.
- Cartan, E., 1942. Sur la Couples de Surfaces Applicables Avec Conservation des Courbures Principales, Bulletin de la Societe Mathematique de France, Volume 66, pp. 55-85.
- Chern, S.S., 1985. Deformation of Surfaces Preserving Principal Curvatures, Differential Geometry and Complex Analysis, H E Rauch Memorial Volume, Springer- Verlag, pp. 155-163.
- Kanbay, F., 2005., Bonnet Ruled Surfaces, Acta Mathematica Sinica, English Series, Volume 21, No 3, pp. 623-630. DOI: 10.1007/s10114-004-0450-z.
- Kim, Y.H., Liu, H., Qian, J., 2016. Some Characterizations of Canal Surfaces, Bulletin of the Korean Mathematical Society, Volume 53, No 2, pp. 461-477. DOI:10.4134/BKMS-2016-53-2-461.
- Maekawa, T., Patrikalakis, N.M., Sakkalis, T., Yu, G., 1998. Analysis and Applications of Pipe Surfaces, Computer Aided Geometric Design, Volume 15, pp. 437-458. DOI:10.1016/S0167-8396(97)-00042-3.
- Xu, Z., Feng, R., Sun, J.G., 2006. Analytic and Algebraic Properties of Canal Surfaces. Journal of Computational and Applied Mathematics, Volume 195, pp. 220-228.
- Dogan, F., Yayli, Y., 2011. On the Curvatures of Tubular Surface with Bishop Frame. Communications Faculty of Science, University of Ankara, Series A1, Volume 60, No 1, pp. 59-69.
- Dogan, F., Yayli, Y., 2017. The Relation Between Parameter Curves and Lines of Curvature on Canal Surfaces, Kuwait Journal of Science, Volume 44, No 1, pp. 29-35.
- Olah-Gol, R., Pal, L., 2009. Some Notes on Drawing Twofolds in 4-Dimensional Euclidean Space, Acta Universitatis Sapientiae, Informatica, Volume 1, No 2, pp. 125-134.
- Soyuçok, Z., 1995. The Problem of Non-Trivial Isometries of Surfaces Preserving Principal Curvatures, Journal of Geometry, Basel, Volume 52, pp. 173-188.
- Eisenhart, L.P., 1960. A Treatise On The Differential Geometry of Curves and Surfaces, Dover Publications, Inc. Newyork 347, pp. 229-231.
- do Carmo, P.M., 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall, USA, 503p.
Abstract
We know that Bonnet surfaces are
surfaces which can admit at least one non-trivial isometry that preserves the
principal curvatures in the Euclidean three-dimensional space. In this study,
firstly, we have examined in which conditions a particular type of swept
surface named canal surfaces may be Bonnet surfaces. Secondly, we have defined
Bonnet canal surfaces in the Euclidean three-dimensional space and have
obtained some special results for the Bonnet canal surfaces. We have studied the Bonnet canal surfaces
which will be formed when the curve generating the canal surface is a special
curve. In addition, we have given an example for Bonnet canal surface by considering
the obtained conditions.
References
- Bonnet, O., 1867. Memoire Sur la Theorie des Surfaces Applicables Sur une Surface Donee, Journal de l’Ecole Polytechnique, Paris, XLII, pp. 72-92.
- Cartan, E., 1942. Sur la Couples de Surfaces Applicables Avec Conservation des Courbures Principales, Bulletin de la Societe Mathematique de France, Volume 66, pp. 55-85.
- Chern, S.S., 1985. Deformation of Surfaces Preserving Principal Curvatures, Differential Geometry and Complex Analysis, H E Rauch Memorial Volume, Springer- Verlag, pp. 155-163.
- Kanbay, F., 2005., Bonnet Ruled Surfaces, Acta Mathematica Sinica, English Series, Volume 21, No 3, pp. 623-630. DOI: 10.1007/s10114-004-0450-z.
- Kim, Y.H., Liu, H., Qian, J., 2016. Some Characterizations of Canal Surfaces, Bulletin of the Korean Mathematical Society, Volume 53, No 2, pp. 461-477. DOI:10.4134/BKMS-2016-53-2-461.
- Maekawa, T., Patrikalakis, N.M., Sakkalis, T., Yu, G., 1998. Analysis and Applications of Pipe Surfaces, Computer Aided Geometric Design, Volume 15, pp. 437-458. DOI:10.1016/S0167-8396(97)-00042-3.
- Xu, Z., Feng, R., Sun, J.G., 2006. Analytic and Algebraic Properties of Canal Surfaces. Journal of Computational and Applied Mathematics, Volume 195, pp. 220-228.
- Dogan, F., Yayli, Y., 2011. On the Curvatures of Tubular Surface with Bishop Frame. Communications Faculty of Science, University of Ankara, Series A1, Volume 60, No 1, pp. 59-69.
- Dogan, F., Yayli, Y., 2017. The Relation Between Parameter Curves and Lines of Curvature on Canal Surfaces, Kuwait Journal of Science, Volume 44, No 1, pp. 29-35.
- Olah-Gol, R., Pal, L., 2009. Some Notes on Drawing Twofolds in 4-Dimensional Euclidean Space, Acta Universitatis Sapientiae, Informatica, Volume 1, No 2, pp. 125-134.
- Soyuçok, Z., 1995. The Problem of Non-Trivial Isometries of Surfaces Preserving Principal Curvatures, Journal of Geometry, Basel, Volume 52, pp. 173-188.
- Eisenhart, L.P., 1960. A Treatise On The Differential Geometry of Curves and Surfaces, Dover Publications, Inc. Newyork 347, pp. 229-231.
- do Carmo, P.M., 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall, USA, 503p.
Details
| Primary Language | English |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | January 15, 2019 |
| Published in Issue | Year 2019 Volume: 21 Issue: 61 |
Cite
Dokuz Eylül Üniversitesi, Mühendislik Fakültesi Dekanlığı Tınaztepe Yerleşkesi, Adatepe Mah. Doğuş Cad. No: 207-I / 35390 Buca-İZMİR.