Non-lightlike Helices Associated with Helical Curves, Relatively Normal-Slant Helices and Isophote Curves in Minkowski 3-space

Year 2023, , 107 - 127, 31.07.2023

Onur Kaya

https://doi.org/10.54974/fcmathsci.1246015

Abstract

In this paper, we introduce a new type of non-lightlike general helix that we name non-lightlike
associated helix which is associated with a non-lightlike special surface curve. By using the Darboux frame
of a surface curve, we generate the position vector of a non-lightlike associated helix in parametric form.
We investigate special cases when the non-lightlike surface curve is a helical curve, a relatively normal-slant
helix or an isophote curve. In every case, we obtain the position vector of the non-lightlike associated helix
by solving differential equations and examples are given for the achieved results.

Keywords

Non-lightlike associated helix, non-lightlike isophote curve, non-lightlike relatively normal-slant helix

References

• Ali A.T., Position vectors of spacelike general helices in Minkowski 3-space, Nonlinear Analysis: Theory, Methods & Applications, 73(4), 1118-1126, 2010.
• Ali A.T., Turgut M., Position vectors of a timelike general helices in Minkowski 3-space, Global Journal of Advanced Research on Classical and Modern Geometries, 2(1), 1-10, 2013.
• Barros M., General helices and a theorem of Lancret, Proceedings of the American Mathematical Society, 125(5), 1503-1509, 1997.
• Chouaieb N., Goriely A., Maddocks J.H., Helices, Proceedings of the National Academy of Sciences, 103(25), 9398-9403, 2006.
• Doğan F., Isophote curves on timelike surfaces in Minkowski 3-space, Analele Stiintifice ale Universitatii Alexandru Ioan Cuza din Iasi - Matematica, 63, 133-143, 2017.
• Doğan F., Yaylı Y., Isophote curves on spacelike surfaces in Lorentz–Minkowski space, Asian-European Journal of Mathematics, 14(10), 2150180, 2021.
• Doğan F., Yaylı Y., On isophote curves and their characterizations, Turkish Journal of Mathematics, 39(5), 650-664, 2015.
• El Haimi A., Chahdi A.O., Parametric equations of special curves lying on a regular surface in Euclidean 3-space, Nonlinear Functional Analysis and Applications, 26(2), 225-236, 2021.
• Hananoi S., Ito N., Izumiya S., Spherical Darboux images of curves on surfaces, Beitrage zur Algebra und Geometrie, 56, 575-585, 2015.
• Izumiya S., Takeuchi N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28, 153-163, 2004.
• Kim K.J., Lee I.K., Computing isophotes of surface of revolution and canal surface, Computer Aided Design, 35(3), 215-223, 2003.
• Lambert J.H., Photometria Sive de Mensura et Gradibus Luminis, Colorum et Umbrae, Klett, 1760.
• López R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, International Electronic Journal of Geometry, 7(1), 44-107, 2014.
• Lucas A.A., Lambin P., Diffraction by DNA, carbon nanotubes and other helical nanostructures, Reports on Progress in Physics, 68(5), 1181, 2005.
• Macit N., Düldül M., Relatively normal-slant helices lying on a surface and their characterization, Hacettepe Journal of Mathematics and Statistics, 46(3), 397-408, 2017.
• Önder M., Helices associated to helical curves, relatively normal-slant helices and isophote curves, arXiv:2201.09684, 2022.
• Öztürk U., Hacısalihoğlu H.H., Helices on a surface in Euclidean 3‐space, Celal Bayar University Journal of Science, 13(1), 113-123, 2017.
• Öztürk U., Nes̆ović E., Koç Öztürk E.B., Numerical computing of isophote curves, general helices, and relatively normal‐slant helices in Minkowski 3‐space, Mathematical Methods in the Applied Sciences, 1-15, 2022.
• Poeschl T., Detecting surface irregularities using isophotes, Computer Aided Geometric Design, 1(2), 163-168, 1984.
• Puig-Pey J., Gálvez A., Iglesias A., Helical Curves on Surfaces for Computer Aided Geometric Design and Manufacturing, International Conference on Computational Science and Its Applications, Springer, 2004.
• Sara R., Local Shading Analysis via Isophotes Properties, Ph.D., Johannes Kepler University, Austria, 1994.
• Toledo-Suárez C.D., On the arithmetic of fractal dimension using hyperhelices, Chaos, Solitons & Fractals, 39(1), 342-349, 2009.
• Yadav A., Pal B., On relatively normal-slant helices and isophotic curves, arXiv:2104.13220, 2021.
• Yadav A., Yadav A.K., Relatively normal-slant helices in Minkowski 3-space, arXiv:2201.03933, 2022.
• Yang X., High accuracy approximation of helices by quintic curves, Computer Aided Geometric Design, 20(6), 303-317, 2003.