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Translation surfaces generating with some partner curve

Yıl 2024, Cilt: 73 Sayı: 1, 165 - 177, 16.03.2024

Aykut Has Beyhan Yılmaz

https://doi.org/10.31801/cfsuasmas.1299533

Öz

In this article, generating curves of translation surfaces are paired with some special curve pairs. With the results obtained from these pairings, the developable and minimal translation surfaces are characterized. In addition, the surface curvatures of the translation surface are obtained. For a better understanding of the results, examples are given and their drawings are made with the help of Mathematica.

Anahtar Kelimeler

Translation surface Bertrand partner curve Mannheim partner curve involute-evolute curves

Kaynakça

  • Bertrand, J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15 (1850), 332–350.
  • Mannheim, A., De l’emploi de la courbe representative de la surface des normales principales d’une courbe gauche pour la d´emonstration de proprietes relatives a cette courbure, Comptes Rendus des Seances de l’Acad´emie des Sciences, 86 (1878), 1254-1256.
  • Do Cormo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976.
  • Eren, K., Ayvacı, K. H., Şenyurt, S., On ruled surfaces constructed by the evolution of a polynomial space curve, Journal of Science and Arts, 23(1) (2023), 77-90. https://doi.org/10.46939/J.Sci.Arts-23.1-a06
  • Has, A., Yılmaz, B., Yaylı, Y., Constant angle ruled surfaces in E31 , Fundamental Journal of Mathematics and Applications, 6 (2023), 78-88. https://doi.org/10.33401/fujma.1218966
  • Güler, E., Yaylı, Y., Local isometry of the generalized helicoidal surfaces family in 4-space, Malaya Journal of Matematik, 11(2) (2023), 210-218. https://doi.org/10.26637/mjm1102/009
  • Li, Y., Eren, K., Ayvacı, K. H., Ersoy, S., The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Mathematics, 8(1) (2023), 2226-2239. https://doi.org/10.3934/math.2023115
  • Turhan, E., Altay, G., Minimal surfaces in three dimensional Lorentzian Heisenberg group, Beitrage zur Algebra und Geometrie / Contributions to Algebra and Geometry, 55 (2014), 1–23. https://doi.org/10.1007/s13366-013-0156-4
  • Çetin, M., Tunçer, Y., Ekmekçi, N., Translation Surfaces in Euclidean 3-Space, International Journal of Mathematical and Computational Sciences, 5(4) (2011), 49-56.
  • Lopez, R., Perdomo, O., Minimal translation surfaces in Euclidean space, The Journal of Geometric Analysis, 27 (2017), 2926–2937. https://doi.org/10.1007/s12220-017-9788-1
  • Ali, A. T., Abdel Aziz, H. S., Sorour, A. H., On curvatures and points of the translation surfaces in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 23 (2015), 167–172. https://doi.org/10.1016/j.joems.2014.02.007
  • Perez Diaz, S., Shen, L., Parameterization of rational translational surfaces, Theoretical Computer Science, 835 (2020), 156-167. https://doi.org/10.1016/j.tcs.2017.08.005
  • Brauner, H., Lehrbuch der Konstruktiven Geometrie, Springer-Verlag, New York, 2013.
  • Schober, H., Transparente Schalen From Topologie Tragwerk, Wilhelm Ernst and Sohn, Berlin, 2015.
  • Verstraelen, L., Walrave, J., Yaprak, S., The minimal translation surfaces in Euclidean space, Soochow Journal of Mathematics, 20(1) (1994), 77-82.
  • Liu, H. L., Translation surfaces with constant mean curvature in 3-dimensional spaces, Journal of Geometry, 64 (1999), 141-149. https://doi.org/10.1007/BF01229219
  • Yoon, D. W., On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese Journal of Mathematics, 6(3) (2002), 389-398. https://www.jstor.org/stable/43833333
  • Lone, M. S., Karacan, M. K., Tuncer, Y., Es, H., Translation surfaces in affine 3-space, Hacet. J. Math. Stat., 49(6) (2020), 1944–1954. https://doi.org/10.15672/hujms.564921
  • Munteanu, M., Nistor, A. I., On the geometry of the second fundamental form of translation surfaces in E3, Houston J. Math., 37(4) (2011), 1087-1102. https://api.semanticscholar.org/CorpusID:15706402
  • Çetin, M., Kocayiğit, H., Önder, M., Translation surfaces according to Frenet frame in Minkowski 3-space, Int. J. Phsy. Sci., 7(47) (2012), 6135-6143. https://doi.org/10.5897/IJPS12.560
  • Yuan, Y., Liu, H. L., Some new translation surfaces in 3-Minkowski space, Journal of Math. Res. and Exp., 31(6) (2011), 1123-1128.
  • Aydın, M. E., Külahçı, M. A., Öğrenmiş, A. O., Constant curvature translation surfaces in Galilean 3-space, International Electronic Journal of Geometry, 12(1) (2019), 9-19. https://doi.org/10.36890/iejg.545741
  • Struik, D. J., Lectures on Classical Differential Geometry, Dover Publications, New York, 1988.
  • Sabuncuoğlu, A., Diferansiyel Geometri, Nobel Academic Publishing, Ankara, 2014.
  • Yüce, S., Öklid Uzayında Diferansiyel Geometri, Pegem Press, Ankara, 2022.
  • O’Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983.

Yıl 2024, Cilt: 73 Sayı: 1, 165 - 177, 16.03.2024

Aykut Has Beyhan Yılmaz

https://doi.org/10.31801/cfsuasmas.1299533

Öz

Kaynakça

  • Bertrand, J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15 (1850), 332–350.
  • Mannheim, A., De l’emploi de la courbe representative de la surface des normales principales d’une courbe gauche pour la d´emonstration de proprietes relatives a cette courbure, Comptes Rendus des Seances de l’Acad´emie des Sciences, 86 (1878), 1254-1256.
  • Do Cormo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976.
  • Eren, K., Ayvacı, K. H., Şenyurt, S., On ruled surfaces constructed by the evolution of a polynomial space curve, Journal of Science and Arts, 23(1) (2023), 77-90. https://doi.org/10.46939/J.Sci.Arts-23.1-a06
  • Has, A., Yılmaz, B., Yaylı, Y., Constant angle ruled surfaces in E31 , Fundamental Journal of Mathematics and Applications, 6 (2023), 78-88. https://doi.org/10.33401/fujma.1218966
  • Güler, E., Yaylı, Y., Local isometry of the generalized helicoidal surfaces family in 4-space, Malaya Journal of Matematik, 11(2) (2023), 210-218. https://doi.org/10.26637/mjm1102/009
  • Li, Y., Eren, K., Ayvacı, K. H., Ersoy, S., The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Mathematics, 8(1) (2023), 2226-2239. https://doi.org/10.3934/math.2023115
  • Turhan, E., Altay, G., Minimal surfaces in three dimensional Lorentzian Heisenberg group, Beitrage zur Algebra und Geometrie / Contributions to Algebra and Geometry, 55 (2014), 1–23. https://doi.org/10.1007/s13366-013-0156-4
  • Çetin, M., Tunçer, Y., Ekmekçi, N., Translation Surfaces in Euclidean 3-Space, International Journal of Mathematical and Computational Sciences, 5(4) (2011), 49-56.
  • Lopez, R., Perdomo, O., Minimal translation surfaces in Euclidean space, The Journal of Geometric Analysis, 27 (2017), 2926–2937. https://doi.org/10.1007/s12220-017-9788-1
  • Ali, A. T., Abdel Aziz, H. S., Sorour, A. H., On curvatures and points of the translation surfaces in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 23 (2015), 167–172. https://doi.org/10.1016/j.joems.2014.02.007
  • Perez Diaz, S., Shen, L., Parameterization of rational translational surfaces, Theoretical Computer Science, 835 (2020), 156-167. https://doi.org/10.1016/j.tcs.2017.08.005
  • Brauner, H., Lehrbuch der Konstruktiven Geometrie, Springer-Verlag, New York, 2013.
  • Schober, H., Transparente Schalen From Topologie Tragwerk, Wilhelm Ernst and Sohn, Berlin, 2015.
  • Verstraelen, L., Walrave, J., Yaprak, S., The minimal translation surfaces in Euclidean space, Soochow Journal of Mathematics, 20(1) (1994), 77-82.
  • Liu, H. L., Translation surfaces with constant mean curvature in 3-dimensional spaces, Journal of Geometry, 64 (1999), 141-149. https://doi.org/10.1007/BF01229219
  • Yoon, D. W., On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese Journal of Mathematics, 6(3) (2002), 389-398. https://www.jstor.org/stable/43833333
  • Lone, M. S., Karacan, M. K., Tuncer, Y., Es, H., Translation surfaces in affine 3-space, Hacet. J. Math. Stat., 49(6) (2020), 1944–1954. https://doi.org/10.15672/hujms.564921
  • Munteanu, M., Nistor, A. I., On the geometry of the second fundamental form of translation surfaces in E3, Houston J. Math., 37(4) (2011), 1087-1102. https://api.semanticscholar.org/CorpusID:15706402
  • Çetin, M., Kocayiğit, H., Önder, M., Translation surfaces according to Frenet frame in Minkowski 3-space, Int. J. Phsy. Sci., 7(47) (2012), 6135-6143. https://doi.org/10.5897/IJPS12.560
  • Yuan, Y., Liu, H. L., Some new translation surfaces in 3-Minkowski space, Journal of Math. Res. and Exp., 31(6) (2011), 1123-1128.
  • Aydın, M. E., Külahçı, M. A., Öğrenmiş, A. O., Constant curvature translation surfaces in Galilean 3-space, International Electronic Journal of Geometry, 12(1) (2019), 9-19. https://doi.org/10.36890/iejg.545741
  • Struik, D. J., Lectures on Classical Differential Geometry, Dover Publications, New York, 1988.
  • Sabuncuoğlu, A., Diferansiyel Geometri, Nobel Academic Publishing, Ankara, 2014.
  • Yüce, S., Öklid Uzayında Diferansiyel Geometri, Pegem Press, Ankara, 2022.
  • O’Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983.
Toplam 26 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Aykut Has 0000-0003-0658-9365

Beyhan Yılmaz 0000-0002-5091-3487

Yayımlanma Tarihi 16 Mart 2024
Gönderilme Tarihi 19 Mayıs 2023
Kabul Tarihi 23 Ekim 2023
Yayımlandığı Sayı Yıl 2024 Cilt: 73 Sayı: 1

Kaynak Göster

APA Has, A., & Yılmaz, B. (2024). Translation surfaces generating with some partner curve. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(1), 165-177. https://doi.org/10.31801/cfsuasmas.1299533
AMA Has A, Yılmaz B. Translation surfaces generating with some partner curve. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Mart 2024;73(1):165-177. doi:10.31801/cfsuasmas.1299533
Chicago Has, Aykut, ve Beyhan Yılmaz. “Translation Surfaces Generating With Some Partner Curve”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, sy. 1 (Mart 2024): 165-77. https://doi.org/10.31801/cfsuasmas.1299533.
EndNote Has A, Yılmaz B (01 Mart 2024) Translation surfaces generating with some partner curve. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 1 165–177.
IEEE A. Has ve B. Yılmaz, “Translation surfaces generating with some partner curve”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 73, sy. 1, ss. 165–177, 2024, doi: 10.31801/cfsuasmas.1299533.
ISNAD Has, Aykut - Yılmaz, Beyhan. “Translation Surfaces Generating With Some Partner Curve”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/1 (Mart 2024), 165-177. https://doi.org/10.31801/cfsuasmas.1299533.
JAMA Has A, Yılmaz B. Translation surfaces generating with some partner curve. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:165–177.
MLA Has, Aykut ve Beyhan Yılmaz. “Translation Surfaces Generating With Some Partner Curve”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 73, sy. 1, 2024, ss. 165-77, doi:10.31801/cfsuasmas.1299533.
Vancouver Has A, Yılmaz B. Translation surfaces generating with some partner curve. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(1):165-77.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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