Family of Surfaces with a Common Special Involute and Evolute Curves

Year 2022, , 160 - 174, 30.04.2022

Süleyman Şenyurt , Kebire Hilal Ayvacı , Davut Canlı

https://doi.org/10.36890/iejg.932757

Cited By: 1

Abstract

In this paper, we define the necessary and sufficient conditions for a parametric surface on which both the involute and evolute of any given curve lie to be geodesic, asymptotic and curvature line. Then, the first and second fundamental forms of these surfaces are calculated. By using the Gaussian and mean curvatures, the developability and minimality assumptions are drawn, as well.
Moreover we extended the idea to the ruled surfaces. Finally, we provide a set of examples to illustrate the corresponding surfaces.

Keywords

Geodesic curve, involue-evolute curves, asymptoticity, curvature line, ruled surfaces.

Project Number

yok

References

• [1] Atalay, G. Ş., Kasap, E.: Surfaces family with common null asymptotic. Applied Mathematics and Computation, 260, 135-139 (2015).
• [2] Atalay, G. Ş. , Kasap, E.: Surfaces family with common Smarandache asymptotic curve. Boletim da Sociedade Paranaense de Matemática, 34(1), 9-20 (2016).
• [3] Atalay, G. Ş., Kasap, E.: Surfaces family with common Smarandache geodesic curve. Journal of Science and Arts, 17(4), 651-664 (2017).
• [4] Bayram, E., Güler, F., Kasap, E.: Parametric representation of a surface pencil with a common asymptotic curve. Computer-Aided Design, 44(7), 637-643 (2012).
• [5] Bayram, E., Bilici, M.: Surface family with a common involute asymptotic curve. 7. International Journal of Geometric Methods in Modern Physics, 13(5), 8 (2016).
• [6] Millman, R. S., Parker, G. D.: Elements of differential geometry (pp. xiv+-265). Englewood Cliffs, NJ: Prentice-Hall.
• [7] Çalışkan, M., Bilici, M.: Some characterizations for the pair of involute-evolute curves in Euclidean space E3. Bulletin of Pure and Applied Sciences, 21(2), 289-294 (2002).
• [8] Do Carmo, M. P.: Differential geometry of curves and surfaces: revised and updated second edition. Courier Dover Publications (2016).
• [9] Fuchs, D.: Evolutes and involutes of spatial curves. The American Mathematical Monthly, 120(3), 217-231 (2013).
• [10] Li, C. Y., Wang, R. H., Zhu, C. G.: Parametric representation of a surface pencil with a common line of curvature. Computer-Aided Design, 43(9), 1110-1117 (2011).
• [11] O’neill, B.: Elementary differential geometry. Elsevier. (2006).
• [12] Paluszny, M.: Cubic polynomial patches through geodesics. Computer-Aided Design, 40(1), 56-61 (2008).
• [13] Sabuncuoğlu, A.: Diferensiyel Geometri. Nobel Yayın Da˘ gıtım, (2010).
• [14] Sánchez-Reyes, J. and Dorado, R.: Constrained design of polynomial surfaces from geodesic curves. Computer-Aided Design, 40(1), 49-55 (2008).
• [15] Ravani, B., Ku, T. S.: Bertrand offsets of ruled and developable surfaces. Computer-Aided Design, 23(2), 145-152 (1991).
• [16] Wang, G. J., Tang, K., Tai, C. L.: Parametric representation of a surface pencil with a common spatial geodesic. Computer-Aided Design, 36(5), 447-459 (2004).
• [17] Zhao, H., Wang, G. A new approach for designing rational bézier surfaces from a given geodesic. Journal of information and computational Science, 4(2), 879-887 (2007).
• [18] Zhao, H., Wang, G.: A new method for designing a developable surface utilizing the surface pencil through a given curve. Progress in Natural Science, 18(1), 105-110 (2008).