Spherical circles and constant angle surfaces
DOI:
https://doi.org/10.31349/RevMexFis.69.041301Keywords:
Constant angle surface; spherical circle; isophote curve; optic; singularityAbstract
In this present paper, we obtain a general version of constant angle surfaces constructed concerning any direction in three dimensional Euclidean space. This constant angle surface is the special case of developable ruled surfaces whose direction is a spherical circle. Here, we obtain the constant angle surfaces by taking the circles (small circles) whose radius is less than the radius of the sphere, as the base curve. Also, the relationship between the isophote curve and this surface and its physical interpretation is mentioned. When we beam from a light source in a constant direction, the intensity of the light will be the same at every point on this constant angle surface. This study is very important in terms of associating optics, a branch of physics, with geometry, a branch of mathematics. Finally, we classify the singular points of these constant angle surfaces.
References
P. Cermelli, A. J. Di Scala, Constant angle surfaces in liquid crystals, Philos. Mag., 87 (2007) 1871-1888
M. I. Munteanu and A. I. Nistor, A new approach on constant angle surfaces in E 3 , Turkish J. Math., 33 (2009) 1-10
A. I. Nistor, Certain constant angle surfaces constructed on curves, International Electronic Journal of Geometry, 4 (2011) 79-87
S. Özkaldi and Y. Yayli, Constant angle surfaces and curves in E 3 , International Electronic Journal of Geometry, 4 (2011) 70-78
A. T. Ali, A constant angle ruled surfaces, International Journal of Geometry, 7 (2018) 69-80
C. Y. Li, C. G. Zhu, Construction of the spacelike constant angle surface family in Minkowski 3−space, AIMS Mathematics, 5 (2020) 6341-6354
S. Özkaldı Karakus, Certain constant angle surfaces constructed on curves in Minkowski 3−Space, International Electronic Journal of Geometry, 11 (2018) 37-47
R. Lopez, M. I. Munteanu, Constant angle surfaces in Minkowski space, Bulletin of the Belgian Mathematical Society-Simon Stevin, 18 (2011) 271-286
F. Dillen, J. Fastenakels, J. Van de Veken , L. Vrancken, Constant angle surfaces in S 2 × R, Monatshefte fur Mathematik, 152 (2007) 89-96
S. Özkaldı Karakus, Quaternionic approach on constant angle surfaces in S 2 × R, Applied Mathematics E-Notes, 19 (2019) 497-506
F. Dillen, M. I. Munteanu, Constant angle surfaces in H 2 × R, Bulletin of the Brazilian Mathematical Society, 40 (2009) 85- 97
J. Fastenakels, M. I. Munteanu and J. Van Der Veken, Constant angle surfaces in the Heisenberg group, Acta Mathematica Sinica, English Series, 27 (2011) 747-756
I. I. Onnis and P. Piu, Constant angle surfaces in the Lorentzian Heisenberg group, Archiv der Mathematik, 109 (2017) 575- 589
F. Doğan and Y. Yayli, On isophote curves and their characterizations, Turkish Journal of Mathematics, 39 (2015) 650-664
C. E. Ordoñez, E. Blotta and J. I. Pastore, Isophote based low computing power eye detection embedded system, IEEE Latin America Transactions, 18 (2020) 336-343
S. Datta, N. Chaki, B. Modak, A novel technique to detect caries lesion using isophote concepts, IRBM, 40 (2019) 174- 182
T. Korpinar, R. C. Demirkol, Z. Korpinar, Polarization of propagated light with optical solitons along the fiber in de-sitter space S 2 1 , Optik-International Journal for Light and Electron Optics, 226 (2021) 165872
T. Korpinar, R.C. Demirkol, Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D Riemannian manifold with Bishop equations, Optik-International Journal for Light and Electron Optics, 200 (2020) 163334
Z. Özdemir, A new calculus for the treatment of Rytov’s law in the optical fiber, Optik-International Journal for Light and Electron Optics, 216 (2020) 164892
B. Yilmaz, A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus, Optik-International Journal for Light and Electron Optics, 247 (2021) 168026
Z. Özdemir, F. N. Ekmekci, Electromagnetic curves and Rytov curves based on the hyperbolic split quaternion algebra, OptikInternational Journal for Light and Electron Optics, 251 (2022) 168359
D. J. Struik, Lectures on classical differential geometry (Massachusetts), (Addison-Wesley Publishing company, Inc, 1961)
S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk J Math, 28 (2004) 153-163
Hacısalihoglu H.H. Diferensiyel Geometri, (Inönü üniversitesi Fen Edebiyat Fakultesi Yayınları, 1983)
A. Sabuncuoğlu, Diferensiyel Geometri, (Nobel Basımevi, Ankara, 2004)
M. Özdemir, Diferansiyel Geometri, (Altı Nokta Yayınevi, Izmir, 2020)
M. P. Do Carmo, Differential Geometry of Curves and Surfaces, (Prentice-Hall, Englewood Cliffs., New Jersey, 1976)
S. Izumiya, N. Takeuchi, Singularities of ruled surfaces in R 3, In Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 130 (2001) 1-11
S. Izumiya, Generating families of developable surfaces in R 3, Hokkaido University preprint series in mathematics, 512 (2001) 1-18. Rev. Mex. Fis. 69 0413
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